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Novel computational methods for inverse heat conduction problems

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Title:
Novel computational methods for inverse heat conduction problems
Creator:
Wang, Yuanlong
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
College of Engineering, Design, and Computing, CU Denver
Degree Disciplines:
Engineering and applied science
Committee Chair:
Premnath, Kannan
Committee Members:
Tadi, Mohsen
Mandel, Jan
Vega, L. Rafael Sanchez
Butler, Troy

Notes

Abstract:
This dissertation is concerned with two computational methods for evaluating five different in verse heat conduction problems. The first problem is the inverse evaluation of the unknown ini tial condition for a parabolic system. Second and third problems are inverse evaluation of ab sorption and diffusion coefficients. The fourth problem is the inverse evaluation of coefficients for a photon diffusion equation, and the fifth problem is the inverse evaluation of the bound ary of tore supra in a Tokamak. The first method is based on proper solution space, which been used to solve all five problems. The algorithm is iterative method in nature, which starts with an initial guess for the unknown function and obtains corrections of the assumed value in each iteration step. The updating part is the new feature of the present algorithm. This method gen erates a set of functions that satisfy some of the boundary condition. It then uses the remaining boundary condition to update the assumed value. The other method is based on optimization which seeks to minimize a cost functional. This method is also an iterative method, and will be used in evaluating the boundary shape of the vacuum in a Tokamak. Both algorithms show good robustness which can be used to obtain a good estimate of the unknown function. A num ber of numerical examples for different inverse problems are used to show the applicability of the methods.

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Copyright Yuanlong Wang. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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NOVEL COMPUTATIONAL METHODS FOR INVERSE HEAT CONDUCTION PROBLEMS by
YUANLONG WANG
B.S., Beijing University of Technology, 2012 M.S.,University of Colorado Denver, 2014 M.S.,University of Colorado Denver, 2018
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Engineering and Applied Science Program
2018


This thesis for the Doctor of Philosophy degree by Yuanlong Wang has been approved for the Engineering and Applied Science Program by
Kannan Premnath, Chair Mohsen Tadi, Advisor Jan Mandel
L. Rafael Sanchez Vega Troy Butler
Date: December 15, 2018


Ill
Wang, Yuanlong (Ph.D., Engineering and Applied Science)
Novel Computational Methods for Inverse Heat Conduction Problems Thesis directed by Associate Professor Mohsen Tadi
ABSTRACT
This dissertation is concerned with two computational methods for evaluating five different inverse heat conduction problems. The first problem is the inverse evaluation of the unknown initial condition for a parabolic system. Second and third problems are inverse evaluation of absorption and diffusion coefficients. The fourth problem is the inverse evaluation of coefficients for a photon diffusion equation, and the fifth problem is the inverse evaluation of the boundary of tore supra in a Tokamak. The first method is based on proper solution space, which been used to solve all five problems. The algorithm is iterative method in nature, which starts with an initial guess for the unknown function and obtains corrections of the assumed value in each iteration step. The updating part is the new feature of the present algorithm. This method generates a set of functions that satisfy some of the boundary condition. It then uses the remaining boundary condition to update the assumed value. The other method is based on optimization which seeks to minimize a cost functional. This method is also an iterative method, and will be used in evaluating the boundary shape of the vacuum in a Tokamak. Both algorithms show good robustness which can be used to obtain a good estimate of the unknown function. A number of numerical examples for different inverse problems are used to show the applicability of the methods.
The form and content of this abstract are approved. I recommend its publication.
Approved: Mohsen Tadi


IV
DEDICATION
I would like to thank Dr. Mohsen Tadi for the supervision of this dissertation. He supported me with several helpful hints and suggestions when I needed them, and he always had time to discuss my work.
Furthermore, I especially wish to express my gratitude to my office-mate Abdalkaleg Hamad, who always had time for discussions on the topic of this dissertation.
Last but not least I want to thank my family for their encouraging help during the whole four years and they never ending patience while listening to me explaining my work. Without their support, it is hard for me to get where I am today.


ACKNOWLEDGEMENTS
I would like to thank Dr. Mohsen Tadi, Abdalkaleg Hamad, and all unnamed referees for their valuable corrections and comments.


vi
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION.......................................................... 1
II. INVERSE PROBLEMS......................................................3
2.1 What Is An Inverse Problem?....................................3
2.2 Well-posed And Ill-posed Inverse Problem.......................4
2.3 Examples of Ill-posed Inverse Problems..........................5
2.3.1 Matrix Inversion........................................5
2.3.2 Differentiation.........................................6
2.3.3 Fredholm Integral Equations of the First Kind...........8
2.3.4 Tomography.............................................10
III. INVERSE HEAT CONDUCTION PROBLEMS....................................16
3.1 Classification of the Inverse Problems.........................17
3.1.1 Evaluation of the Boundary Conditions (Cauchy Problem).17
3.1.2 Evaluation of the Initial Condition....................18
3.1.3 Evaluation of Thermophysical Properties................18
3.1.4 Inverse Source Problem.................................18
3.1.5 Evaluation of the Domain...............................18
3.2 Methods for Solving Inverse Heat Conduction Problems...........19
3.2.1 Method of Fundamental Solution.........................19
3.2.2 Truncated SVD Regularization...........................20
3.2.3 Tikhonov Regularization................................22
3.2.4 Optimization Method....................................22


vii
3.2.5 D-bar Method...............................................23
3.2.6 Method of Multiple Forward.................................26
3.2.7 Active Subspace Method.....................................29
3.3 General Identification Algorithm...................................30
3.3.1 Identification Algorithm...................................30
3.3.2 Proper Solution Space Method...............................30
3.4 General Contribution of This Dissertation.........................31
IV. EVALUATION OF THE INITIAL CONDITION.......................................32
4.1 Problem Statement and Identification Algorithm.....................32
4.2 Proper Solution Space..............................................33
4.3 Numerical Examples.................................................35
4.4 Remark.............................................................41
V. EVALUATION OF ABSORPTION COEFFICIENT......................................42
5.1 Problem Statement and the Identification Algorithm.................42
5.2 Proper Solution Space..............................................44
5.3 Numerical Examples.................................................46
5.4 Remark.............................................................54
VI. EVALUATION OF DIFFUSION COEFFICIENT; CALDERON'S PROBLEM . . .56
6.1 Problem Statement and the Identification Algorithm.................56
6.2 Proper Solution Space..............................................58
6.3 Numerical Examples.................................................59
6.4 Remark.............................................................68
VII. EVALUATION OF ABSORPTION AND DIFFUSION COEFFICIENT
72


viii
7.1 Problem Statement and the Identification Algorithm................72
7.2 Proper Solution Space.............................................75
7.2.1 Details of Step 3.........................................75
7.2.2 Details of Step 6.........................................77
7.3 Numerical Examples................................................78
7.4 Remark............................................................87
VIII. EVALUATION OF THE BOUNDARY OF THE VACUUM IN A TOKAMAK .88
8.1 Problem Statement.................................................89
8.2 Adjoint Method....................................................91
8.3 Interpolation.....................................................93
8.4 Proper Solution Space.............................................95
8.5 Numerical Results.................................................96
8.5.1 Adjoint Method............................................96
8.5.2 Interpolation and Proper Solution Space...................99
8.6 Remark...........................................................105
IX. CONCLUSION............................................................110
BIBLIOGRAPHY
112


IX
LIST OF FIGURES
FIGURE
2.1 Mathematical model of a physical process.....................................4
2.2 Actual g and noisy g5........................................................8
2.3 Result of actual g and g5....................................................8
2.4 Electrical impedance tomography.............................................11
2.5 CT scanner..................................................................13
2.6 Geometry of data acquisition in CT..........................................13
3.1 Hi-posed problem............................................................27
3.2 Well-posed problem I........................................................28
3.3 Well-posed problem II.......................................................28
4.1 The recovered initial condition for Example 1. The figure compares the final value to
the actual function..........................................................37
4.2 The reduction in error for Example 1 as a function of the number of iterations .... 38
4.3 The recovered initial condition for Example 2. The figure presents the recovered
function at a few intermediate iterations and compares the final value to the actual function.....................................................................39
4.4 The reduction in error for Example 2 as a function of the number of iterations .... 39
4.5 The recovered initial condition for Example 3. The figure presents the recovered
function at a few intermediate iterations and compares the final value to the actual function.....................................................................40
4.6 The reduction in error for Example 3 as a function of the number of iterations .... 40


5.1 The recovered absorption for the Example 1. The figure compares the final value to
the actual function....................................................................48
5.2 The reduction in error for the Example 1 as a function of the number of iterations . 49
5.3 The actual absorption coefficient for Example 2......................................50
5.4 The recovered absorption coefficient for the Example 2, after 1000 iterations........51
5.5 The reduction in error for the Example 2 as a function of the number of iterations . 52
5.6 The recovered absorption coefficient for the Example 3, after 150 iterations.........52
5.7 The reduction in error for the Example 3 as a function of the number of iterations . 53
5.8 The actual absorption coefficient for the Example 4..................................54
5.9 The recovered absorption coefficient for the Example 4, after 1000 iterations with
full measurements..........................................................55
5.10 The recovered absorption coefficient for the Example 4, after 1000 iterations with partial measurements. The data is collected within the interval [0.3, 0.7] on each
side............................................................................55
6.1 Thermal conductivity for example 1..............................................60
6.2 Recovered conductivity after 174 iterations for Example 1.......................62
6.3 The reduction in error for Example 1 as a function of the number of iteration..63
6.4 Recovered conductivity after 150 iteration for Example 2 with more sets of data . . .64
6.5 Comparison of the recovered and the actual conductivity at the diagonal cross section
for Example 2 for different levels of noise.....................................65
6.6 Comparison the reduction error for Example 2 for different levels of noise......65
6.7 Thermal conductivity for Example 3..............................................66
6.8 Recovered conductivity after 700 iteration for Example 3........................67


XI
6.9 Error reduction for Example 3 as a function of the number of iteration..............67
6.10 Comparison of the recovered and the actual conductivity at the diagonal cross
section for Example 3..............................................................68
6.11 Comparison of the recovered and the actual conductivity at the diagonal cross
section for Example 3 for different levels of noise................................69
6.12 Thermal conductivity for Example 4................................................69
6.13 Thermal conductivity for Example 4................................................70
6.14 Comparison of the recovered and the actual conductivity at the diagonal cross
section for Example 4 for two values of............................................70
6.15 Error reduction for Example 3 as a function of the number of iteration for two
values of..........................................................................71
7.1 The actual diffusion coefficient k for Example 1.................................78
7.2 The actual absorption coefficient /J.a for Example 1.............................79
7.3 Recovered diffusion coefficient k for Example 1....................................82
7.4 Recovered absorption coefficient /J.a for Example 1................................82
7.5 Error reduction for diffusion coefficient k as a function of the number of iteration . 83
7.6 Error reduction absorption coefficient jia as a function of the number of iteration . .83
7.7 The actual diffusion coefficient k for Example 2.................................84
7.8 The actual absorption coefficient /J.a for Example 2.............................85
7.9 Recovered diffusion coefficient k for Example 2....................................85
7.10 Recovered absorption coefficient /J.a for Example 2...............................86
7.11 Error reduction for diffusion coefficient k as a function of the number of iteration. 86


xii
7.12 Error reduction for diffusion coefficient jia as a function of the number of
iteration....................................................................87
8.1 Toroidal geometry of the tokamak Tore Supra..................................89
8.2 A cross section..............................................................90
8.3 A cross section..............................................................94
8.4 Normal derivative at the boundary (data), and noisy data.....................98
8.5 Recover unknown interior boundary for the example 1..........................99
8.6 Reduction in error for the example 1........................................100
8.7 Normalized error in the recovered interior boundary for the example 1.......100
8.8 Recovered unknown interior boundary: example 1, effect of noise.............101
8.9 Reduction in error, example 1, effect of noise..............................102
8.10 Normalized error in the recovered interior boundary, example 1, effect of noise . 102
8.11 Normal derivative at the boundary (data), and noisy data...................105
8.12 Recovered unknown interior boundary for different value of s...............106
8.13 Reduction in error for different value of s.................................107
8.14 The relative error for the recovered functions for different value of s....107
8.15 Recovered unknown interior boundary for different value of noisy...........108
8.16 Reduction in error for different value of noisy.............................109
8.17 The relative error for the recovered functions for different value of noisy.109


1
CHAPTER I INTRODUCTION
The problems of the determination of the unknown functions involved in mathematical models describing various physical systems are refereed to inverse problems. This identification is based on exciting the physical system by external inputs and recording the outputs. Such problems include the determination of diffusivity and conductivity profiles [f, 2], absorption coefficient [3], source terms [4], geometry of scatterers [5] and prior temperature distributions [6].
The field of inverse problems has developed rapidly in recent years. One finds numerous applications of inverse analysis in the physical and mechanical sciences, such as the determination of earthquake hypocenters [7], groundwater hydrology [8], polymer processing [9], material cooling [10], optical tomography [11], nondestructive evaluation [12], ocean circulation [13], sig-nal/image processing [14], weather prediction [15], data analysis [16], computer vision [17], machine learning [18] and many more. It turns out that many such inverse problems are ill-posed, while the corresponding direct problems are well-posed. Chapter II provides more details about the definition of inverse problem and introduces difficulties encountered with an ill-posed problem.
This dissertation focuses on inverse heat conduction problems. In a heat conduction problem, if the boundary conditions are known, and all parameters of the system are known, then the temperature distribution of the domain can be evaluated, which is termed as a direct problem. However in many heat transfer situations, the surface heat flux and/or temperature must be determined from temperature measurements at one or more interior locations, which is termed as an inverse problem. Chapter III briefly introduces the classification of inverse heat conduction


2
problems and several existing methods for solving such problems.
The inverse problems can be divided into five categories based on the type of the estimation that is needed. We concentrate on computational methods for three classes of problems. Chapter IV is concerned with the evaluation of the initial condition for a parabolic system, where the unknown function is the initial condition. The method starts with an initial guess for the unknown function and obtains corrections to the assumed value in every iteration. In heat transfer, thermal conductivity is a measure of how fast the heat flows through the medium. The absorption coefficient is how heat is absorbed or generated. Both coefficients are important in heat transfer analysis. Chapter V is concerned with the evaluation of absorption coefficient of a parabolic system. Chapter VI focuses on evaluation of the thermal conductivity of an elliptic equation. The unknown coefficient is the positive thermal conductivity. In both chapters, we assume that the unknown function can be measured at the boundary. In Chapter VII, we consider an inverse problem for an elliptic system with two unknown functions. Chapter VIII is concerned with two methods for the inverse evaluation of the interior boundary location of a vacuum in a Tokamak. By interior boundary we are referring to the interior boundary of the vacuum which is the same as the outer boundary of the plasma. The first method is based on optimization which seeks to minimize a cost functional, and the second method uses the method of proper solution space to generate the value of the poloidal flux within two concentric circles. It then interpolates for the location of the boundary.


3
CHAPTER II
INVERSE PROBLEMS
Inverse problems have been one of the fastest growing areas in various application fields [19].
The mathematical description of inverse problem may be broadly described as the problem of determining the unknown internal structure or unrecorded past state of a system by using indirect measurements. The major difficulty in the treatment of inverse problems is the instability of the solution in the presence of noise in the observed (measured) data.
2.1 What Is An Inverse Problem?
Typically, when we describe a physical process via the mathematical model (see Fig. 2.1), it contains the following three different components: the system, the inputs, and the outputs. The analysis of a given process can be subdivided into three types of problems:
(1) The direct problem: given the input data and system parameters, compute the output.
(2) The reconstruction problem: given the system parameters and the output of the process, compute the input that leads to the given output.
(3) The identification problem: given the input and the output, compute the unknown system parameters which is in agreement with the relation between the input and the output.
The type (1) is called direct problem since it is oriented along a cause-effect consequence. Type (2) and (3) are usually treated as an inverse problem. We provide a mathematical description of


4
input system output
(parameters)
FIGURE 2.1: Mathematical model of a physical process
this process. Assume that the given data is g £ Q and the unknown quantity is u £ U. Consider a simple operator equation given by
Au = g. (2.1.1)
The data g and the unknown u can be either vectors or functions. Here, A : U —>• Q denotes an operator mapping from the space U C M” to the space Q C M”. Therefore, the inverse problem is to determine u £ U such that u = A~lg when arbitrary g £ Q is given. An inverse problem can be described as the solution of the operator equation produced by some phenomenon from physics, and the direct theory is the mathematical description of the physical phenomenon.
2.2 Well-posed And Ill-posed Inverse Problem
Consider the definition of inverse problem from previous section:
Find u £ U that satisfies Eq. (2.1.1) where the given input g £ Q is arbitrary. For a well-posed inverse problem we must have the following
(1) The problem must possess a solution for all input data, i.e., for every g £ Q there have u £ U such that g = Au. In other words, the operator mapping A needs to be a surjection.
(2) The problem must have a unique solution, i.e., suppose ui,U2 £ U are two solutions satisfying Aui = Aua = g £ Q then m = u-2 must hold. In the other words, the operator mapping A: U -£ Q needs to be an injection.


5
(3) When the operator A is both injective and surjective, then it is a bijection and the inverse mapping A~l :Q ^t-U exists. Then the inverse operator A~l must be continuous, and the solution of Eq. (2.1.1) must be stable.
If any of these requirements is not fulfilled, the inverse problem is called ill-posed. The small changes in the data will lead to large deviation in the solution, which means the solution is unstable. In practice, the given data g is never exactly the same as in the mathematical formulation. There are several reasons for this, such as the limited accuracy of the measurement equipment, the incompleteness of the mathematical model, the external disturbances in the environment, and the round-off error in the numerical computation. Some useful remarks on the inverse and ill-posed problems can be found in [20]. Some classic examples of inverse and ill-posed problem are shown in next section.
2.3 Examples of Ill-posed Inverse Problems
2.3.1 Matrix Inversion
Consider the solution of a linear system. It can be written in the form of Eq. (2.1.1) with u £ R™ and g £ R™ being n-dimensional real vectors, and A £ Rraxra being a matrix with real entries. Consider the perturbed system given by
Au = g + fig, (2.3.1)
where 5g is an added disturbance (or noise). The relative error of data and of solution can be defined as
e
9
and eu
(2.3.2)
Since we have
A(u — u) = 5g
u — u
A-^g,
(2.3.3)
then the relative error of solution can be expressed as
u~u|| = IIA 1dg|| < ||A 11|||dg|| Hull ||u|| — ||u||
(2.3.4)


6
We also have another relationship which is
Au
lAull
|g|
u
>
<
(2.3.5)
Therefore, the relative error can be rewritten as
O’n N
A-1||Pg||
< IIA
-ii
Pgll
u
k(A)
Pgll
(2.3.6)
where k(A) = ||A_1||||A|| denotes the condition number of the matrix A. It is well known from numerical linear algebra that the condition number is a measure of stability of Eq. (2.1.1). Hence we observe that the relative error eg of data will be amplified by a large condition number k(A). A matrix with a large value of k is called ill-conditioned.
2.3.2 Differentiation
Another classic inverse problem is the differentiation. Suppose we have a function g e C'1([0,1]) with g{0) = 0 for which we want to compute u = g'. These conditions are satisfied if and only if u and g satisfy the operator equation
ry
g(y) = / u{x)dx, (2.3.7)
Jo
which can be rewritten as the operator equation Au = g (same as Eq. (2.1.1)) with the linear operator. Now we consider gs as the noisy data substitute with g for which we assume that the perturbation is additive , i.e. gs = g + ns. It is obvious that the derivative u only exists if the noise is also differentiable. However, even in case is differentiable, the error in the derivative can be arbitrarily large. Consider a sequence (Sj)j^ with 5j —> 0 and the sequence can be defined as
nSj(x) := 5j sin (2.3.8)
for a fixed but arbitrary number k. We can obtain the solution of Eq. (1.1.1) with perturbation as
uSj (x)
g1 + k cos
kx
1
(2.3.9)


7
and therefore obtain the estimation
\u ~ 3 IIl°°([0,1]) — IIC^yilL-ao,!])
, . kx
kcos ( — dj
= k,
(2.3.10)
L°°([0,1])
where L°° norm is defined as the maximum of the absolute values of its components. Thus, despite the fact that the noise in the data is becoming arbitrarily small, the error in the derivative can become arbitrarily large. Consider a given function g £ C'1([0,1]) as
g(x) = x3 and g( 0) = 0.
(2.3.11)
The direct problem in this condition should be determine the function g £ C'1([0,1]) when its derivative g'{x) = 3a;2 with the initial value g(0) = 0. The inverse problem should be determine the derivative of g £ C1([0,1]) when g{x) = x3.
The exact result of this inverse problem should be
u = g'{x) = 3a;2.
(2.3.12)
However, some difficulties arise when the given data g is inaccurate. Instead of g, we assume the observe data gs is given as
gs(x) = g{x) + sin (200a;) = x3 + sin (200a;), (2.3.13)
which has derivative
us = g'{x) = 3a;2 + cos (200a;).
(2.3.14)
The error of u is
\\u - ■u‘5||LTC([0)i]) = ||cos (200a;)||Loo([0,i]) = 1- (2.3.15)
For this example, the disturbed data gs is close to g which is shown in Fig. 2.2. However, corresponding derivatives are pretty different, which is shown in Fig 2.3. The solutions of ill-posed inverse problems often show strong dependency on small perturbation (noise) in the data.


8
FIGURE 2.2: Actual g and noisy g&
FIGURE 2.3: Result of actual g and g&
2.3.3 Fredholm Integral Equations of the First Kind
Consider a Fredholm integral equation of the first, kind given by
K(s,t)f(t)dt = g(s),

(2.3.16)


9
where K is a given kernel function, g is the known right-hand side function, and the unknown is the function f(t). This problem is ill-posed in the sense that
(f) the solution may not exist, and/or may not be unique,
(2) even if a unique solution exists an arbitrarily small perturbation of the known data g can give rise to a large perturbation of the solution.
A function K is square integrable if its norm exists. Every square integrable kernel has a singular value expansion given by
CO
K(s,t) = ^mui(s)vi(t), (2.3.17)
i= 1
and
/ K(s,t)vi(t)dt = HiUi(s), for * = 1,2,... (2.3.18)
J Qt
where (ui,Uj) = (vi,Vj) = 5ij, and /*i > /*2 > /*3 > ... > 0. Now suppose Eq. (2.3.17) is an absolutely and uniformly convergent expansion for K(s,t), then
K(s, t)f(t)dt
CO CO
5~2iMUi(s)vi(t)f(t)dt = ^2tM(vi(t),f(t))ui(s) =g(s)
i= 1 i= 1
On the other hand {«j(s)}^1 is an orthogonal basis and thus
(2.3.19)
g{s) = ^2(ui(s),g(s))ui(s). (2.3.20)
i= 1
Comparing coefficients of Ui(s) in Eq. (2.3.19) and Eq. (2.3.20) gives gi(vi(t), f(t)) = (Ui(s),g(s)), which gives the solution to Eq. (2.3.16) as
m = f>(*), mw) = f; NkNM)
=1 =1 A4*
(2.3.21)
i= 1 i= 1
The Eq. (2.3.16) has a solution if and only if the right hand side g satisfies the Picard condition
(2.3.22)
The Picard condition implies that the absolute value of the coefficients (ui,g) must decay faster than the corresponding singular values /q in order that a solution exists. The main difficulty is


10
when the noisy data g does not satisfy the Picard condition. Another aspect to be considered is the scale of perturbation in the data that will affect the solution. Consider the function
f(t) = sin(2n7rt), n = 1,2,... (2.3.23)
then, for n —> oo and arbitrary K we have
g(s) = ( K(s,t)f(t)dt= f K(s, t) sm(2mrt)dt —> 0. (2.3.24)
Jo Jo
This shows the function f(t) has been smoothed by K. High frequencies are damped in the mapping / h->â–  g. However, the mapping from g to f amplify the high frequencies. From the above example, we can distinguish the difficulties while recovering / from noisy data g.
2.3.4 Tomography
The non-destructive evaluation (NDE) is a wide group of analytical and experimental methods that are used to probe structures such as bridges and buildings [21]. These techniques are widely used in medical imaging, including Computed Tomography (CT) [22], Positron Emission Tomography (PET) [23], Magnetic Resonance Imaging (MRI) [22], Electric Impedance Tomography (EIT) [24], Microwave imaging [25], and Acousto-electromagnetic tomography [26]. The availability of computational resources also makes it possible for investigators to develop accurate computational algorithms for such problems.
2.3.4.1 Electrical impedance tomography (EIT)
Electrical impedance tomography (EIT) is a medical imaging system, which seeks to produce a picture of the interior structure of an object, i.e., human body, by recovering its conductivity distribution from electrical measurements on the boundary of the object [27, 28, 29]. For the measurement, an array of electrodes is attached to the skin (boundary). In this procedure, electric currents are fed through the electrodes and the voltage at the electrodes are measured as shown in Fig 2.4.


11
Constant Current Source
—KD1--------
FIGURE 2.4: Electrical impedance tomography
Currently the most accurate mathematical model for the electric impedance tomography is given by the complete electrode model [24, 30]. Consider a bounded domain Q in R2 or R3, with a continuous boundary <9Q. Attached to the boundary dQ are a set of electrodes denoted by {ee}]j>=1, which are open connected and are disjoint from each other, i.e., e* fl ey = 0, if i / j Let It denote the applied current on the I-th electrode ec, and in addition, the current vector I = [Ii, 1-2,..., Ie\T satisfy Ym<=i U = 0 (conservation of charge). Let Uc denote the voltage at the I-th electrode with U = [U\, U2,Ur]T, together with a grounding condition given by Sr=i Ut = 0. Given a set of input currents U, the forward problem for the complete electrode


12
model is to find out the potential u and the electrode voltages Ug (for £ = 1,2,L) such that

du
£(7 dn du
r —(is on
= 0, for xeQ,
£ on eg,£ = 1,2,...
- h, for £ = 1,2,
= 0 on dQ\ uf=1 e£,
(2.3.25)
du a dn
where The electric impedance tomography (EIT) technology has widely used in various fields including early detection of breast cancer [31], head imaging [28], noninvasive testing [32, 33], and process monitoring in industry [34],
2.3.4.2 X-ray Computed Tomography (CT)
In computerized tomography (CT) [35, 36], one makes use of computer-processed combinations of many X-ray measurements to produce cross-sectional images of specific areas of a scanned object. In general, a three-dimensional body is measured from an outer curve, where the goal is to recover a two-dimensional slice from the collected data. Mathematically, the exact measurement is a collection of line integrals of the non-negative attenuation coefficient function along the paths of the X-rays. X-ray Tomography has been widely used in Medical diagnosis [37, 38]. Due to the fact that different tissues absorb different amounts of x-ray radiation, their absorption coefficients have different values. Therefore, when the total x-ray absorption across the body is measured in different directions, one can reconstruct the absorption coefficient across the body. A modern CT scanner is shown in Fig. 2.5.
The simplest mathematical model of CT assumes that the scanner measures the line integrals of the absorption coefficient a(x). We assume that there is a source ,S, emitting an X-ray beam,


13
FIGURE 2.5: CT scanner
FIGURE 2.6: Geometry of data acquisition in CT
which crosses the body through path L, and its intensity is measured by a detector at the end. The attenuation of the X-rays cross the body at location x (see Fig. 2.6) can be described by the following simple model
dT
— (x) = —a(x)I(x), (2.3.26)
where I is the intensity measured by D and u is coordinate in Fig. 2.6. It follows that
(Jj^j = ~ J a(x)du.
In
(2.3.27)


14
By moving the S — D system along two parallel lines and measuring the intensity at all possible positions, we can get the projection of the unknown coefficient a(x) in direction 9, which is given by
{Poa){s) = J a(s9 + u9')du. (2.3.28)
Here 9' denotes a vector orthogonal to 9. By rotating the S — D system and repeating the above linear scanning for all possible angles and positions we can obtain all possible projections, which gives rise to the Radon transform
(7Za){s,9) = {Poa){s) = J a(s9 + u9')du. (2.3.29)
Then it is possible to simplify the system as
g(s,9) = (Ka)(s,9), (2.3.30)
where g(s,9) is the measurement data. Therefore, the inverse problem of computerized tomography is the determination of a based on the knowledge of its integral g.
2.3.4.3 bioluminescence tomography
In bioluminescence tomography [39, 40, 41], biological test subjects (e.g. tumour cells) are tagged with luciferase enzymes and implanted in a small animal. This technology provides a way to reveal cellular and molecular features in biology and disease in real time. In essence, the goal of bioluminescence tomography (BLT) is to locate the distribution and quantitatively reconstruct the intensity of the internal light source using the transmitted and scattered bioluminescent signal on the surface of the small animal. One approach to model the transfer of light in biophotonics is the Radiative Transfer Equation (RTE) approximated from the Maxwells Equations.
In many applications, it is sufficient to consider the diffusion equation given by
- V-(D v (x)) +AE where 0(x) denotes the photon density, D is the diffusion coefficient and ga represents absorption coefficients. The term S'(x) is the source distribution of gene expression. The appropriate


15
boundary condition is given by
D Vn (x) — c«/>(x) = 0, x G <9Q, (2.3.32)
where n is the unit outer normal to the boundary and a is a coefficient related to the internal reflection at the boundary. In this problem, the parameters D, /j,a and a are known, and the bioluminescence tomography (BLT) is composed of recovering the source distribution from the collected data at the boundary. Mathematically, BLT is the source inversion problem that recovers S'(x) from optical measurement on the domain boundary dQ. In addition to being highly ill-posed, the solution to the above (BLT) inverse problem is in general not unique [41].


16
CHAPTER III
INVERSE HEAT CONDUCTION PROBLEMS
Loosely speaking, inverse heat conduction methods can be used to determine heat flux and temperature on an in-accessible surface of a wall by measuring temperature on an accessible boundary. In general, many of the mathematical models encountered in heat transfer problems are either elliptic or parabolic. For a full description of a heat transfer problem, one needs a specific set of boundary/initial conditions, and a full knowledge of the thermo-physical properties of the domain. An inverse problem arises when some of the required boundary/initial conditions, or material properties are unknown. Such problems appear naturally in various fields, including the reentering heat shield [42], remote sensing of climates [43], oil exploration [44], nondestructive evaluation of material [45] and the determination of the earths interior structure [46, 47]. The inverse heat conduction problems (IHCP) is highly ill-posed [48] in the sense that any small noise in the measurements can result in a drastic change to the solution. In the past decades, many numerical methods have been developed for solving the IHCP [49, 48, 50, 51]. Consider a general form of governing equation with possible boundary and initial conditions as du(n. t)
pc—^=v(KV«(x/)) + ft, xeQ, te(0,tf], (3.0.1)
■u(x, t) = Ub(yi, t), x g dQ, t € (0, tf], (3.0.2)
â– u(x, 0) = -Uo(x), x G Q, (3.0.3)
where -u(x, t) is the temperature ([A]), p is the mass density ([kg/m3]), c is the constant-volume specific heat ([J/kg K]), k is the thermal conductivity {[W/m K]), Qv is the rate of heat generation per unit volume ([W/m3]), usually denotes as source term, Ub is the given boundary condition, uq is the initial temperature; tf is the finial time. In order to recover the unknown part,


17
some additional conditions can be given as
= x G te(0,tf], (3.0.4)
—k^U^ ^ = hc [~u(x, t) — ~ue(x, t)] , x G <9Q, te(0,t/], (3.0.5)
where % and ue are given functions on the boundary, /ic is the heat transfer coefficient ([W/m2 ill]). Equations (3.0.2), (3.0.4) and (3.0.5) are different types of boundary condition which can be regarded as Dirichlet, Neumann and Robin boundary condition. Equation (3.0.3) denotes the initial condition.
3.1 Classification of the Inverse Problems
Broadly speaking, the inverse heat transfer problems can be divided with the following categories
(1) evaluation of the boundary conditions,
(2) evaluation of the initial condition,
(3) evaluation of the thermophysical properties,
(4) inverse source problem,
(5) evaluation of the domain .
The inverse heat conduction problem can be either linear or nonlinear. More details will be presented in the following section.
3.1.1 Evaluation of the Boundary Conditions (Cauchy Problem)
In this problem, the governing equation is known, however, part of the boundary condition is unknown. In order to solve the inverse problem additional measurements, such as the temperature or heat flux, are needed. They can be measured on the accessible part of the boundary or


18
in a discrete set of points inside the domain. In a transient heat conduction problem this particular situation problem is referred to as sideway heat conduction problem [52, 53, 54], For a steady-state heat conduction, the additional condition is in the form of flux at the boundary which is the classical Cauchy problem [55, 56].
3.1.2 Evaluation of the Initial Condition
In this case initial conditions of are unknown, i.e. the function uo is unknown in Eq. (3.0.3).
The initial condition needs to be recovered based on additional information. In some applications, the temperature filed can be measured in the whole domain for fixed t > 0 and provided as data [57, 58].
3.1.3 Evaluation of Thermophysical Properties
In this problem, the thermal conductivity n in Eq. (3.0.1) is the unknown coefficient. The extra information concerning temperature or heat flux in the domain [59, 60], or on the boundary have to be measured.
3.1.4 Inverse Source Problem
In the problem of inverse source term, the term Qv in Eq. (3.0.1) is unknown. The intensity of source, its location or both can be unknown. In many cases, additional information, such as the temperature history, are given at chosen points location inside the domain Q, or on parts of the boundary [61].
3.1.5 Evaluation of the Domain
In shape optimization problem, the location of the domain is unknown. To recover the unknown (or optimal) location of the boundary, additional information needs to be provided on the known part of the boundary, which can be treated as Eqs. (3.0.4) and (3.0.5). In particular, the boundary conditions are over-specified on the known part. For the unknown part of the


19
boundary, it should be determined by imposing a specific boundary condition on it.
3.2 Methods for Solving Inverse Heat Conduction Problems
In recent years, many analytical and numerical methods have been developed for solving the inverse heat conduction problems. Explicit analytical solutions are limited to simple geometries, and are not suitable for most practical problems. Recent results on solving inverse problem includes a method based on Quazi-Reversibility [62], Neural network [63], a method based on conjugate gradient minimization [64, 65, 66], minimization of a cost functional [67, 68], multiple forward problem [69], statistical approaches [70], regularized Gauss Newton method [71, 72], the D-bar method [73], and the wavelet multi-scale method [74], Additional methods have also been reviewed in [75]. In the next section, several existing methods are presented.
3.2.1 Method of Fundamental Solution
In this method, the fundamental solution of the corresponding heat equation is used to create a basis for approximating the solution of the problem. Consider a system given by
= A«(x,i), xefi, and, t € (0, tf], (3.2.1)
with boundary conditions
du(n. t)
u(x,t) = gi(x,t), —^—= g2(x,t) xeffi, and, t e (3.2.2)
and initial condition
u{x., 0) = h(x), x G Q. (3.2.3)
The fundamental solution of Eq. (3.2.1) is given by
F(X,t)= (4vrfV/2 eXP(~^)’ XGQ’ t>0’ (3'2'4)
where n is the dimension of domain. Assuming that tf

20
and initial condition can by expressed by the following linear combination:
m
u(x,t) = '^2\j(x-xj,t-tj), (3.2.5)
j=i
where 0(x, t) = F(-x.,t + t*), m is the total measurement points, F is given by Eq. (3.2.4) and Aj are unknown coefficients to be determined. For the choice of basis functions (f), the approximated solution T must satisfy the original system. Using the boundary and initial condition which are given in Eq. (3.2.2) and Eq. (3.2.3), we can obtain a simple linear system for the unknown coefficients A j given by
AA = b, (3.2.6)
where
0(xj Xy. / ,; i j)
g^(xfc — xj'Ufc — tj)
and b
^â– (xi) ti)
p(xk,tk)_
In the above equations, Mi is the measurement which is polluted with noise.
Mi
(3.2.7)
Techniques for Regularization of Ill-posed Problems
A regularized solution to an ill-posed problem can be obtained by replacing the original ill-posed problem with a different problem where additional conditions on the solution are imposed [76]. Over the years, investigators have developed various methods to regularize an ill-posed inverse problem. In this section, a number of such methods are discussed.
3.2.2 Truncated SVD Regularization
Consider the linear system of equations given by:
Ax = y, (3.2.8)
where, A e Rmxra (m > n) is a matrix, x e Râ„¢ and y G Rm.


21
If the matrix A is ill-conditioned, the solution for the system (3.2.8) is inaccurate and unstable. The truncated SVD regularization method is widely used to solve stably and accurately ill-conditioned matrix equations. Singular value decomposition SVD of a matrix A is given by:
A = U£Vt, (3.2.9)
where U = [ui, U2,..., um] and V = [vi, V2,..., vra] are left and right matrices whose column are orthonormal. The matrix S= diag ( X = I]—V, (3-2.10)
ti ai
where k is the rank of the matrix A. Since A is an ill-conditioned matrix, there are many small singular values which are close to zero. An approximate solution can be obtained according to: [77, 61, 78]:
N
^approx = 'yW (3.2.11)
“ Vi
1=1
where, IV e N is the truncation parameter determined when small singular values are left.
The solution for an ill-posed problem by the truncated SVD regularization method depends the value of N. If the truncation parameter N is equal to k, no regularization is applied. If the truncated parameter N is equal to zero, this leads to neglect of all the singular values, and there is no approximation solution for the system (3.2.8). The truncated SVD regularization with too small N value results a more oscillatory solution [79, 80, 81]. Many investigators have studied several approaches to choose the appropriate truncated SVD regularization including the generalized cross-validation (GCV) [81, 82] , discrepancy principle [61, 83], and L-curve criterion [81, 84, 85].


22
3.2.3 Tikhonov Regularization
Tikhonov regularization is widely used to stabilize various ill-posed problems . In this method, the solution to the linear system given in Eq. (3.2.8) is obtained by minimizing the cost functional given by:
min J = || Ax — y||2 + /3||x||2, (3.2.12)
where a bound on the solution is introduced by (5 > 0. A stable solution for x e X can then be obtained by least-square minimization. In case of a differential operator, one can consider the minimization problem given by:
min J = ||Ax-y||2 + /?||’Tx||2 Vxex " "
where Ik often represents the first order derivative.
(3.2.13)
3.2.4 Optimization Method
The optimization methods are widely used to solve many inverse problems. Typically, one seeks to recover a model x based on observations of a field u, where u is related to x by a forward problem. The forward problem in practice can be written as:
A(x)u = y, (3.2.14)
where, A e Rraxra is a nonsingular matrix, and y is a right-hand side that consisting of source terms and boundary condition values. The data on the boundary is given by:
Bu = g, (3.2.15)
where the vector g contains the data, and the matrix B represents the measurement operator that selects fields at data measurement locations. A solution to x can be obtained by minimizing the error differences:
min J
U,X
- ||Bu — g| 2II bi
such that A(x)u
(3.2.16)


23
Since the inverse problem generally is ill-posed, the regularization is needed to stabilize the solution. Then the constrained optimization problem with Tikhonov regularization is given by:
min J = ^||Bu - g||2 + ^/?||^x||2. (3.2.17)
u,x 2 2
Since the system (3.2.14) is linear in u, then one can arrive at:
u = (A(x))-1y. (3.2.18)
This leads to the unconstrained optimization problem:
min J = ^||B(A(x))_1y — g||2 + ^/?||^x||2. (3.2.19)
This unconstrained optimization problem can be solved by Gauss-Newton method [86, 87]. The optimization methods work well for most hyperbolic systems, but it has poor performance for elliptic and parabolic systems. In general , there may be many local minimums and, the convergence can also be very poor.
3.2.5 D-bar Method
Nachman’s uniqueness proof for the 2-D inverse conductivity problem outlines a direct procedure for reconstructing the conductivity profile from knowledge of the Dirichlet-to-Neumann map [80, 88]. Consider Calderon’s problem which is given by
V-(ff(x)v«)=0, xellcR2, (3.2.20)
where, â– u(x) = g(x), x e dQ. (3.2.21)
The function g(x) is a given voltage. The goal is to recover the dn
/(x), x e dQ.
(3.2.22)


Using the function q = and u = ^fou, one can transform the conductivity equation to the Schrodinger equation as
(- A +q(z))u(z) = 0, zgR2. (3.2.23)
Introduce a complex parameter k and look for solution k) of the Schrodinger equation
(- A+q)iP(-,k) = 0, (3.2.24)
satisfying the asymptotic condition
e~ikztp(z, k)-le IU1,P(R2), (3.2.25)
for any 2 < p < oo. The space IU1,P(R2) is the Sobolev space consisting of LP(R2) functions whose week derivatives belong to LP(R2) as well. Note that ip(z,k) is asymptotic to the exponentially growing function etkz. Therefore it is possible to define a bounded function m(z,k) by
m(z, k) = e~ikzip(z, k).
(3.2.26)
Using the d operators
B=+%) • a=d*=- %) ■
(3.2.27)
and Eq. (3.2.26) to verify
q{z)tl){z,k) = A tp(z,k),
q(z)elkzm(z,k) = Add(elkzm(z,k)),
ikZry
(3.2.28)
= elkz(Aikd + A)m(z, k).
This implies
(— A —Aikd + q(z))m(z, k) = 0,
(3.2.29)
i.e. m is the solution of a PDE involving the d operator.
The D-bar method for solving the Calderon’s problem consists four steps as following:


25
Use the DN map Aa to construct the complex geometrical optics (CGO) solution ip(z,k). For all k e C\{0} solve the boundary integral equation
k)\an = etkz\an - Sk(Aa - Aik), (3.2.30)
where
Sk{K - Ai)u{z,k) = [ Gk{z ~t){Aa - Ai)ip(£,k)ds(£). (3.2.31)
Jan
(2) tp(-, k)\an —► t(k).
Evaluate the scattering transform t(k) by using Alessandrini’s identity [89] as
t(k) = f q(z)elkztp(z,k)dA = f etkz(Aa — Ai)ip(-, k)ds. (3.2.32)
Jn Jan
(3) t(k) —> m(z, k).
For each z -^m(z,k) = ^y-e_fc(z) m(z, k), (3.2.33)
ok 47ik
where, e-k(z) = exp(—i(kz + kz)), and m(z, •) —1 e LPonL°°. The Eq. (3.2.33) is generated by differentiating the Lippman-Schwinger equation m = 1 — gk * (qm) with respect to k. Nachman verified that Eq (3.2.33) has a unique solution for m(z, •) — 1 G LPo n L°°(C) for some po > 2 [90].
(4) m(z, k) —> a.
Examine k —> 0 in Eq. (3.2.29) and using q = ^pj§~ implies
- Am(z,0) = ^0-m(z,O). (3.2.34)
Thus m and solve the same Schrodinger equation
Am(-, 0) = / • m(-, 0) and A y/a = f • y/a, (3.2.35)
with f(z) = . The decay condition in m — 1 e W1,p° and a = 1
m\z>V) <7 (z)
on R2\Q now implies m(-,0) = yfa. Recover the conductivity by
\fo\z) = lim m(z, k) = m(z, 0).
k—s-0
(3.2.36)


26
3.2.6 Method of Multiple Forward
Consider a closed bounded domain Q G M2 and a 2-D Helmholtz equation given by [91]
Au + k2g(x)u = 0, x 6 H c l2, (3.2.37)
with Dirichlet boundary conditions and additional boundary measurement in the form of the Neumann condition are
(3.2.38)
â– u(x) = /(x), x G dQ,
Vn«(x)=/l(x), X G Q,
where /(x) is the given function. The variable -u(x) denotes the electric field, the parameter k denotes the frequency of the incident wave and the function g(x) is a physical parameter. The goal is to recover the function g(x) based on boundary measurement. The algorithm is iterative in nature and it consists of the following steps.
(1) Assume an initial guess of the unknown function g(x) as g(x). The initial guess together with the given Dirichlet conditions generate a background field, A(x), which satisfies the system given by
A u + k2g(-x.)u = 0, xeHc M2,
â– u(x) = /(x), x G dQ.
The Neumann boundary conditions can be estimated on the boundary.
(3.2.39)
Vnfi(x) = A(x), X G Q.
(3.2.40)
(2) Subtract the actual and background filed to obtain an equation for the error, according to e(x) = -u(x) — fi(x), which is given by
Ae + k2g(x)u — k2g(x)u = 0, xgHc M2, e(x) =0, x G dQ,
Vrae(x) = A(x) — A(x), xgH.
(3.2.41)


It is also possible to write g(x) = y(x) + g(x) where correction term g(x) is unknown. It follows that
27
Ae + fc2g(x)e + k2q(x.)u = 0, xeHc R2,
e(x) =0, x e <9Q, (3.2.42)
V?ie(x) = h(x) — /?.(x), x g Q.
The next step is to linearize around the background field and arrive at
Ae + fc2y(x)e + fc2g(x)fi = 0, xeHc R2,
e(x) =0, x e <9Q, (3.2.43)
V«e(x) = /),(x) - ft(x), x G Q,
which has been shown in Fig. 3.1. Now, there is only one unknown /?,(x) in the
e(x, 1) = 0
= hy(x, 1)
e(0,y) = 0
Ur(n>y) = M°U/)
e(x, 0) = 0
ey(j;,0) = fiy(:r,0)
FIGURE 3.1: Ill-posed problem
Ae + k'2g(x)e + k2q(x.)u = 0 Ill-posed problem over specified at boundary
e(l ,y) = n ^(1,2/) = M1.?/)
system need to be calculated. However the inverse problem (3.2.43) is over specified at boundary. In the multiple forward method, it separates the ill-posed problem into two well-posed problems to solve the unknown function.


28
e(x, 1) = 0
e*(0 ,y) = hx(0, y)
Ae 4- k'2g(x.)e + k'2q(x)u = 0 Well-posed problem 1
e(Ly) = 0
e,,(:r,0) = /i„(:r,0)
FIGURE 3.2: Well-posed problem I
ey(x, 1) = hy(x, 1)
e(0,y) — n
Ae + /,:2y(x)e + k'2q{x)u = 0 Well-posed problem 2
e*(l ,V) = Ml>»)
e(x, 0) = 0
FIGURE 3.3: Well-posed problem II
(3) Two well-posed problems have the same equation but different boundary condition and measurements, which are shown in the Fig. 3.2 and Fig. 3.3. Combining two well-posed problem and using least square minimization method, the unknown correction term y(x) can be estimated.


29
This computational method is quite versatile and can be applied to various system. Our new computational method for the inverse heat conduction problem is based on it.
3.2.7 Active Subspace Method
Active subspace method examines various directions of gradients and obtains the direction of the strongest variability. It then exploits these directions to obtain the active subspace (the direction with most variability) [92, 93]. Consider a function / with m continuous inputs
/ = /(x), xefcr (3.2.44)
Let A be equipped with a bounded probability density function p : Rm —>• R+, where
p(x)>0,xeA and p(x) = 0,x^A. (3.2.45)
where Vx/(x) =
df df
dx± dXm
l T
Define the m x m matrix C by
C = E[(Vx/)(Vx/)T], (3.2.46)
and E[-] is the expectation. The matrix C is a covariance-like matrix of the gradient, which is used to determine the directions of variability by factorization. Since the matrix C is symmetric and positive semi-definite, it can be decomposed into
C = WAW
T
(3.2.47)
where W denotes eigenvectors and A = diag(Ai, A2,..., Am), for Ai > A2 > ... Xm. With decreasing eigenvalues, the active subspace method separates components of the rotated coordinate system into a set y, corresponding to greater average variation and a set z, corresponding to smaller variation. Then, the matrix of eigenvectors W can be partitioned as
A
and W = [Wi, W2]
(3.2.48)
where Ai = diag(Ai, A2,..., Ara) with n < m, A2 = diag(Ara, Ara+i,..., Am), Wi is m x n, and W2 is m x (m — n).


30
Two new sets of coordinates y and z can be defined as
y = Wfx, yeP,
(3.2.49)
z = x, ye Rra-m.
The choice of eigenvectors that construct the subspace associate with the practical consideration and the scale of eigenvalues. The real advantage of this method is that one can construct response surfaces with reduced number of variables x e R™ instead of f’s natural variables x g Rm, which lower the dimension of problem from m to n.
3.3 General Identification Algorithm
3.3.1 Identification Algorithm
In this dissertation, a general identification algorithm has been used to solve five different ill-posed inverse problems. The present algorithm is iterative in nature which corrects the initial guess with a correction term in each iteration. The algorithm can be described as following steps.
(1) Construct the initial guess for the unknown function and, using the given boundary condition, obtain a background field.
(2) Subtract the background field from actual field, and obtain the error field.
(3) Apply the proper solution space method to obtain the correction term to the assumed value for the unknown function. Repeat the above steps until a significant reduction in error is achieved.
3.3.2 Proper Solution Space Method
The third step of the algorithm involves the identification of the correction to the assumed value of the unknown function. We mainly adopt the proper solution space method in this dissertation, which is a type of subspace method for choosing particular subspace to reconstruct


31
the solution space. In general, the proper solution space can be thought of as a subspace method where the subspace is specifically generated for the problem at hand, i.e., error field. This method consists of following steps:
(1) Choose a linearly independent set of functions q(x), for i = 1,2,... ,N over the domain Q, and assume that the correction term can be expressed as a linear combination of this linearly independent set of functions c/s. The q(x) should satisfy a part of given boundary conditions.
(2) Generate a set of functions eg, for £ = 1,2,... ,N, that span the error field e(x) and satisfy part of the given boundary conditions (the Dirichlet part).
(3) Expand the actual error field e(x) in the span of the space generated by eg, £ = 1,2,..., N.
(4) Use the remaining boundary condition to obtain the correction.
In this dissertation, we assume that the unknown coefficient can be measured on the boundary. This is not general necessary. It is only done for convenience. Therefore, the correction term for the assumed value needs to satisfy the zero (Dirichlet) boundary condition. For numerical considerations a global approximation space such as sine functions are more appropriate.
3.4 General Contribution of This Dissertation
The contribution of this dissertation is threefold. First, we apply the proper solution space method for solving inverse problems for parabolic and elliptic systems. Second, we use our algorithm to reconstruct simultaneously two unknown functions involved in the photon diffusion equation. Finally, we try to do the identification of location of the domain for a elliptic system, which is a shape optimization problem.


32
CHAPTER IV
EVALUATION OF THE INITIAL CONDITION
In this chapter we use the proper solution space method for the inverse evaluation of the initial condition for a parabolic system. It is well known that this problem is highly ill-posed [94], In section 4.1, we present the iterative algorithm. It assumes an initial value for the unknown initial condition and obtains corrections to the assumed value. The new feature of the present algorithm is the updating stage which is presented in section 4.2. In section 4.3, we use several numerical examples to study the applicability of the method.
4.1 Problem Statement and Identification Algorithm
Let Q = {(t,x),x £ [0, l],f £ [0,tOl and consider a 1-D heat conduction equation given by
Uf; — UXX? (t; x') £ 14, u(t,0) = go(t), u(t, 1) =gi(t),
(4.1.1)
where u(t, x) is the temperature, t? is the finial time, thermal conductivity and diffusivity are assumed to be equal to one for simplicity, and Dirichlet boundary conditions are imposed. The unknown condition is the initial condition u(0,x). This ill-posed problem is supplemented by additional conditions, namely, the Neumann boundary conditions on the boundaries, i.e.
Ux(t, 0) = fo(t), Ux(t, 1) = fi(t).
(4.1.2)
The inverse problem of interest here is to recover the unknown initial condition u(0, x) =

(1) Assume a function for the unknown initial condition, u(0,x) = 33
'U't — ^xx)
u(t, 0) =go(t),
(t, X) £ fl,
u(t,l) =gi(t).
(4.1.3)
(2) Subtract the background field from Eq. (4.1.1), and obtain the error field, e(t,x) = u(t,x) — u(t,x), given by
Ct — &xxi (t, x) £ Q, e(t, 0) = e(t, 1) = 0.
The error field is required to satisfy additional conditions given by
(4.1.4)
(3)
ex(t, 0) = f0(t) - ux(t, 0) = f0(t),
(4.1.5)
ex(t, 1) = /i(t) - ux(t, 1) = fi(t).
To recover the initial condition, the correction term e(0, x) need to be computed. Use the additional boundary conditions in Eq. (4.1.5) and obtain the initial condition for the error field. Update the assumed value for the initial condition and go to step one.
The above three steps iterations is a basic algorithm that we used for a similar problems [95]. The novel feature of the present method is the third step in the above algorithm which is the evaluation of the unknown initial condition using a new method. We will described the third step in detail in next section.
4.2 Proper Solution Space
The third step of the algorithm involves the identification of the correction of the assumed value of the unknown initial value. The new method was first developed for the inverse evaluation of a boundary condition in steady heat conduction problems [96]. The actual value of the unknown function is related to the background field according to u(0,x) = u(0,x) + e(0, x). In


34
order to recover e(0,a;) , we consider a linearly independent set of functions cg(x), £ = 1, 2,N over x £ [0, f] and assume that the unknown functions e(0, x) can be expressed as a linear combination of these functions, i.e., e(0, x) £ {c\,C2,..., cn}- Then, we need to generate a set of functions eg(t,x) that satisfy the error field equation with the known (zero) Dirichlet boundary condition, i.e.,
e£t = eixx1 S 0
ee(t,0) = ee(t,l)=0, (4-2.4)
e(0, x) = cg(x).
Therefore, every function eg(t,x) satisfies the (zero) Dirichlet boundary conditions. It is then possible to expand the actual (and unknown) error field e(t, x) in the span of the space generated by eg(t,x), £ = 1,2,.., N, according to
N
e(t, x) = ^ Tgeg(t, x), (4.2.2)
e=i
where the functions eg(t,x) can be obtained from Eq. (4.2.1), but the constants Tg are unknown. We next argue that the error field e(t, x) must satisfy the gradient condition that is furnished by the measurements and are given in Eq. (4.1.5). The gradient conditions can be expressed by the operators Bq and B\. The error field is required to satisfy the conditions given by
N N
B0e(t, 0) = Bo ^2 T£e£(t, 0) = ^ TgB0eg{t, 0) = f0(t) (4.2.3)
e=i e=i
N N
1) = Bi^TgegXt, 1) = ^TgBieg(t, 1) = fi(t) (4.2.4)
e=i e=i
The above equation can be used to obtain the unknown coefficients Tg for £ = 1, 2,..., N. Once the unknown coefficients are obtained the unknown initial condition can be obtained according to
N N
e(0,a;) = E rgeg( 0,x) = ^rgcg(x). (4.2.5)
e=i e=i
The last equality in Eq. (4.2.5) holds because of the chosen initial condition given in Eq. (4.2.1) which completes the updating stage. After solving for e(0, x), the assumed value of the un-


35
known initial condition can be updated according to u(0,x) = u(0,x) + e(0,a;). and the three steps in the algorithm can be repeated.
4.3 Numerical Examples
In this section, three numerical examples have been used to investigate the applicability of the method. For the initial condition problem we consider 1-D dimension, however, we will estimate different types of the unknown function. In the first example, a simple 1-D heat condition equation with uniform thermal conductivity and one hump unknown initial condition are considered. Then the unknown initial condition will be modified to a two hump function in second example. Complex unknown function is harder to recover in usual. In the third example, we consider a parabolic system with non-uniform thermal conductivity.
Example 1. Consider the evaluation of the initial condition for a 1-D parabolic equation with uniform thermal conductivity given in Eq. (4.1.1) and assume that the actual initial condition is given by
u(0, x) = exp
(x — 0.5)2 0.03
(4.3.1)
with the boundary conditions u(t, 0) = u(t, 1) = 0 for all t. To recover the initial condition, the Neumann condition at x = 0 and x = 1 are provided for t e [0, 0.25). The above three step algorithm can be started by assuming a guess for the initial condition, collecting data at the boundaries, then arriving at the error field in the step 3. In order to obtain the correction of the assumed value in the third step of the algorithm, an appropriate set of linearly independent functions can be considered according to
ce(x) = sin (lirx), £ = 1,2,...,N, (4.3.2)
where cg(0) = cg( 1) = 0 for £ = 1, 2,..., N. The solution set can be obtained by using the individual functions c^ix) as the initial condition for the error field in Eq. (4.2.1), for £ = 1, 2,..., N. The exact solution for these functions are given by
— (in )2t
ee(t,x) = e
sin(Dtx), £ = l,2,...,N.
(4.3.3)


36
and, it is easy to note that these functions are linearly independent. We can denote the nodal values of ee(t, x) by the vectors ey for ^ = 1,2,N. We then argue that the error field must satisfy the additional boundary conditions that is provided by the measurements given in Eqs. (4.2.3) and (4.2.4), according to
N N
B0e(t, 0) = B0 ^ reee(t, 0) = ^ TeB0ee(t, 0) = b0, (4.3.4)
e=i e=i
N
N
Bie(t, f) = Bi ^Tgseit, f) = ^r^Bie^t, f) = bi, e=i e=i
where vectors bo and bi are the measurement data at boundary x = 0 and x = 1. Grouping these two equations leads to
(4.3.5)
Boey b0
• T =
Bi ey bi
(4.3.6)
where r is the N dimensional vector containing the values of the unknown parameters T£ for £ = 1,2, ..,N. The vectors bo and bi contain the values of /o(t) and f\(t) at different times. The above coefficient matrix is non-square and rank-deficient. It is possible to stabilize the inversion by imposing some form of regularization according
Boey b0
Bi ey â–  T = bi
0
(4.3.7)
where, the matrix represents the first-order operator given by
N
£<* tt)t£ cos(Gra;fc),

dce(x)
dx
x=xk
(4.3.8)
e=i
evaluated at discrete locations Xk along the domain. The constant (3 > 0 , and the above linear system can be solved for the unknown coefficients T£ for (. = 1,2,..., N.
Dividing the domain [0 : 1] into ne = 100 equally intervals leads to n = ne + 1 = 101 nodes
in the x direction. Dividing the time interval [0 : 0.25] into rit = 100 leads to 101 time intervals. Choosing IV = 20 sine functions in Eq. (4.3.2) leads to the proper solution space having


1
T
t---------r
Actual
Recovered
37
FIGURE 4.1: The recovered initial condition for Example 1. The figure compares the final value to the actual function
N = 20 vectors, with each vector having the dimension (101) x (101). The collected data at the boundaries are contaminated with a zero mean randomly generated noise with noise-to-signal ratio of 4%. The data is then filtered using a simple polynomial filter. The algorithm is iterative, and we are also using a form of under-relaxation in the sense that the updating is achieved ii,(0,x) = ii,(0,x) + ae(0,x) where a = 0.1. Figure 4.1 compares the recovered initial condition with the actual initial condition after 8830 iterations with [3 = 0.04 and figure 4.2 shows the reduction in the error. The error is define as
Error
rtf Jt=0 UD~ ' ./,(/)2 dt
rtf Jt=0 fo{t) 2 + flit)2 dt |first iteration
(4.3.9)
where, fo(t) and f\(t) are the difference between the measurement data and the calculated gradients at the boundaries given by Ecp (4.1.5).
Example 2.
given by
We now consider the evaluation of 2 hump of the initial condition, and the u(0, x)
u( 0, x)
[ (x — 0.26)41 [ (x — 0.74)4~|
exp 0.007 + exp 0.007
(4.3.10)


38
1 c
0.1 r
0.01 r
o
LU
0.001 T
0.0001 T
1e-05
111
1
10
100
1000
10000
Number of Iterations
FIGURE 4.2: The reduction in error for Example 1 as a function of the number of iterations.
For this example, we set (3 = 25. Figure 4.3 compares the actual initial condition with the recovered value after 90,000 iterations. It also shows a few intermediate value. The present method relies on Tikhonov regularization to stabilize the inversion in Ecp (4.3.7). The regularization penalizes large values of the first, derivative. For initial conditions with large gradients, a larger value of (3 is needed for stable and accurate inverse evaluation of the initial condition. The function is not known a prior, in general, a large value of (3 can be used to start, the evaluation. The definition of error is sames as before which shown in Fig. 4.4.
Example 3. We next, consider a. parabolic system with nonuniform thermal conductivity. Consider the evaluation of the initial condition for a. system given by
ut = (k(x)ux)x, (fijell, u(t, 0) = u(t, 1) = 0,
(4.3.11)
with
(4.3.12)


39
X
FIGURE 4.3: The recovered initial condition for Example 2. The figure presents the recovered function at a few intermediate iterations and compares the final value to the actual function.
FIGURE 4.4: The reduction in error for Example 2 as a function of the number of iterations.
and the u(0, x) is given by
u( 0, x)
exp
(x - 0.26)2 0.007
+ 1.5 exp
(x - 0.74)2 0.007
(4.3.13)


40
We fixed the value of /3 = 25. In the figure 4.5, we compares the actual initial condition with
X
FIGURE 4.5: The recovered initial condition for Example 3. The figure presents the recovered function at a few intermediate iterations and compares the final value to the actual function.
FIGURE 4.6: The reduction in error for Example 3 as a function of the number of iterations.
the recovered value after 90,000 iterations. It also shows a few intermediate value. Figure 4.6


41
presents the reduction in the error as a function of the number of iterations.
4.4 Remark
In this chapter, we used a computational method based on proper solution space to solve the inverse evaluation of the initial condition for a parabolic equation. Three examples have been implemented to study the applicability of this method which demonstrate good robustness to noise. No linearization is required in the algorithm in this chapter. This method also can be applied to heat conduction problems with variable material properties. Evaluation of absorption coefficient for a parabolic system based on proper solution space method will be considered in the next chapter.


42
CHAPTER V
EVALUATION OF ABSORPTION COEFFICIENT
In this chapter, we use proper solution space method to recover absorption coefficient for a parabolic system. We consider the absorption coefficient in one and two dimensions of a bounded domain. In both cases we assume that the unknown function can be measured at the boundary, and for simplicity, we assume that it is equal to one. In Section 5.1, we present the iterative algorithm. It assumes an initial value for the unknown function and obtains corrections to the assumed value. The new feature of the present algorithm is the updating stage which is presented in Section 5.2. In Section 5.3, we use a number of numerical examples to study the applicability of the method.
5.1 Problem Statement and the Identification Algorithm
Let Q = {(t,x),x £ [0,1],t £ [0,ff)} and consider a 1-D parabolic equation with absorption coefficient given by
ut = uxx + a(x)u, (t, x) £ SI,
u(t, 0) = g0(t), u(t, 1) = gi(t), (5.1.1)
â– u(0, x) = uo(x),
where u(t, x) is the temperature, t? is the final time, a(x) is the absorption coefficient and Dirich-let boundary conditions u(t, 0) and u(t, 1) are imposed. The unknown function is the absorption coefficient a(x). This ill-posed problem is supplemented by additional the Neumann boundary conditions at the boundaries, i.e.
ux(0,t) = /0(t), ux(l,t) = fi(t).
(5.1.2)


43
The inverse problem of interest here is to recover a(x) based on the additional given conditions at the boundaries Eq. (5.1.2). For the purpose of inversion, we can define u(t,x) = ev(-t,x) [97] (since u(t,x) > 0) and rewrite Eq. (5.1.1) according to
vt = vxx + v2x + a(x), (t,x)eQ,
v(t, 0) = \n(go(t)), v(t, 1) = \n(gi(t)), (5.1.3)
v(0, x) = ln(«o(*))-
The given data are transformed according to vx = ux/u at the boundaries with u > 0 for all t £ [0,^]. This formulation is suitable because the unknown function is isolated in Eq. (5.1.3). The present algorithm is similar as chapter 4. Due the unknown function is in the governing equation, the detail of the system is not exactly same as previous chapter. The different is the linearization need to be used in this algorithm. The method is iterative in nature and consists of three steps.
(1) Assume a value for the unknown absorption coefficient a(x) and, using the given Dirichlet boundary conditions go(t) and gi(t), obtain a background field satisfying the system
vt = vxx + v2x + a(x), (t,x)eQ, v{t, 0) = \n(go(t)), v(t, 1) = In(gi(t)), (5.1.4)
■0(0, x) = ln(«o(*))-
(2) Subtract the background field from Eq. (5.1.3), and obtain the error field, e(t,x) = v(t,x) —v(t,x), given by
(5.1.5)
e* = exx + vx — + (a(x) — a(x)), (t, x) £ Q,
e(t, 0) = e(t, 1) = e(0, x) = 0.
The error field is required to satisfy additional conditions. The given data is in the form of the gradient of the field at the boundaries
ex(°’ ^ = S§)~ = /«(*)>
~vx(l,t) = fi(t).
(5.1.6)


44
(3) Assume that the unknown function is related to the assumed value according to a(x) = a(x) + q(x), where q{x) is still an unknown function. Then the Eq. (5.4.5) changes to
et = exx+v2x-v2x + q{x), {t,x) £Vt,
(5.4.7)
e(t, 0) = e(t, 4) = e(0, x) = 0.
Use the additional measurements on the boundaries in Eq. (5.4.6) and obtain the unknown correction term q(x), to the assumed value of a(x). Update the assumed value, a(x), and go to step 4.
The basic step of algorithm is same as before. The different part of this method is the third step which is the evaluation of the unknown correction term q(x). In the previous chapter, the error in the initial condition can be treated as correction term to update the assumed value. In this chapter, the unknown part is absorption coefficient. Therefore, the correction term should be in the governing equation for error field. The way to estimate the unknown coefficient base on proper solution space is more difficult than before. The details will be shown in the next section.
5.2 Proper Solution Space
The third step of the algorithm involves the identification of the correction to the assumed value of the absorption coefficient a(x). We first linearize the quadratic terms in Eq. (5.1.7) according to
et = exx + v2x-v2x + q(x)
= exx + {vx +vx)(vx -vx) +q{x) (5.2.1)
= exx + 2vxex + q(x).
Our goal in this chapter is to recover q{x). Same as before, we can find the linearly independent set of functions ce, i = 1,2,..., N over x £ [0,1] and assume the unknown functions q(x) £ {ci, C2,..., cn}- We are assuming that it is possible to evaluate the unknown a(x) at the


45
boundaries, and as a result, we can assume that q{0) = g(l) = 0. Next, generate a set of functions that satisfy the error field equation with the known (zero) Dirichlet boundary condition,
i.e.,
(5.2.2)
Qt = eexx + 2vxegx + cg(x), (t, x) e Q, ee(t, 0) = eg(t, f) = e^(0, x) = 0.
Therefore, every function eg(t,x) must satisfy the (zero) Dirichlet boundary conditions at x = 0 and x = 1, and zero initial condition. It is then possible to expand the actual (and unknown) error field e(t, x) in the span of the space generated by eg(t, x), £ = 1,2,, N, according to
N
e{t,x) = yTgeg(t,x),
(5.2.3)
e=i
where the functions eg(t,x) are known from Eq. (5.2.2), however the constants Tg are still unknown. We next argue that the error field e(t, x) must satisfy the gradient condition that is furnished by the measurements and are given in Eq. (5.1.6). The gradient conditions can be expressed by the operators Bq and B\. The error field is required to satisfy the conditions given by
N N
B0e(t, 0) = Bo ^2 Tgeg(t, 0) = J2 Te&o£e(t, 0) = f0(t) (5.2.4)
e=i e=i
N N
Bie(t, 1) = B1'$2 Teee(t, I) = s^JTgB\eg(t, I) = fi(t) (5.2.5)
e=i e=i
The above two equations can be used to obtain the unknown constants Tg for ^ = 1,2,..., N. This step is presented in more details in the next numerical example section. Once the unknown coefficients are obtained, the unknown q{x)can be computed by substituting Eq. (5.2.3) in Eq. (5.2.1) which leads to
N
“ eN, - 2vxegx] = q{x). (5.2.6)
e=i
Using Eq. (5.2.2) and assuming an expansion for q{x) = Y^e=i aece(x)i and subtracting into Eq. (5.2.6) leads to
N
Y, cg(rg - ag) = 0 e=i
(5.2.7)


46
Due to the functions ctix) are linearly independent and can not be zero, we can conclude that at = Tt, £ = 1,2, and q{x) = J2tLi T£ce(x)■ By using Eqs. (5.2.4) and (5.2.5), we can get
the value of unknown constants 77. This completes the third step of the algorithm. After solving for q(x), the assumed value of the unknown absorption coefficient can be updated according to a(x) = a(x) + q(x)and the three steps in the algorithm can be repeated.
5.3 Numerical Examples
In this section a number of 1-D and 2-D examples have been used to study the applicability of this method in inverse coefficient evaluation problem.
Example 1. First, we will consider the evaluation of the absorption coefficient for a 1-D parabolic equation given in Eq. (5.1.1) and assume that the actual function is given by
(x — 0.7)2
a(x) = 1 + exp
0.01
(5.3.1)
Also assume the initial condition is u(0,x) = 0.5. Two Dirichlet boundary condition are
(t — 0.4)2
go(t) = 0.5 + 0.24 exp
0.03
(5.3.2)
and
gi(t) = 0.5 + 0.24 sin2 (OAnt).
(5.3.3)
The algorithm will start from assuming a value of absorption coefficient for a = 1, obtaining the background field of the transformed system v(t,x), and providing the boundary conditions for the error field in Eq. (5.1.6) for t £ [0, 0.25). An appropriate set of linearly independent functions can be considered according to
ct(x) = sin (£ttx), q(0) = q(1) = 0, £ = 1, 2,..., N. (5.3.4)
The solution set ei can be solved by using the individual functions cg(x) in Eq. (5.2.2), for £ =
1,2,..., N. We can denote the nodal values of ee(t, x) by the vectors ey for £ = 1,2,..., N. We


47
next argue that the error field must satisfy the additional boundary conditions that is provided
by the measurements given in Eqs. (5.2.4) and (5.2.5), according to
N N
B0e(t, 0) = Bo^ Tf£e(t, 0) = ^ T£B0ee(t, 0) = b0
£=1
£=1
(5.3.5)
N
N
Bie(t, 1) = Bi ^ reee(t, 1) = ^ reBiee(t, 1) = bi
(5.3.6)
e=i
e=i
Grouping these two equations leads to
Bq££ b0
â–  T =
Bq££ bi
(5.3.7)
where r is the N dimensional vector containing the values of the unknown parameters 77 for £ = 1,2, ..,N. The vectors bo and bi contain the values of /o(t) and f\(t) at different times. The above coefficient matrix is non-square and rank-deficient. It is possible to stabilize the inversion by imposing some form of regularization according
Boey b0
Boey â–  T = bi
0
(5.3.8)
where, the matrix represents the first-order operator given by
N
x=xk

dci(x, y)
dx
'^(£tt)t£ cos(£irXk)
(5.3.9)
e=i
evaluated at discrete locations Xk along the domain. The constant (5 > 0 is set by the designer, and the above linear system can be solved for the unknown coefficients 7y for 1 = 1,2,..., N. Dividing the domain [0 : 1] into ne = 100 equal intervals leads to n = ne + 1 = 101 nodes in the x direction. Dividing the time interval [0 : 0.5] into rit = 100 leads to 101 time intervals. Choosing IV = 20 sine functions in Eq. (5.2.3) leads to the proper solution space having N = 20 vectors, with each vector having the dimension (101) x (101). The collected data at the boundaries are contaminated with a zero mean randomly generated noise with noise-to-signal ratio of


48
FIGURE 5.1: The recovered absorption for the Example 1. The figure compares the final value to the actual function
4%. The data is then filtered using a simple 7-th order polynomial. Figure 5.1 compares the recovered absorption coefficient with the actual function after 150 iterations with (5 = 0.00005, and figure 5.2 shows the reduction in the error. In this note the error is define as Eq.(4.3.9), where fo(t) and f\(t) are the difference between the given data and the calculated gradients at the boundaries given in Eq. (5.1.6).
Example 2. Next consider the evaluation of an absorption coefficient within a 2-D domain (x, y) e Q = [0,1] x [0,1] given by
a(x, y)
1 + exp
1" (x — 0.65)2] r (y - 0.35)2 j
0.02 exp 0.02
(5.3.10)
which is shown in Fig. 5.3. The boundary is accessible and can be used to impose temperature according to
u(0, y, t) = 0.5 + 0.02 exp
u(l,y,t) = 0.5 + 0.02 exp
c*+. 1 O to 1 0 to
0.002 exp 0.002
(t — 0.4)s
x sin2 (7xy).
(5.3.11)
0.002
(5.3.12)


49
FIGURE 5.2: The reduction in error for the Example 1 as a function of the number of iterations
u(x, 0, t) = 0.5 + 0.02 exp u(l,x, t) = 0.5 + 0.02 exp
c*+. 1 o to r (x — o.5)21
0.002 exp 0.002
(t — 0.4)s 0.002
x sin2 (2irx).
(5.3.13)
(5.3.14)
For this example, we let Ax = Ay = 1/60, and divide the time interval [0 : 0.5] into n£ = 60 equal intervals (i.e., At = 0.5/60). The difficult part, of this problem is the unknown correction term is a 2-D function. So we need to find the linearly independent function depends on x and y. In order to generate the solution space, a set of linearly independent functions can be considered according to
c£(x, y) = sin(wra:) sin(jny)
£ = (i-l)M + j, i 1.2...........\l. j 1.2........\l.
And it should satisfy the Dirichlet. boundary conditions as
(5.3.15)
ce( 0, y) = ce(l,y) = c£(x, 0) = c£(x, 1) = 0.
(5.3.16)


50
For M = 20, the proper solution space has the dimension N = M2 = 400. Each vector should have the dimension 61 x 61 x 61. Due to the system is still non-square and rank-deficient. It is possible to stabilize the inversion by imposing some form of regularization according to
Bo £t b0
Bo £t • T = bi
0
1 •6 1 0
(5.3.17)
where, the matrix \I>T and y represents the first-order operator depend on x and y given by
^.7

dce{x, y)
dx
dce(x, y)
N N
dy
x=Xi,y=yj
x=Xi,y=yj
EE l\ir cos(£iirXi) sin^vrj/j)
e1=ie2=i
N N
EE £‘2Tt sin(^i7ra;i) cos^nyj)
(5.3.18)
(5.3.19)
e1=ie2=i
where Xi and yj are the discrete location in the domain. The constant f3 is larger than zero, which is set by the designer, and the above linear system can be solved for unknown coefficient


51
FIGURE 5.4: The recovered absorption coefficient for the Example 2, after 1000 iterations.
Tt for E = 1,2, 3,..., N. The data is again collected in the form of the gradient at the boundaries and the same level of noise (4%) is added. Figure 5.4 shows the recovered function after 1000 iterations, and figure 5.5 shows the reduction in error as a function of the number of iterations. The parameter (3 is set at (3 = 0.00001. For the inverse problem, we usually can not get the whole measurement on the boundary. Therefore, next two examples will be used partial measurement to recover the unknown absorption coefficient.
Example 3. In this example, we consider the absorption coefficient is same as the one in Example 1 and use the same boundary conditions as given from Eqs. (5.3.11) to (5.3.14). However, in this example, partial measurements will be considered. Assume data would be collected on two sides x = 0 and x = 1.
For this example, the domain [0 : 1] will be divided into ne = 60 equal intervals leads to n = ne + 1 = 61 nodes in both x and y directions. Dividing the time interval [0 : 0.5] in to nt = 60 leads to 61 time intervals. Choosing M = 20 sine functions in Eqs. (5.3.18) and (5.3.19)


52
FIGURE 5.5: The reduction in error for the Example 2 as a function of the number of iterations.
FIGURE 5.6: The recovered absorption coefficient for the Example 3, after 150 iterations.


53
lead to the proper solution space having N = M2 = 400 vectors, with each vector having the dimension 61 x 61 x 61. Figure 5.6 shows the recovered result after 150 iterations. The parameter f3 is again set at f3 = 0.0002. The reduction in error as a function of the number of iteration is presented in Fig. 5.7. Even we collect the data only on two sides, this method also can recover the approximate shape and correct location of the object function. In order to improve the result with partial measurements, collecting data on four sides of the domain is worth a try. In the next example, a two humps object function will be recovered by using the partial measurements on four sides.
FIGURE 5.7: The reduction in error for the Example 3 as a function of the number of iterations.
Example 4. Next consider the evaluation of an absorption coefficient given by
(x — 0.3)21 r (y — 0.7)21 r (x — 0.7)2] [ (y - 0.3)2"
a(x, y) = 1 + exp
0.015
exp
0.015
+ exp
0.015
exp
0.015
(5.3.20)
which is shown in Fig. 5.8. Imposing the same boundary conditions as given in Eqs. (5.3.11) to (5.3.14) and setting (5 = 0.00001, figure 5.9 shows the recovered function after 1000 itera-


54
FIGURE 5.8: The actual absorption coefficient for the Example 4.
t.ions with full measurements. For partial measurement, we are collecting data with the interval [0.3 : 0.7] on each side. Figure 5.10 shows the recovered function after 1000 iterations for partial measurements. Compare with two results, this method shows good robustness to noise even we use partial measurements.
5.4 Remark
In this chapter, we used the proper solution space method to recover unknown absorption coefficients. Several 1-D and 2-D examples were used to study the applicability of this method. The effect of noise and the applicability of the method for partial data were also studied.


55
FIGURE 5.9: The recovered absorption coefficient for the Example 4, after 1000 iterations with full measurements.

1.5
1.4
1.3
1.2
1.1
1
0.9
FIGURE 5.10: The recovered absorption coefficient for the Example 4, after 1000 iterations with partial measurements. The data is collected within the interval [0.3,0.7] on each side.


56
CHAPTER VI
EVALUATION OF DIFFUSION COEFFICIENT; CALDERON’S
PROBLEM
In this chapter, we consider inverse identification of the diffusion coefficient for an elliptic equation. This problem, which is often referred to as the Calderon’s problem [98], appears very naturally in various applications. It is well known that this problem is highly ill-posed [99] and various method have been developed to overcome such difficulties. In Section 6.1, we present the iterative algorithm. In Section 6.2, we present updating stage. In Section 6.3, we use a number of numerical examples to study the applicability of this method.
6.1 Problem Statement and the Identification Algorithm
Let Q be the bounded domain in R2 or R3 with smooth boundary dQ. In the absence of sinks or sources, the temperature field -u(x) is given by
(6.1.1)
V • (k(x) \/u) = 0, x G Q, u(x) = g(x), x e dQ,
where, k(x) > 0 is the positive unknown thermal conductivity. The Dirichlet boundary condition g(x) are imposed. The extra measurements are the temperature gradient given by
V»«(x) = 7^ = Z(x) X G dQ,
(6.1.2)
where, n is the outward normal.The inverse problem is to determine k(x) based on the additional given conditions at the boundary (6.1.2). The basic algorithm is same as before, which is iterative and consists of three steps. However, due to the unknown part is diffusion coefficient, the details of the algorithm are different. The iterative algorithm have been shown below


57
(1) Assume a initial value for the unknown coefficient function k(x) and using the given Dirichlet boundary conditions obtain a background field satisfying the system
k(x) A u + \/k(x) ' = 0> xefi
fi(x) = g(x) xg90
(6-1.3)
for «(x) = (2) Subtract the background field from Eq. (6.1.1), and obtain the error field, e(x) = u(x) — u(x), given by
k(x) A u + Vk(x) ' V'u ~ (^(x) A u + \/k(x) ' VA) = 0) xeH, (6.1.4)
for x g Q, where the error field is required to satisfy e(x) = 0,Vx g dQ. The error field is required to satisfy additional conditions given by
Vne(x) = /(x) - Vne(x) = /(x) X G dQ. (6.1.5)
(3) The assumed function, k(x), is related to the actual value according to k(x) = k(x) + q(x), where q(x) is still an unknown function. Use the additional boundary conditions in Eq. (6.1.5) to obtain the unknown correction term q(x). Then we can use the correction q(x) to update the assumed value, k(x), and go to step (1).
we have applied this algorithm to an elliptic problem in [96] and a parabolic problem in [100]. Also we try to recover the absorption coefficient in the chapter 5. The detail of step 3 in the algorithm will be shown in the next section.


58
6.2 Proper Solution Space
The third step of the algorithm includes the identification of the correction,i.e. q(x), to the assumed value of the diffusion coefficient. Substituting for k(x) in Eq. (6.1.4) lead to
k A e + xjh â–  xje + q(x) A u + v (6.2.1)
e(x) =0 x G <9Q.
Note that u(x) = u(x) + e(x)and the terms that are quadratic in unknowns are dropped. After linearizing the above equation around the background field it leads to
k A e + \/k â–  xje + q(x) A u + \/q(x) â–  \ju = 0, xGfi
(6.2.2)
e(x) =0 x G <9Q.
In order to recover q(x) for x G Q we can proceed as follow. Consider a linearly independent set of functions cg(x),£ = 1,2,3, ...,N over x G and assume that the unknown functions q(x) can be expressed as a linear combination of functions q(x), i.e. q(x) G {ci, C2, C3,..., cn}- We are assuming that it is possible to evaluate the unknown function k(x) at the boundaries. For simplify, we assume that it is equal to one. As a result we can get that k A e£ + \/k ■ VQ + C£ A u + VQ • X/u = 0, x G Q,
(6.2.3)
along with e(x),Vx G dQ. Therefore, every function q(x) satisfies the Dirichlet boundary conditions. It is then possible to expand the actual error field e(x) in the span of the space generated by ee(x),£ = 1, 2, 3,..., N according to
N
e(x) = ^r^(x), xGfi (6.2.4)
e=i
where the functions q(x) are known, but the constants T£ are unknown. We next argue that the error field e(x) must satisfy the gradient condition are given in Eq. (6.1.5). The gradient conditions can be expressed by the operator B. The error field is required to satisfy the condition given by


59
N N
Be(x) = B^receix) = ^r^^(x) = /(x) x G <9Q, (6.2.5)
£=\ £=\
The above equation can be used to obtain the unknown coefficients T£ for i = 1, 2, 3,..., N. More detail of this part will be shown in the next section. Once the unknown coefficients are obtained the unknown function q(x) can by computed by substituting Eq. (6.2.4) in Eq. (6.2.2) which leads to
N
y, Tg(k Ae< + \/k â–  yq) + r=i
Next we need to use Eq. (6.2.3) and simplify above equation to get
N
y^n[~{ct A u + VQ • V*)] + r=i
Assuming the the unknown function q(x) can be expressed as linear combination of functions
ce(x) according to q(x) = J2eLi N
^(e-Te)(ceAu + S7Ce-S7u) = 0, xeH. (6.2.8)
r=i
Next we argue that in general u(x) / 0 and q(x) are linearly independent. Then we can conclude that N
£=l
This completes the third step of the algorithm. After solving the correction term q(x), the unknown diffusion coefficient can be updated according to k(x) = k(x) + q(x). Then the three steps in the whole algorithm need to be repeated.
6.3 Numerical Examples
In this section we use a number of examples to study the applicability of this method.
Example 1. Consider the evaluation of the of the diffusion coefficient for a 2-D elliptic equation given in Eq. (6.1.1) and assume that the actual function of n{x,y) is given by
k(x, y) = 1 + exp
\—{x — 0.35)21 \—(y — 0.65)2!
0.02 exp 0.02
(6.3.1)


60
which is shown in Fig. 6.1, and the boundary condition are given by
FIGURE 6.1: Thermal conductivity for example 1.
g(x,y) = cos(k(.rcos6 + ysin6)) (x,y) e Q.
(6.3.2)
In this example we use 16 sets of data with k = 1, 2,3,..., 8 and 6 = 7t/3.2, 7t/4.2. We can start, the algorithm by assuming the value for k(x,y) = 1, obtaining the background field u(x,y), and providing the gradient boundary conditions for the error field for Eq. (6.1.5). In order to obtain the correction term q(x, y) in the third step of the algorithm, an appropriate set of linearly independent functions need to be considered according to
ce{x,y) = sin(t?i7r.'r) sin^vry), Cf(x) = 0, x e dQ, l\ = 1,2,3 ,...M, l2 = 1,2,3 ,...,M, and £ = (h- 1 )M + 4, N = M2.
(6.3.3)


61
The solution set can be obtained by using the individual functions cg(x,y) in Eq. (6.2.3), for £ = 1, 2, 3,..., N. The finite difference approximation to the elliptic system simplifies to linear systems given by
Aey + Fcg = 0, or, eg = — A~lTcg, W = 1, 2, 3,..., N, (6.3.4)
where A is finite dimensional approximation of the 2D diffusion operator depends on k with unique inverse (since the problem is well-posed), £g and eg are the vector containing the nodal values of the function eg(x,y) and cg{x,y). The matrix T is the finite-difference approximation of the operator (Au + \ju • v) that acts on q. Since the functions ci are linear independent, it follows that ei are also linearly independent for i = 1,2,..., N. The error field needs to be expanded in the space generated by ei according to
N
e(aJ, y) = Y1 Te€e(x, V) (6.3.5)
e=i
where Tg is unknown coefficients for £ = 1, 2, 3,..., N. In order to compute the value of Tg, the additional gradient boundary condition for the error field need to be used. According Eq. (6.2.5) and denoting the appropriate the finite dimensional approximation to the gradient operator by B leads to
N N
Be = B s^rg£g = s^jTgY$£g = f or, [Bey] r = f, (6.3.6)
e=i e=i
where r is the N dimensional vector containing the values of the unknown coefficients 17 for £ = 1,2,3, ...,N2. In Eq. (6.3.6), the vector f is known, the vectors in the solution space, £g for £ = 1, 2, 3,..., N are also known from Eq. (6.3.4). The square matrix B is singular and the matrix, [B£g\, is non-square and rank-deficient and therefore the above system cannot directly be solved for the coefficients Tg. As shown in Eq. (6.2.9), the correction term can be expanded as a linear combinations of cg(x, y), for ^ = 1,2,..., N. In order to compute the unknown coefficient,


62
it is possible to impose some form of regularization to stabilize the inversion according to
1 U) PQ i f
T = 0
1 •6 l 0
where the matrix \I/T and \I/y represent the first, order differential operator given by
vEf,

dce{x, y)
N N
dx
dce(x, y)
dy
x=Xi,y=yj
x=xiyy=yj
EE £\it cos(£iirXi) sin^vryj),
£i=i£2=i N N
EE £‘2Tt s\n(linxi) cos{£‘2'xyj),
(6.3.7)
(6.3.8)
£l = l£2 = l
where Xi and yj are the discrete location in the domain. The constant f3 is larger than zero, which is set by the designer, and the above linear system can be solved for unknown coefficient
T£ for £ = 1,2, 3,
.tv.
1.6
1.4

1
0.8
1
1
1.5
1.4
1.3
1.2
1.1
1
FIGURE 6.2: Recovered conductivity after 174 iterations for Example I


63
Dividing the unit square domain into ne = 60 equal intervals leads to n = ne + 1 = 61 nodes in the x and y direction. Choosing M = 20 sine functions in Eq. (5.3.3) leads to the proper solution space having N = M2 = 400 vectors, with each vector having the dimension n x n. The collected data at the boundaries are contaminated with a zero mean randomly generated noise with noise-to-signal ratio of 4%. The data is filtered by the seventh order polynomial curve fitting. Figure 6.2 shows the recovered diffusion coefficient after 174 iterations with (3 = 0.003. We also study the reduction in the error, which is defined as
/ [v«(x) - v»-(x)]2dx
Error =
/ [v»-(x) - V"»-(x)] «xlfirst iteration
(6.3.9)
and the result is shown in Fig.6.3.
FIGURE 6.3: The reduction in error for Example 1 as a function of the number of iteration
Example 2. In order to improve the result in Example 1, more set of data have been used which means k = 1,2, 3, 4, 5 and 6 = 7t/2.3, 7t/3.2, 7t/4.2, 7t/5.2, 7t/6.2. Figure 6.4 shows the recovered diffusion coefficient after 150 iterations with f3 = 0.0001. In this example different noisy


64
level have been considered. Figure 6.5 compares the actual and recovered values of the diffusion coefficient along the diagonal cross-section for three different values of noise. And Figure 6.6 presents the reduction in the error as a function after 450 iteration for different noisy. In the next example, a 2 hump object function will be considered.
FIGURE 6.4: Recovered conductivity after 150 iteration for Example 2 with more sets of data
Example 3. Next consider the evaluation of the diffusion coefficient given by
' — (x — 0.3)2] \-(y- 0.7)2'
k,(x, y) = 1 + exp + exp
0.015
—(x - 0.7)2
0.015
exp
6XP 0.015
—(y — o.3)2"
0.015
(6.3.10)
which is shown in Figure 6.7. The same boundary conditions as given in Eq. (6.3.2) are imposed. For this example we set the (5 = 0.001. The recovered result after 700 iterations is presented in Fig. 6.8. Figure 6.9 shows the reduction in the error as a function of the number of iterations. Figure 6.10 compares the recovered and the actual values of the conductivity function along diagonal cross-section for the example 3. Figure 6.11 compares the actual and the recovered values of the conductivity function along the diagonal cross-section for three different


65
FIGURE 6.5: Comparison of the recovered and the actual conductivity at the diagonal cross section for Example 2 for different levels of noise
FIGURE 6.6: Comparison the reduction error for Example 2 for different levels of noise


66
values of noise level. For increased level of noise, the accuracy of the recovered function drops, but the method can still recover a good estimate of the 2 hump unknown function.
FIGURE 6.7: Thermal conductivity for Example 3.
Example 4. Consider the evaluation of the diffusion coefficient given by
'—(x — 0.25)4] \—{y ~ 0.75)4
k(x, y) = 1 + exp
+ exp
0.0001
—(x - 0.75)4
0.0001
exp
exp
P 0.0001 ~(y — 0.25)4
0.0001
(6.3.11)
which is presented in Fig. 6.12. The unknown conductivity function in this example has regions with relatively large gradients. Figure 6.13 shows the recovered function after 151 iterations with p = 0.0005 and 4% noise. Figure 38 compares the actual function and the recovered function along the diagonal for two different values of ft. The present method relies on Tikhonov regularization to invert the matrix in Ecp (5.3.7). It penalizes large gradients, and by reducing p a better value of the magnitude of the unknown function can be obtained. Figure 38 presents the reduction in the error for two different p1 s as functions of number of iterations.


K(x,y)
67
2
1.8
1.6
1.4
1.2
1
0.8
0.6
1
Y 0 0 x
1
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
FIGURE 6.8: Recovered conductivity after 700 iteration for Example 3.
FIGURE 6.9: Error reduction for Example 3 as a function of the number of iteration


68
FIGURE 6.10: Comparison of the recovered and the actual conductivity at the diagonal cross section for Example 3.
6.4 Remark
In this chapter, we used the proper solution space method to evaluate the unknown diffusion coefficient of 2-D elliptic problem. We use three 2-D examples to study the applicability of this method. The accuracy of the recovered function drops as the level of noise is increased.


69
FIGURE 6.11: Comparison of the recovered and the actual conductivity at the diagonal cross section for Example 3 for different levels of noise
FIGURE 6.12: Thermal conductivity for Example 4.


70
FIGURE 6.13: Thermal conductivity for Example 4.
FIGURE 6.14: Comparison of the recovered and the actual conductivity at the diagonal cross section for Example 4 for two values of f3.


71
FIGURE 6.15: Error reduction for Example 3 as a function of the number of iteration for two values of f3.


72
CHAPTER VII
EVALUATION OF ABSORPTION AND DIFFUSION COEFFICIENT
In this chapter, we consider the inverse identification of the diffusion and absorption coefficient for a photon diffusion equation [101]. It is well known that this problem is highly ill-posed [36, 102], In Section 7.1, we present the iterative algorithm. It assumes initial guesses for the two unknown functions and obtains the corrections to the assumed values at every iteration. Section
7.2 presents the updating parts of this algorithm in detail. Two numerical examples have been used to study applicability of this method, which is shown in Section 7.3.
7.1 Problem Statement and the Identification Algorithm
Let 0 be a bounded domain in R2 with smooth boundary dQ. In the frequency domain the diffusion approximation (DA) equation is expressed by
where, T(x, u) is the photon density at position x and modulation frequency to, k(x) denotes the diffusion coefficient, /x«(x) is the absorption coefficient, c is the speed of light in the medium
frequently used boundary condition is Robin boundary condition, which is given as follows:
where g(x, to) is the boundary source, 7 is a dimension-dependent constant (72 = 1/7773 = 1/4) and v is the outer normal at boundary domain.
The unknown function in this problem are the diffusion coefficient k(x) and absorption coefficient /xa(x). The measurements in this paper consist of the complex intensity in terms of phase
(7.1.1)
and i = \/—1- In order to solve Eq. (6.1.1), boundary conditions need to be imposed. The most
(7.1.2)


73
shift and amplitude of received optical signal with adding 4% Gaussian noise to both, which are measured as
4>(x, w) = /(x) x G dQ. (7.4.3)
The inverse problem is to determine k(x) and /na(x) based on the additional measurement at the boundary (7.4.3). Two unknown functions need to be evaluated based on proper solution space. The algorithm is similar to previous chapters, however, it should conclude more steps than before. The algorithm is still iterative and consists of seven steps
f. Assume initial values for both unknown coefficients as k(x) and /fa(x). First, treat /na(x) = /x«(x) and use the given boundary conditions to obtain a background field satisfying the system
-(k(x) A i> + v«(x) • V$) + 0A(x) + 'lJf) 3> = 0, x G Q,
(7-1-4)
T(x) + =5(x), xedQ.
2. Subtract the background field from Eq. (7.4.f) and obtain the error field, e(x) = T(x) — T(x), given by
-(k(x) A T + v«(x) ' V$) + («(x) A + v«(x) ' V$)
(7.4.5)
+ (/xa(x) + yr) e = 0, x G Q,
where, the error field is required to satisfy e(x) + ^k(x)d(Q^ =0, Vxs dQ. The error field is also required to satisfy additional conditions given by
e(x) = /(x) — T(x) = /(x) x G dQ. (7.4.6)
3. Assume the unknown function k(x) is related to the assumed value according to k(x) = k(x) +p(x), where p(x) is still an unknown function. Use the additional boundary conditions in Eq. (7.4.6) to obtain the unknown correction term p(x). Then we can use the correction p(x) to update the assumed value, k(x), and go to step f. The iteration form step f to 3 is called iteration I.


74
Using iteration I, the unknown coefficient k(x) can be updated. However, the value of /7a(x) is still initial value. Therefore, the previous algorithm need to be expanded and use updated k(x) to recover unknown coefficient /xa(x).
4. Consider k(x) = k(x), which is obtained from iteration I and, using the given boundary conditions, obtain a different background field satisfying the system
-(k(x) A $ + v«(x) • V$) + (/7a (x) + f) $ = 0, x e Q,
(7.1
^(x) + ^(x)Â¥=j(X)> xg9Q.
5. Subtract the background field from Eq. (7.1.1) and Eq. (7.1.2), and obtain a new error field, e(x) = $(x) — T(x), given by
-(«(x) A e + v«(x) • Ve) + (/u(x) + ^) $,
(7.1.8)
- (/7a (x) + ^) T = 0, xefi,
where, the error field is required to satisfy e(x) + ^/^(x)9^ =0, Vxs <9Q. The new error field is required to satisfy additional conditions given by
e(x) = /(x) - $(x) = /(x) x £ dQ. (7.1.9)
6. Assume the unknown function /x«(x) is related to the assumed value according to /xa(x) = /7a(x) + 7. Repeat iteration II, the assumed absorption coefficient /7a(x) can be updated. Then we need to go back to step 1 and do iteration I and iteration II again to improve the result of k(x) and /7a(x), which is called iteration III.
The whole algorithm in this section consists of two parts. The iteration I is the basic algorithm used to recover absorption coefficient in chapter 5, and the iteration II is the algorithm used


75
to evaluate of diffusion coefficient in chapter 6. The difficult part of this problem is two unknowns in the system and it requires to recover them in the same time. Therefore, we can’t recover the unknown coefficients by a single iteration. Two assumed values need to be use in this algorithm. More details about step 3 and 6 will be shown in next section.
7.2 Proper Solution Space
7.2.1 Details of Step 3
The third step of the algorithm includes the determination of the correction to the assumed value of the diffusion coefficient k(x), which is one of the new features of this algorithm. Substituting for k(x) = k(x) ~hp(x) in Eq. (7.1.5) leads to
-k A e - vk • Ve+ ^a(x) + ^ e - p(x) A T - VT(X) • V$ = 0, x e Q, (7.2.1)
where, the error field satisfies e(x) + Wk{x)deg^ =0, Vxs dQ. Linearizing the equation around the background field leads to
-k A e - vk • Ve+ ^«(x) + ^ e - p(x) A T — \/p(x) ■ = 0, xeH. (7.2.2)
In order to recover p(x) for x e Q we can use the same assumption as before. Consider a linearly independent set of functions q(x), £ = 1,2,3, ...,N over x e Q and assume that the unknown functions p(x) can be expressed as a linear combination of functions q(x), i.e. p(x) e {ci, C2, C3,..., cn}- We are assuming that it is possible to evaluate the unknown function p(x) at the boundaries. For simplify, we assume that it is equal to one. As a result we can get that q(x) = 0 Vx e dQ. The next step is to generate a set of functions that satisfy the error field according to
-k A q - v« • Ver + ^0(x) + “) e* “ c* A $ - = 0, x G Q, (7.2.3)
with the known Robin boundary condition e^(x) + ^k(x)= 0, Vx e dQ. It is then possible to expand the actual error field e(x) in the span of the space generated by e^(x), £ =


76
1, 2, 3,N according to
N
e(x)=^r^(x), x e Q,
(7.2.4)
£=1
where, the functions e^(x) are known, but the constants T£ are unknown. We next argue that the error field e(x) must satisfy the Dirichlet condition are given in Eq. (7.4.6). The Dirichlet conditions can be expressed by the operator B. The error field is required to satisfy the condition given by
N N
Be(x.) = B^Teceix) = ^r^Se^(x) = /(x) x e <9Q. (7.2.5)
t=l t=l
The above equation can be used to obtain the unknown coefficients T£ for i = 1, 2, 3,..., N. More detail of this part will be shown in the next section. Once the unknown coefficients are obtained the unknown function p(x) can be computed by substituting Eq. (7.2.4) in Eq. (7.2.2) which leads to
N
t=i
-k A ez - xjk • yq + ft«(x) 4----------a
1LO
- p(x) AT- VP(X) ' = 0, (7.2.6)
for Vx e 0. Next we need to use Eq. (7.2.3) and simplify above equation to get
N
^q(qA + VA • V$) - P(x) AT- \/p(x) ■ = 0,
x £ Q.
(7.2.7)
e=i
Assuming the unknown function p(x) can be expressed as linear combination of functions q(x)
according to p(x) = Ym>=i CrA(x)) and simplifying lead to
N
X](a-6)(a AT +VA-V^5) = 0, xeQ.
(7.2.8)
e=i
Next we argue that in general T(x) / 0 and q(x) are linearly independent. Then we can conclude that (g = T£ for all ^ = 1,2,3,..., N and
N
p(x) = ^%(x) x £ Q.
(7.2.9)
£=1
This completes the third step of the algorithm. After solving the correction term p(x), the unknown diffusion coefficient can be updated according to k(x) = k(x) +p(x). Then the iteration I need to repeat to approximate the diffusion coefficient k(x).


77
7.2.2 Details of Step 6
In the sixth step, the same method need to be used to express the unknown function g(x) by the same linear combination of functions cg(x). Substituting for /x«(x) = /xa(x) + q(x) in Eq.
(7.1.8) and linearizing the new error field equation lead to
-(k(x) A e + VK(X) ' Ve) + (Va(x) + e + where the new error filed also need to satisfy e(x) + ^k(x)^^- = 0, x G <9Q. Use the same set of functions cg(x) and assume that q(x) can be expressed as a linear combination of functions cg(x). In order to evaluate the correction term -(k(x) A eg + VK(X) • Vh) + ^/4(x) + ^ eg + cg(x)$ = 0, x G Q, (7.2.11)
with ig(x) + ^k(x) = 0, Vx 6 <9Q. Then it is also possible to expand the error field e(x)
in the span of the space generated by eg(x), £ = 1, 2, 3,..., IV according to
N
e(x) = ^f^(x), x G Q, (7.2.12)
r=i
where q(x) can be obtained from Eq. (7.2.11) and the constant fg are still unknown. In order
to calculate fg, the same measurement /(x) need to be used according to
N N
Be(x) fg€g(x) = ^ fgBeg(x) = /(x) x G <9Q, (7.2.13)
e=i e=i
where, operator B is the same as before. Using the same analytical method, we can conclude that
N
q(x) = ^2fgcg(x) xGfi. (7.2.14)
e=i
This completes the sixth step of the algorithm. Solving for the correction term q(x), the unknown absorption coefficient can be updated according to /x«(x) = /fa(x) + q(x). Then the II need to be repeated to update the unknown coefficient /xa(x). Obtaining updated value of absorption coefficient, the algorithm should go back to step 1 and do iteration III to improve both k(x) and /xa(x).


78
7.3 Numerical Examples
In this section we use two examples to study the applicability of this method in this problem.
Example 1. Consider the evaluation of the of absorption coefficient and the diffusion coefficient. for a 2-D elliptic equation given in Ecp (7.1.1), and assume that the actual function of K(x,y) and ga(x, y) are given by
k(x, y) = 1 + exp —(x — 0.35)2 exp —(y — 0.65)2
0.02 '—(x — 0.65)2 0.02 \—(y — 0.3 5)21
Va{x, y) = 1 + exp
exp
0.02 0.02
which are shown in Fig. 7.1 and Fig. 7.2, and the boundary condition are given by
(7.3.1)
g(x,y) = cos(k(.r cos# + ysinO)).
(7.3.2)
Y 0 0 x
FIGURE 7.1: The actual diffusion coefficient k for Example 1.
Recover k(x)


79
Y 0 0 x
FIGURE 7.2: The actual absorption coefficient fia for Example 1.
In this example we use data with k = 5, 7,10,13,15 and 6 = 7t/5.2, 7t/7.2, 7t/9.2, 7t/11.2, 7t/13.2,7t/15.2. We can start, the algorithm by assuming the value for k(x,y) = 1 and jia{x,y) = 1, obtaining the background field $(x,y), and providing the boundary conditions for the error field for Eq. (7.1.6). In order to obtain the correction term p(x,y) in step 3, an appropriate set of linearly independent functions need to be considered according to
c/?(x, y) = sin(t?i7r.'r) sin^vry), cc = 0, Vx e dQ,
(7.3.3)
£\ = 1,2,3£2 = 1,2,3,...,M,
£ = (£i- 1 )N + l2, and M2 = N.
The solution set e£ can be obtained by using the individual functions cc(x,y) in Eq. (7.2.3), for £ = 1,2, 3,..., N. The finite difference approximation to the elliptic system simplifies to linear systems given by
Ak£e + Ace& = 0, or, ee = -Ak1ACe&, for T = 1,2,3, ...,1V,
(7.3.4)


80
where A% and ACl are finite dimensional approximation of the 2D diffusion operator depends on k and eg individually, eg and 4> are the vector containing the nodal values of the function eg(x,y) and Since the functions eg are linear independent, it follows that eg are also lin-
ear independent. In order to compute the unknown value re in Eq. (7.2.4), the measurement boundary condition for the error field need to be used. According Eq. (7.2.5) and denoting the appropriate the finite dimensional approximation to the gradient operator by B leads to
N N
Be = B ^ Tgeg = ^ TgBeg = f, or, [Beg] r = [f], (7.3.5)
e=i £=i
where, r is the N dimensional vector containing the values of the unknown coefficients Tg for ^ = 1,2,3,..., N. In the above equation, the vector f is known, the solution space eg for £ = 1,2,3, ...,N2 are also known from Eq. (7.3.4). The above coefficient matrix, [Bey], is non-square and rank-deficient. As shown in Eq. (7.2.9), the correction term can be expanded by cg(x,y). It is possible to stabilize the inversion by imposing some form of regularization according to
1 U) CP 1 f
Pi&x T = 0
0
(7.3.6)
where the matrix i$’x and y represent the first order differential operator given by
’iE

dcg(x, y)
dx
dcg{x,y) dy
N N
x=Xi,y=yj
x=Xi,y=yj
EE £itt cos(£i7TXi) sin^vrj/j),
£i = 1£2 = 1 N N
EE sin(t?i7r ay) cos^nyj),
(7.3.7)
£1 = 1£2 = 1
where ay and yj are the discrete location in the domain. The constant f3\ is larger than zero, which is set by the designer, and the above linear system can be solved for unknown coefficient Tg for ^ = 1,2, 3,..., N. Substituting Tg in Eq. (7.2.9), we can get the correction term p(x) and update the diffusion coefficient according to /ta(x) = /fa(x) + q(x).
Recover /xa(x)


81
Similar as previous iteration, the updated absorption coefficient need to be used in this part. The finite difference approximation to Eq. (7.2.11) to linear systems given by
= 0, or, eg = -A^cg®, for l = 1, 2, 3,..., N, (7.3.8)
where Aya is finite dimensional approximation of the 2D diffusion operator depends on /la with unique inverse, £g and 4 are the vector containing the nodal values of the function eg(x, y) and and eg is a vector depends on the functions cg(x,y) = sin(t?i7ra;) sin(t?2vry). Since ci are linear independent, it shows that eg are also linear independent. According to equation (7.2.13) and adding regularization term, the new system is given by
1 <10 CP 1 {
Ih* x T = 0
0
By solving the above system, the unknown constants fg can be obtained. Substituting fg in Eq. (7.2.14), we can get the correction term q(x) and update the absorption coefficient according to Mx) = /4(x) + Considering a two-dimensional domain (x, y) € Q = [0,1] x [0,1] and dividing the domain into ne = 60 equal intervals leads to n = ne + 1 = 61 nodes in the x and y direction. Choosing M = 20 sine functions in Eqs. (7.2.4) and (7.2.12) lead to the proper solution space having N = M2 = 400 vectors, with each vector having the dimension n x n. The collected data at the boundaries are contaminated with a zero mean randomly generated noise with noise-to-signal ratio of 4 %. The data is filtered by the seventh order polynomial curve fitting. Figure 7.3 and Figure 7.4 show the recovered diffusion coefficient and absorption coefficient. The iteration I for updating k(x) with (5\ = 8e — 7 is 5, the iteration II for recovering /x«(x) with /?2 = le — 5 is 40, and the iteration III is 10. We also study the reduction in the error, which is defined as Eq.
(6.3.9). The error result of diffusion coefficient is shown in Fig. 7.5 and result of absorption is shown in Fig. 7.6.


Ma(x’V) K(x,y)
82
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
FIGURE 7.3: Recovered diffusion coefficient n for Example 1
1
1.4
1.3
1.2
1.1
1
FIGURE 7.4: Recovered absorption coefficient for Example 1


83
Number of iterations
FIGURE 7.5: Error reduction for diffusion coefficient n as a function of the number of iteration
Number of iterations
FIGURE 7.6: Error reduction absorption coefficient fia as a function of the number of iteration


84
Example 2. Next consider the evaluation of the diffusion coefficient given by k(x, y) = 1 + exp
1 o bo to \—(y — o. 7)21 1 o to \—(y — o. 3)21
0.015 exp 0.015 + exp 0.015 exp 0.015
(7.3.10)
and absorption coefficient given by '—(x — 0.3)2
Lia(x,y) = 1 + exp
0.015
exp
'-(y- Q-3)2
0.015
+ exp
"—(x - 0.7)s 0.015
exp
'-(V- 0-7)'
0.015
(7.3.11)
which are shown in Fig. 7.7 and Fig. 7.8, and the same boundary condition given in Eq. (7.3.2) For this example we set the f3\ = 5e — 7, /?2 = 5e — 5 and M = 20. The iteration I is 10, the iteration II is 40, and the iteration III is 10. The recovered results for k(x) and /xa(x) are presented in Fig. 7.9 and Fig. 7.10. Figure 7.11 and Figure 7.12 show the reduction in the error as a function of the number of iterations for k(x) and /xa(x).
FIGURE 7.7: The actual diffusion coefficient n for Example 2.


K(x,y) A*a(x>y)
85
2
1.8
1.6
1.4
1.2
1
1
Y 0 0
X
2
1.8
1.6
1.4
1.2
1
FIGURE 7.8: The actual absorption coefficient fia for Example 2.
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
FIGURE 7.9: Recovered diffusion coefficient n for Example 2.


86
x"
1.1
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
FIGURE 7.10: Recovered absorption coefficient f.ia for Example 2.
Number of iterations
FIGURE 7.11: Error reduction for diffusion coefficient k as a function of the number of iteration.


87
FIGURE 7.12: Error reduction for diffusion coefficient fia as a function of the number of iteration.
7.4 Remark
In this chapter, we studied the inverse evaluation of the diffusion and absorption coefficient in a 2-D elliptic problem by using proper solution space method. We used two 2-D examples to study the applicability of this method.


CHAPTER VIII
EVALUATION OF THE BOUNDARY OF THE VACUUM IN A
TOKAMAK
In this chapter, we study the inverse evaluation of the internal boundary of the tore supra in a tokamak. This is essential to the stable operation of a magnetic confinement device with toroidal geometry as shown in Fig 8.1. The exact location of the plasma within the confinement is essential for the design of feedback control laws which are necessary for the stable operation of a fusion based energy source. Fusion based energy production has a potential to provide unlimited energy with little negative byproducts such as waste, or pollution [103]. In this chapter, we investigate an inverse problem for the poloidal flux which acts as a buffer between the plasma and the boundary. Analytical issues on this problem have been presented in [104], and a review of recent works on this inverse problem can also be found in [105].
Two computational methods have been used for evaluating of the interior boundary of the tore supra (or plasma boundary inside a Tokamak). By interior boundary we are referring to the interior boundary of the vacuum which is the same as the outer boundary of the plasma. The first one is based on the adjoint method, which seeks to minimize a cost functional. This method is computationally feasible, and fast. However, we have no result on the convexity of the minimization. The second method is somewhat indirect. In the sense that we first obtain the value of the poloidal flux within a ring which includes the tore supra (or the vacuum). In other words, we extend the domain and let the domain of interest be within two concentric circles. Then using the fact that the poloidal flux has a constant value of say c at the interior boundary, we can interpolate the location of the boundary. Basically, the second method involves the solution of a Cauchy problem for an annular domain. This problem which is still ill-posed has been con-


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NOVELCOMPUTATIONALMETHODSFORINVERSEHEAT CONDUCTIONPROBLEMS by YUANLONGWANG B.S.,BeijingUniversityofTechnology,2012 M.S.,UniversityofColoradoDenver,2014 M.S.,UniversityofColoradoDenver,2018 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy EngineeringandAppliedScienceProgram 2018

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ii ThisthesisfortheDoctorofPhilosophydegreeby YuanlongWang hasbeenapprovedforthe EngineeringandAppliedScienceProgram by KannanPremnath,Chair MohsenTadi,Advisor JanMandel L.RafaelSanchezVega TroyButler Date:December15,2018

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iii Wang,YuanlongPh.D.,EngineeringandAppliedScience NovelComputationalMethodsforInverseHeatConductionProblems ThesisdirectedbyAssociateProfessorMohsenTadi ABSTRACT Thisdissertationisconcernedwithtwocomputationalmethodsforevaluatingvedierentinverseheatconductionproblems.Therstproblemistheinverseevaluationoftheunknowninitialconditionforaparabolicsystem.Secondandthirdproblemsareinverseevaluationofabsorptionanddiusioncoecients.Thefourthproblemistheinverseevaluationofcoecients foraphotondiusionequation,andthefthproblemistheinverseevaluationoftheboundaryoftoresuprainaTokamak.Therstmethodisbasedonpropersolutionspace,whichbeen usedtosolveallveproblems.Thealgorithmisiterativemethodinnature,whichstartswith aninitialguessfortheunknownfunctionandobtainscorrectionsoftheassumedvalueineach iterationstep.Theupdatingpartisthenewfeatureofthepresentalgorithm.Thismethodgeneratesasetoffunctionsthatsatisfysomeoftheboundarycondition.Itthenusestheremaining boundaryconditiontoupdatetheassumedvalue.Theothermethodisbasedonoptimization whichseekstominimizeacostfunctional.Thismethodisalsoaniterativemethod,andwill beusedinevaluatingtheboundaryshapeofthevacuuminaTokamak.Bothalgorithmsshow goodrobustnesswhichcanbeusedtoobtainagoodestimateoftheunknownfunction.Anumberofnumericalexamplesfordierentinverseproblemsareusedtoshowtheapplicabilityof themethods. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:MohsenTadi

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iv DEDICATION I would like to thank Dr. Mohsen Tadi for the supervision of this dissertation. He supported me with several helpful hints and suggestions when I needed them, and he always had time to discuss my work. Furthermore, I especially wish to express my gratitude to my office mate Abdalkaleg Hamad, who always had time for discussions on the topic of this dissertation. Last but not least I want to thank my family for their encoura ging help during the whole four years and they never ending patience while listening to me ex plaining my work. Without their support, it is hard for me to get where I am today.

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v ACKNOWLEDGEMENTS I would like to thank Dr. Mohsen Tadi, Abdalkaleg Hamad, and all unnamed referees for their valuable corrections and comments.

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vi TABL E OF CONTENTS CHAPTER I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II . INVERSE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 What Is An Inverse Problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Well posed And Ill posed Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Examples of Ill posed Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.1 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.2 Diff erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.3 Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . 8 2.3.4 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 III. INVERSE HEAT CONDUCTION PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Classi fi cation of the Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Evaluation of the Boundary Conditions (Cauc hy Problem) . . . . . . 17 3.1.2 Evaluation of the Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.3 Evaluation of Thermophysical Properties . . . . . . . . . . . . . . . . . . . . 18 3.1.4 Inverse Source Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.5 Evaluation of the Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Methods for Solving Inverse Heat Conduction Problems . . . . . . . . . . . . . . . 19 3.2.1 Method of Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.2 Truncated SVD Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.3 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.4 Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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vii 3.2.5 D bar Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.6 Method of Multiple Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.7 Active Subspace Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 General Identifi cation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Identifi cation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Proper Solution Space Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 General Contribution of This Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 IV. EVALUATION OF THE INITIAL CONDITION . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1 Problem Statement and Identi fi cation Algorithm . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Proper Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Numer ical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 V. EVALUATION OF ABSORPTION COEFFICIENT . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1 Problem Statement and the Identi fi cation Algorithm . . . . . . . . . . . . . . . . . . . 42 5.2 Proper Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 .4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 VI. EVALUATION OF DIFFUSION COEFFICIENT; CALDERON'S PROBLEM . . . 56 6.1 Problem Statement and the Identi fi cation Algorithm . . . . . . . . . . . . . . . . . . . 56 6.2 Pro per Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 VII. EVALUATION OF ABSORPTION AND DIFFUSION COEFFICIENT . . . . . . . . 72

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viii 7.1 Problem Statement and the Identi fi cation Algorithm . . . . . . . . . . . . . . . . . . . 72 7.2 Proper Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.1 Details of Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.2 Details of Step 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 VIII. EVALUATION OF THE BOUNDARY OF THE VACUUM IN A TOKAMAK . 88 8.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2 Adjoint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.4 Proper Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.5.1 Adjoint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.5.2 Interpolation and Proper Solution Space . . . . . . . . . . . . . . . . . . . . . 99 8.6 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 IX. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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ix LIST OF FIGURES FIGURE 2.1 Mathematical model of a physical process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Actual g and noisy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Result of act ual and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Electrical impedance tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 CT scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Geometry of data acquisition in CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 Ill posed problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Well posed problem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Well posed problem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1 The recovered initial condition for Example 1. The fi gure compares the fi na l value to the actual function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 The reduction in error for Example 1 as a function of the number of iterations . . . . 38 4.3 The recovered initial condition for Example 2. The fi gure presents the r ecovered function at a fe w intermediate iterations and compares the fi nal value to the actual function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 The reduction in error for Example 2 as a function of the number of iterations . . . . 39 4.5 The recovered initial condition for Example 3. The figure presents the r ecovered function at a few intermediate iterations and compares the fi nal value to the actual function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.6 The reduction in error for Example 3 as a function of the number of iterations . . . . 40

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x 5.1 The recovered absorption for the Example 1. The fi gure compares the fi nal v alue to the actual function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 The reduction in error for the Example 1 as a function of the number of iterations . 49 5.3 The actual absorption coe f f i cient for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 The recovered absorption coe ffi cient for the Example 2, after 1000 iterations . . . . . 51 5.5 The reduction in error for the Example 2 as a function of the number of iterations . 52 5.6 The recovered absorption coe ffi cie nt for the Example 3, after 150 iterations . . . . . . 52 5.7 The reduction in error for the Example 3 as a function of the number of iterations . 53 5.8 The actual absorption coe ffi cient for the Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.9 The recovered absorption coe ffi cient for the Example 4, after 1000 iterations w ith full measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.10 The recovered absorption coe ffi cient for the Example 4, after 1000 iterations w ith partial measurements. The data is collected within the interval [0.3, 0.7] on each side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.1 Thermal conductivity for example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 Recovered conductivity after 174 iterations for Exampl e 1 . . . . . . . . . . . . . . . . . . . . 62 6.3 The reduction in error for Example 1 as a function o f the number of iteration . . . . . 63 6.4 Recovered conductivity after 150 iteration for Example 2 with more sets of data . . . 64 6.5 Comparison of the recovered and the actual conductivity at the diagonal cross section for Example 2 for diff erent levels of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.6 Comparison the reduc tion error for Example 2 for diff erent levels of noise . . . . . . . 65 6.7 Thermal conductivity for Example 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.8 Recovered conductivity after 700 iteration for Example 3. . . . . . . . . . . . . . . . . . . . . 67

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xi 6.9 Error reduction for Example 3 as a function of the number of iteration . . . . . . . . . . . 67 6.10 Comparison of the recovered and the actual con ductivity at the diagona l cross section for Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.11 Comparison of the recovered and the actual conductivity at the diagonal cross section for Example 3 for diff erent levels of noise . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.12 Thermal conductivity for Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.13 Thermal conductivity for Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.14 Comparison of the recovered and the actual con ductivity at the diagonal cross section for Example 4 for two values of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.15 Error reduction for Example 3 as a function of t he number of iteration for two values of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.1 The actual di ff usion coe fficient for Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.2 The actual absorption coe ffi cient for Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.3 Recovered di ff usion coe fficient for Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.4 Recovered absorption coe fficient for Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.5 Error reducti on for di ff usion coe fficient as a functi on of the number of iteration . 83 7.6 Error reduction absorption coe fficient as a function of the number of iteration . . 83 7.7 The actual diff usion coe fficient for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.8 The actual absorption coe fficient for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.9 Recovered diff usion coe fficient for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.10 Recovered absorption coe fficient for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . 86 7.11 Error reduction for diff usion coe fficient as a funct ion of the number of iteration. 86

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xii 7.12 Error reduction for diff usion coe fficient as a funct ion of the number of i teration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.1 Toroidal geometry of the tokamak Tore Supra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2 A cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.3 A cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.4 Normal derivative at the boundary (data), and noisy data. . . . . . . . . . . . . . . . . . . . . . 98 8.5 Recover unknown interior boundary for the example 1 . . . . . . . . . . . . . . . . . . . . . . . 99 8.6 Reduction in error for the example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.7 Normalized error in the recovered interior boundary for the example 1 . . . . . . . . . 100 8.8 Recovered unknown interior boundary: example 1, eff ect of noise . . . . . . . . . . . . . 101 8.9 R eduction in error, example 1, eff ect of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.10 Normalized error in the recovered interior boundary, example 1 , effect of noise . 102 8.11 Normal derivative at the boundary (data), and noisy data. . . . . . . . . . . . . . . . . . . . 105 8.12 Recovered u nknown interior boundary for different value of . . . . . . . . . . . . . . . 106 8.13 Reduction in error for different value of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.14 The relative error for the rec overed functions for different value of . . . . . . . . . . 107 8.15 Recovered u nknown interior boundary for diff erent value of noisy . . . . . . . . . . . . 108 8.16 Reduction in error for diff erent value of noisy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8 .17 The relative error for the recovered functions for diff erent value of noisy . . . . . . 109

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1 CHAPTERI INTRODUCTION Theproblemsofthedeterminationoftheunknownfunctionsinvolvedinmathematicalmodels describingvariousphysicalsystemsarerefereedto inverseproblems .Thisidenticationisbased onexcitingthephysicalsystembyexternalinputsandrecordingtheoutputs.Suchproblems includethedeterminationofdiusivityandconductivityproles[1,2],absorptioncoecient [3],sourceterms[4],geometryofscatterers[5]andpriortemperaturedistributions[6]. Theeldofinverseproblemshasdevelopedrapidlyinrecentyears.Onendsnumerousapplicationsofinverseanalysisinthephysicalandmechanicalsciences,suchasthedeterminationofearthquakehypocenters[7],groundwaterhydrology[8],polymerprocessing[9],material cooling[10],opticaltomography[11],nondestructiveevaluation[12],oceancirculation[13],signal/imageprocessing[14],weatherprediction[15],dataanalysis[16],computervision[17],machinelearning[18]andmanymore.Itturnsoutthatmanysuchinverseproblemsareill-posed, whilethecorrespondingdirectproblemsarewell-posed.ChapterIIprovidesmoredetailsabout thedenitionofinverseproblemandintroducesdicultiesencounteredwithanill-posedproblem. Thisdissertationfocusesoninverseheatconductionproblems.Inaheatconductionproblem, iftheboundaryconditionsareknown,andallparametersofthesystemareknown,thenthe temperaturedistributionofthedomaincanbeevaluated,whichistermedasadirectproblem. Howeverinmanyheattransfersituations,thesurfaceheatuxand/ortemperaturemustbedeterminedfromtemperaturemeasurementsatoneormoreinteriorlocations,whichistermedas aninverseproblem.ChapterIIIbrieyintroducestheclassicationofinverseheatconduction

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2 problemsandseveralexistingmethodsforsolvingsuchproblems. Theinverseproblemscanbedividedintovecategoriesbasedonthetypeoftheestimation thatisneeded.Weconcentrateoncomputationalmethodsforthreeclassesofproblems.ChapterIVisconcernedwiththeevaluationoftheinitialconditionforaparabolicsystem,wherethe unknownfunctionistheinitialcondition.Themethodstartswithaninitialguessfortheunknownfunctionandobtainscorrectionstotheassumedvalueineveryiteration.Inheattransfer,thermalconductivityisameasureofhowfasttheheatowsthroughthemedium.The absorptioncoecientishowheatisabsorbedorgenerated.Bothcoecientsareimportantin heattransferanalysis.ChapterVisconcernedwiththeevaluationofabsorptioncoecientof aparabolicsystem.ChapterVIfocusesonevaluationofthethermalconductivityofanellipticequation.Theunknowncoecientisthepositivethermalconductivity.Inbothchapters, weassumethattheunknownfunctioncanbemeasuredattheboundary.InChapterVII,we consideraninverseproblemforanellipticsystemwithtwounknownfunctions.ChapterVIII isconcernedwithtwomethodsfortheinverseevaluationoftheinteriorboundarylocationof avacuuminaTokamak.Byinteriorboundarywearereferringtotheinteriorboundaryof thevacuumwhichisthesameastheouterboundaryoftheplasma.Therstmethodisbased onoptimizationwhichseekstominimizeacostfunctional,andthesecondmethodusesthe methodofpropersolutionspacetogeneratethevalueofthepoloidaluxwithintwoconcentriccircles.Ittheninterpolatesforthelocationoftheboundary.

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3 CHAPTERII INVERSEPROBLEMS Inverseproblemshavebeenoneofthefastestgrowingareasinvariousapplicationelds[19]. Themathematicaldescriptionofinverseproblemmaybebroadlydescribedastheproblemof determiningtheunknowninternalstructureorunrecordedpaststateofasystembyusingindirectmeasurements.Themajordicultyinthetreatmentofinverseproblemsistheinstability ofthesolutioninthepresenceofnoiseintheobservedmeasureddata. 2.1WhatIsAnInverseProblem? Typically,whenwedescribeaphysicalprocessviathemathematicalmodelseeFig.2.1,it containsthefollowingthreedierentcomponents:thesystem,theinputs,andtheoutputs.The analysisofagivenprocesscanbesubdividedintothreetypesofproblems: Thedirectproblem:giventheinputdataandsystemparameters,computethe output. Thereconstructionproblem:giventhesystemparametersandtheoutputofthe process,computetheinputthatleadstothegivenoutput. Theidenticationproblem:giventheinputandtheoutput,computetheunknownsystemparameterswhichisinagreementwiththerelationbetweenthe inputandtheoutput. Thetypeiscalleddirectproblemsinceitisorientedalongacause-eectconsequence.Type andareusuallytreatedasaninverseproblem.Weprovideamathematicaldescriptionof

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4 FIGURE2.1:Mathematicalmodelofaphysicalprocess thisprocess.Assumethatthegivendatais g 2G andtheunknownquantityis u 2U .Consider asimpleoperatorequationgivenby A u = g: .1.1 Thedata g andtheunknown u canbeeithervectorsorfunctions.Here, A : U!G denotesan operatormappingfromthespace U R n tothespace G R n .Therefore,theinverseproblem istodetermine u 2U suchthat u = A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 g whenarbitrary g 2G isgiven.Aninverseproblem canbedescribedasthesolutionoftheoperatorequationproducedbysomephenomenonfrom physics,andthedirecttheoryisthemathematicaldescriptionofthephysicalphenomenon. 2.2Well-posedAndIll-posedInverseProblem Considerthedenitionofinverseproblemfromprevioussection: Find u 2U thatsatisesEq..1.1wherethegiveninput g 2G isarbitrary.Forawell-posed inverseproblemwemusthavethefollowing Theproblemmustpossessasolutionforallinputdata,i.e.,forevery g 2G there have u 2U suchthat g = A u .Inotherwords,theoperatormapping A needsto beasurjection. Theproblemmusthaveauniquesolution,i.e.,suppose u 1 ;u 2 2U aretwosolutionssatisfying A u 1 = A u 2 = g 2G then u 1 = u 2 musthold.Intheotherwords, theoperatormapping A : U!G needstobeaninjection.

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5 Whentheoperator A isbothinjectiveandsurjective,thenitisabijectionand theinversemapping A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 : G!U exists.Thentheinverseoperator A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 mustbe continuous,andthesolutionofEq..1.1mustbestable. Ifanyoftheserequirementsisnotfullled,theinverseproblemiscalledill-posed.Thesmall changesinthedatawillleadtolargedeviationinthesolution,whichmeansthesolutionis unstable.Inpractice,thegivendata g isneverexactlythesameasinthemathematicalformulation.Thereareseveralreasonsforthis,suchasthelimitedaccuracyofthemeasurement equipment,theincompletenessofthemathematicalmodel,theexternaldisturbancesintheenvironment,andtheround-oerrorinthenumericalcomputation.Someusefulremarksonthe inverseandill-posedproblemscanbefoundin[20].Someclassicexamplesofinverseandillposedproblemareshowninnextsection. 2.3ExamplesofIll-posedInverseProblems 2.3.1MatrixInversion Considerthesolutionofalinearsystem.ItcanbewrittenintheformofEq..1.1with u 2 R n and g 2 R n being n -dimensionalrealvectors,and A 2 R n n beingamatrixwithrealentries.Considertheperturbedsystemgivenby A ^ u = g + g ; .3.1 where g isanaddeddisturbanceornoise.Therelativeerrorofdataandofsolutioncanbe denedas e g = k g k k g k and e u = k u k k u k = k u )]TJ/F15 10.9091 Tf 11.667 0.152 Td [(^ u k k u k : .3.2 Sincewehave A u )]TJ/F15 10.9091 Tf 11.666 0.152 Td [(^ u = g = u )]TJ/F15 10.9091 Tf 11.666 0.152 Td [(^ u = A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 g ; .3.3 thentherelativeerrorofsolutioncanbeexpressedas e u = k u )]TJ/F15 10.9091 Tf 11.667 0.152 Td [(^ u k k u k = k A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 g k k u k k A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 kk g k k u k : .3.4

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6 Wealsohaveanotherrelationshipwhichis Au = g = k Au k = k g k = k A kk u kk g k = 1 k u k k A k k g k : .3.5 Therefore,therelativeerrorcanberewrittenas e u k A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 kk g k k u k k A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 kk A k k g k k g k = A k g k k g k .3.6 where A = k A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 kk A k denotestheconditionnumberofthematrix A .Itiswellknown fromnumericallinearalgebrathattheconditionnumberisameasureofstabilityofEq..1.1. Henceweobservethattherelativeerror e g ofdatawillbeampliedbyalargeconditionnumber A .Amatrixwithalargevalueof iscalledill-conditioned. 2.3.2Dierentiation Anotherclassicinverseproblemisthedierentiation.Supposewehaveafunction g 2 C 1 [0 ; 1] with g =0forwhichwewanttocompute u = g 0 .Theseconditionsaresatisedifandonlyif u and g satisfytheoperatorequation g y = Z y 0 u x dx; .3.7 whichcanberewrittenastheoperatorequation Au = g sameasEq..1.1withthelinear operator.Nowweconsider g asthenoisydatasubstitutewith g forwhichweassumethatthe perturbationisadditive,i.e. g = g + n .Itisobviousthatthederivative u onlyexistsifthe noise n isalsodierentiable.However,evenincase n isdierentiable,theerrorinthederivativecanbearbitrarilylarge.Considerasequence j j 2 N with j ! 0andthesequencecanbe denedas n j x := j sin kx j .3.8 foraxedbutarbitrarynumber k .WecanobtainthesolutionofEq..1.1withperturbation as u j x = g 0 + k cos kx j ; .3.9

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7 andthereforeobtaintheestimation k u )]TJ/F25 10.9091 Tf 10.909 0 Td [(u j k L 1 [0 ; 1] = k n j 0 k L 1 [0 ; 1] = k cos kx j L 1 [0 ; 1] = k; .3.10 where L 1 normisdenedasthemaximumoftheabsolutevaluesofitscomponents.Thus,despitethefactthatthenoiseinthedataisbecomingarbitrarilysmall,theerrorinthederivative canbecomearbitrarilylarge.Consideragivenfunction g 2 C 1 [0 ; 1]as g x = x 3 and g =0 : .3.11 Thedirectprobleminthisconditionshouldbedeterminethefunction g 2 C 1 [0 ; 1]whenits derivative g 0 x =3 x 2 withtheinitialvalue g =0.Theinverseproblemshouldbedetermine thederivativeof g 2 C 1 [0 ; 1]when g x = x 3 . Theexactresultofthisinverseproblemshouldbe u = g 0 x =3 x 2 : .3.12 However,somedicultiesarisewhenthegivendata g isinaccurate.Insteadofg,weassume theobservedata g isgivenas g x = g x + 1 200 sin x = x 3 + 1 200 sin x ; .3.13 whichhasderivative u = g 0 x =3 x 2 +cos x : .3.14 Theerrorof u is k u )]TJ/F25 10.9091 Tf 10.909 0 Td [(u k L 1 [0 ; 1] = k cos x k L 1 [0 ; 1] =1 : .3.15 Forthisexample,thedisturbeddata g iscloseto g whichisshowninFig.2.2.However,correspondingderivativesareprettydierent,whichisshowninFig2.3.Thesolutionsofill-posed inverseproblemsoftenshowstrongdependencyonsmallperturbationnoiseinthedata.

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8 FIGURE2.2:Actual g andnoisy g FIGURE2.3:Resultofactual g and g 2.3.3FredholmIntegralEquationsoftheFirstKind ConsideraFredholmintegralequationoftherstkindgivenby Z t K s;t f t dt = g s ;s 2 s ; .3.16

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9 where K isagivenkernelfunction, g istheknownright-handsidefunction,andtheunknownis thefunction f t .Thisproblemisill-posedinthesensethat thesolutionmaynotexist,and/ormaynotbeunique, evenifauniquesolutionexistsanarbitrarilysmallperturbationoftheknown data g cangiverisetoalargeperturbationofthesolution. Afunction K issquareintegrableifitsnormexists.Everysquareintegrablekernelhasasingularvalueexpansiongivenby K s;t = 1 X i =1 i u i s v i t ; .3.17 and Z t K s;t v i t dt = i u i s ; for i =1 ; 2 ;::: .3.18 where h u i ;u j i = h v i ;v j i = ij ,and 1 2 3 ::: 0.NowsupposeEq..3.17isan absolutelyanduniformlyconvergentexpansionfor K s;t ,then Z t K s;t f t dt = Z t 1 X i =1 i u i s v i t f t dt = 1 X i =1 i h v i t ;f t i u i s = g s : .3.19 Ontheotherhand f u i s g 1 i =1 isanorthogonalbasisandthus g s = 1 X i =1 h u i s ;g s i u i s : .3.20 Comparingcoecientsof u i s inEq..3.19andEq..3.20gives i h v i t ;f t i = h u i s ;g s i , whichgivesthesolutiontoEq..3.16as f t = 1 X i =1 h v i t ;f t i v i t = 1 X i =1 h u i s ;g s i i v i t : .3.21 TheEq..3.16hasasolutionifandonlyiftherighthandside g satisesthePicardcondition 1 X i =1 h u i ;g i i 2 < 1 : .3.22 ThePicardconditionimpliesthattheabsolutevalueofthecoecients h u i ;g i mustdecayfaster thanthecorrespondingsingularvalues i inorderthatasolutionexists.Themaindicultyis

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10 whenthenoisydata g doesnotsatisfythePicardcondition.Anotheraspecttobeconsideredis thescaleofperturbationinthedatathatwillaectthesolution.Considerthefunction f t =sin2 nt ;n =1 ; 2 ;::: .3.23 then,for n !1 andarbitrary K wehave g s = Z 1 0 K s;t f t dt = Z 1 0 K s;t sin nt dt ! 0 : .3.24 Thisshowsthefunction f t hasbeensmoothedby K .Highfrequenciesaredampedinthe mapping f 7! g .However,themappingfrom g to f amplifythehighfrequencies.Fromthe aboveexample,wecandistinguishthedicultieswhilerecovering f fromnoisydata g . 2.3.4Tomography Thenon-destructiveevaluationNDEisawidegroupofanalyticalandexperimentalmethodsthatareusedtoprobestructuressuchasbridgesandbuildings[21].Thesetechniquesare widelyusedinmedicalimaging,includingComputedTomographyCT[22],PositronEmission TomographyPET[23],MagneticResonanceImagingMRI[22],ElectricImpedanceTomographyEIT[24],Microwaveimaging[25],andAcousto-electromagnetictomography[26].The availabilityofcomputationalresourcesalsomakesitpossibleforinvestigatorstodevelopaccuratecomputationalalgorithmsforsuchproblems. 2.3.4.1ElectricalimpedancetomographyEIT ElectricalimpedancetomographyEITisamedicalimagingsystem,whichseekstoproduce apictureoftheinteriorstructureofanobject,i.e.,humanbody,byrecoveringitsconductivity distributionfromelectricalmeasurementsontheboundaryoftheobject[27,28,29].Forthe measurement,anarrayofelectrodesisattachedtotheskinboundary.Inthisprocedure,electriccurrentsarefedthroughtheelectrodesandthevoltageattheelectrodesaremeasuredas showninFig2.4.

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11 FIGURE2.4:Electricalimpedancetomography Currentlythemostaccuratemathematicalmodelfortheelectricimpedancetomographyis givenbythecompleteelectrodemodel[24,30].Consideraboundeddomainin R 2 or R 3 , withacontinuousboundary @ .Attachedtotheboundary @ areasetofelectrodesdenoted by f e ` g L ` =1 ,whichareopenconnectedandaredisjointfromeachother,i.e., e i e j =0,if i 6 = j . Let I ` denotetheappliedcurrentonthe ` -thelectrode e ` ,andinaddition,thecurrentvector I =[ I 1 ;I 2 ;:::;I ` ] T satisfy P L ` =1 I ` =0conservationofcharge.Let U ` denotethevoltage atthe ` -thelectrodewith U =[ U 1 ;U 2 ;:::;U ` ] T ,togetherwithagroundingconditiongivenby P L ` =1 U ` =0.Givenasetofinputcurrents I ` ,theforwardproblemforthecompleteelectrode

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12 modelistondoutthepotential u andtheelectrodevoltages U ` for ` =1 ; 2 ;:::;L suchthat 5 x 5 u x =0 ; for x 2 ; u + z ` @u @n = U ` ; on e ` ;` =1 ; 2 ;:::;L; Z e ` @u @n ds = I ` ; for ` =1 ; 2 ;:::;L; @u @n =0on @ n[ L ` =1 e ` ; .3.25 where x istheelectricconductivity, n istheouternormalvector,and f z ` g L ` =1 arepositive constantimpedances.Theappliedcurrentspattern I ` isknown,anditispossibletomeasure thevoltagesattheelectrodes U ` .Electricalimpedancetomographyistheinverseproblemof determiningtheelectricconductivityintheinteriorofdomain,basedontheappliedcurrent, andthemeasuredvoltagesattheelectrodes. TheelectricimpedancetomographyEITtechnologyhaswidelyusedinvariouseldsincludingearlydetectionofbreastcancer[31],headimaging[28],noninvasivetesting[32,33],and processmonitoringinindustry[34]. 2.3.4.2X-rayComputedTomographyCT IncomputerizedtomographyCT[35,36],onemakesuseofcomputer-processedcombinations ofmanyX-raymeasurementstoproducecross-sectionalimagesofspecicareasofascanned object.Ingeneral,athree-dimensionalbodyismeasuredfromanoutercurve,wherethegoalis torecoveratwo-dimensionalslicefromthecollecteddata.Mathematically,theexactmeasurementisacollectionoflineintegralsofthenon-negativeattenuationcoecientfunctionalong thepathsoftheX-rays.X-rayTomographyhasbeenwidelyusedinMedicaldiagnosis[37,38]. Duetothefactthatdierenttissuesabsorbdierentamountsofx-rayradiation,theirabsorptioncoecientshavedierentvalues.Therefore,whenthetotalx-rayabsorptionacrossthe bodyismeasuredindierentdirections,onecanreconstructtheabsorptioncoecientacross thebody.AmodernCTscannerisshowninFig.2.5. ThesimplestmathematicalmodelofCTassumesthatthescannermeasuresthelineintegralsof theabsorptioncoecient a x .Weassumethatthereisasource, S ,emittinganX-raybeam,

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13 FIGURE2.5:CTscanner FIGURE2.6:GeometryofdataacquisitioninCT whichcrossesthebodythroughpath L ,anditsintensityismeasuredbyadetectorattheend. TheattenuationoftheX-rayscrossthebodyatlocation x seeFig.2.6canbedescribedby thefollowingsimplemodel dI du x = )]TJ/F25 10.9091 Tf 8.485 0 Td [(a x I x ; .3.26 where I istheintensitymeasuredby D and u iscoordinateinFig.2.6.Itfollowsthat ln I 0 I = )]TJ/F31 10.9091 Tf 10.303 14.849 Td [(Z L a x du: .3.27

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14 Bymovingthe S )]TJ/F25 10.9091 Tf 11.463 0 Td [(D systemalongtwoparallellinesandmeasuringtheintensityatallpossiblepositions,wecangettheprojectionoftheunknowncoecient a x indirection ,whichis givenby P a s = Z a s + u 0 du: .3.28 Here 0 denotesavectororthogonalto .Byrotatingthe S )]TJ/F25 10.9091 Tf 12.334 0 Td [(D systemandrepeatingthe abovelinearscanningforallpossibleanglesandpositionswecanobtainallpossibleprojections, whichgivesrisetotheRadontransform R a s; = P a s = Z a s + u 0 du: .3.29 Thenitispossibletosimplifythesystemas g s; = R a s; ; .3.30 where g s; isthemeasurementdata.Therefore,theinverseproblemofcomputerizedtomographyisthedeterminationof a basedontheknowledgeofitsintegral g . 2.3.4.3bioluminescencetomography Inbioluminescencetomography[39,40,41],biologicaltestsubjectse.g.tumourcellsaretagged withluciferaseenzymesandimplantedinasmallanimal.Thistechnologyprovidesawaytorevealcellularandmolecularfeaturesinbiologyanddiseaseinrealtime.Inessence,thegoalof bioluminescencetomographyBLTistolocatethedistributionandquantitativelyreconstruct theintensityoftheinternallightsourceusingthetransmittedandscatteredbioluminescentsignalonthesurfaceofthesmallanimal.OneapproachtomodelthetransferoflightinbiophotonicsistheRadiativeTransferEquationRTEapproximatedfromtheMaxwellsEquations. Inmanyapplications,itissucienttoconsiderthediusionequationgivenby )-222(5 D 5 x + a x = S x ; x 2 ; .3.31 where x denotesthephotondensity, D isthediusioncoecientand a representsabsorptioncoecients.Theterm S x isthesourcedistributionofgeneexpression.Theappropriate

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15 boundaryconditionisgivenby D 5 n x )]TJ/F25 10.9091 Tf 10.909 0 Td [( x =0 ; x 2 @ ; .3.32 where n istheunitouternormaltotheboundaryand isacoecientrelatedtotheinternal reectionattheboundary.Inthisproblem,theparameters D , a and areknown,andthe bioluminescencetomographyBLTiscomposedofrecoveringthesourcedistributionfromthe collecteddataattheboundary.Mathematically,BLTisthesourceinversionproblemthatrecovers S x fromopticalmeasurementonthedomainboundary @ .Inadditiontobeinghighly ill-posed,thesolutiontotheaboveBLTinverseproblemisingeneralnotunique[41].

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16 CHAPTERIII INVERSEHEATCONDUCTIONPROBLEMS Looselyspeaking,inverseheatconductionmethodscanbeusedtodetermineheatuxandtemperatureonanin-accessiblesurfaceofawallbymeasuringtemperatureonanaccessibleboundary.Ingeneral,manyofthemathematicalmodelsencounteredinheattransferproblemsare eitherellipticorparabolic.Forafulldescriptionofaheattransferproblem,oneneedsaspecicsetofboundary/initialconditions,andafullknowledgeofthethermo-physicalproperties ofthedomain.Aninverseproblemariseswhensomeoftherequiredboundary/initialconditions,ormaterialpropertiesareunknown.Suchproblemsappearnaturallyinvariouselds, includingthereenteringheatshield[42],remotesensingofclimates[43],oilexploration[44], nondestructiveevaluationofmaterial[45]andthedeterminationoftheearthsinteriorstructure [46,47].TheinverseheatconductionproblemsIHCPishighlyill-posed[48]inthesensethat anysmallnoiseinthemeasurementscanresultinadrasticchangetothesolution.Inthepast decades,manynumericalmethodshavebeendevelopedforsolvingtheIHCP[49,48,50,51]. Considerageneralformofgoverningequationwithpossibleboundaryandinitialconditionsas c @u x ;t @t = 5 5 u x ;t + _ Q v ; x 2 ;t 2 ;t f ] ; .0.1 u x ;t = u b x ;t ; x 2 @ ;t 2 ;t f ] ; .0.2 u x ; 0= u 0 x ; x 2 ; .0.3 where u x ;t isthetemperature[ K ], isthemassdensity[ kg=m 3 ], c istheconstant-volume specicheat[ J=kgK ], isthethermalconductivity[ W=mK ], _ Q v istherateofheatgenerationperunitvolume[ W=m 3 ],usuallydenotesassourceterm, u b isthegivenboundarycondition, u 0 istheinitialtemperature; t f isthenialtime.Inordertorecovertheunknownpart,

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17 someadditionalconditionscanbegivenas )]TJ/F25 10.9091 Tf 8.485 0 Td [( @u x ;t @n = q b x ;t ; x 2 @ ;t 2 ;t f ] ; .0.4 )]TJ/F25 10.9091 Tf 8.485 0 Td [( @u x ;t @n = h c [ u x ;t )]TJ/F25 10.9091 Tf 10.909 0 Td [(u e x ;t ] ; x 2 @ ;t 2 ;t f ] ; .0.5 where q b and u e aregivenfunctionsontheboundary, h c istheheattransfercoecient[ W=m 2 K ]. Equations.0.2,.0.4and.0.5aredierenttypesofboundaryconditionwhichcanbe regardedasDirichlet,NeumannandRobinboundarycondition.Equation.0.3denotesthe initialcondition. 3.1ClassicationoftheInverseProblems Broadlyspeaking,theinverseheattransferproblemscanbedividedwiththefollowingcategories evaluationoftheboundaryconditions, evaluationoftheinitialcondition, evaluationofthethermophysicalproperties, inversesourceproblem, evaluationofthedomain. Theinverseheatconductionproblemcanbeeitherlinearornonlinear.Moredetailswillbepresentedinthefollowingsection. 3.1.1EvaluationoftheBoundaryConditionsCauchyProblem Inthisproblem,thegoverningequationisknown,however,partoftheboundaryconditionis unknown.Inordertosolvetheinverseproblemadditionalmeasurements,suchasthetemperatureorheatux,areneeded.Theycanbemeasuredontheaccessiblepartoftheboundaryor

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18 inadiscretesetofpointsinsidethedomain.Inatransientheatconductionproblemthisparticularsituationproblemisreferredtoas sidewayheatconductionproblem [52,53,54].Fora steady-stateheatconduction,theadditionalconditionisintheformofuxattheboundary whichistheclassical Cauchyproblem [55,56]. 3.1.2EvaluationoftheInitialCondition Inthiscaseinitialconditionsofareunknown,i.e.thefunction u 0 isunknowninEq..0.3. Theinitialconditionneedstoberecoveredbasedonadditionalinformation.Insomeapplications,thetemperatureledcanbemeasuredinthewholedomainforxed t> 0andprovided asdata[57,58]. 3.1.3EvaluationofThermophysicalProperties Inthisproblem,thethermalconductivity inEq..0.1istheunknowncoecient.Theextra informationconcerningtemperatureorheatuxinthedomain[59,60],orontheboundary havetobemeasured. 3.1.4InverseSourceProblem Intheproblemofinversesourceterm,theterm _ Q v inEq..0.1isunknown.Theintensityof source,itslocationorbothcanbeunknown.Inmanycases,additionalinformation,suchasthe temperaturehistory,aregivenatchosenpointslocationinsidethedomain,oronpartsofthe boundary[61]. 3.1.5EvaluationoftheDomain Inshapeoptimizationproblem,thelocationofthedomainisunknown.Torecovertheunknownoroptimallocationoftheboundary,additionalinformationneedstobeprovidedon theknownpartoftheboundary,whichcanbetreatedasEqs..0.4and.0.5.Inparticular,theboundaryconditionsareover-speciedontheknownpart.Fortheunknownpartofthe

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19 boundary,itshouldbedeterminedbyimposingaspecicboundaryconditiononit. 3.2MethodsforSolvingInverseHeatConductionProblems Inrecentyears,manyanalyticalandnumericalmethodshavebeendevelopedforsolvingtheinverseheatconductionproblems.Explicitanalyticalsolutionsarelimitedtosimplegeometries, andarenotsuitableformostpracticalproblems.RecentresultsonsolvinginverseproblemincludesamethodbasedonQuazi-Reversibility[62],Neuralnetwork[63],amethodbasedonconjugategradientminimization[64,65,66],minimizationofacostfunctional[67,68],multiple forwardproblem[69],statisticalapproaches[70],regularizedGaussNewtonmethod[71,72],the D-barmethod[73],andthewaveletmulti-scalemethod[74].Additionalmethodshavealsobeen reviewedin[75].Inthenextsection,severalexistingmethodsarepresented. 3.2.1MethodofFundamentalSolution Inthismethod,thefundamentalsolutionofthecorrespondingheatequationisusedtocreatea basisforapproximatingthesolutionoftheproblem.Considerasystemgivenby @u x ;t @t = 4 u x ;t ; x 2 ; and ;t 2 ;t f ] ; .2.1 withboundaryconditions u x ;t = g 1 x ;t ; @u x ;t @n = g 2 x ;t x 2 @ ; and ;t 2 ;t f ] ; .2.2 andinitialcondition u x ; 0= h x ; x 2 : .2.3 ThefundamentalsolutionofEq..2.1isgivenby F x ;t = 1 t n= 2 exp )]TJ 9.681 7.38 Td [(j x j 2 4 t ; x 2 ;t> 0 ; .2.4 where n isthedimensionofdomain.Assumingthat t f
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20 andinitialconditioncanbyexpressedbythefollowinglinearcombination: ~ u x ;t = m X j =1 ~ j x )]TJ/F19 10.9091 Tf 10.909 0 Td [(x j ;t )]TJ/F25 10.9091 Tf 10.909 0 Td [(t j ; .2.5 where x ;t = F x ;t + t , m isthetotalmeasurementpoints, F isgivenbyEq..2.4and j areunknowncoecientstobedetermined.Forthechoiceofbasisfunctions ,theapproximatedsolution ~ T mustsatisfytheoriginalsystem.Usingtheboundaryandinitialcondition whicharegiveninEq..2.2andEq..2.3,wecanobtainasimplelinearsystemfortheunknowncoecients ~ j givenby A ~ = ~ b ; .2.6 where A = 2 6 4 x i )]TJ/F19 10.9091 Tf 10.909 0 Td [(x j ;t i )]TJ/F25 10.9091 Tf 10.909 0 Td [(t j @ @n x k )]TJ/F19 10.9091 Tf 10.909 0 Td [(x j ;t k )]TJ/F25 10.9091 Tf 10.91 0 Td [(t j 3 7 5 and ~ b = 2 6 6 6 6 6 6 6 6 4 ~ M i h x i ;t i g 1 x i ;t i g 2 x k ;t k 3 7 7 7 7 7 7 7 7 5 : .2.7 Intheaboveequations, ~ M i isthemeasurementwhichispollutedwithnoise. TechniquesforRegularizationofIll-posedProblems Aregularizedsolutiontoanill-posedproblemcanbeobtainedbyreplacingtheoriginalillposedproblemwithadierentproblemwhereadditionalconditionsonthesolutionareimposed [76].Overtheyears,investigatorshavedevelopedvariousmethodstoregularizeanill-posedinverseproblem.Inthissection,anumberofsuchmethodsarediscussed. 3.2.2TruncatedSVDRegularization Considerthelinearsystemofequationsgivenby: Ax = y ; .2.8 where, A 2 R m n m n isamatrix, x 2 R n and y 2 R m .

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21 Ifthematrix A isill-conditioned,thesolutionforthesystem.2.8isinaccurateandunstable.ThetruncatedSVDregularizationmethodiswidelyusedtosolvestablyandaccurately ill-conditionedmatrixequations.SingularvaluedecompositionSVDofamatrix A isgivenby: A = UV T ; .2.9 where U =[ u 1 ; u 2 ;:::; u m ]and V =[ v 1 ; v 2 ;:::; v n ]areleftandrightmatriceswhosecolumn areorthonormal.Thematrix =diag 1 ; 2 ;:::; n isadiagonalmatrixwithnon-negative diagonalelementsinnon-increasingorder.Thediagonalelementsof arethesingularvalues of A ,whicharethesquarerootofeigenvaluesof A T A .ThesolutiontoEq..2.8canbe obtainedasalinearcombinationoftherightandleftsingularvectors: x = k X i =1 u i y i v i ; .2.10 where k istherankofthematrix A .Since A isanill-conditionedmatrix,therearemanysmall singularvalueswhichareclosetozero.Anapproximatesolutioncanbeobtainedaccordingto: [77,61,78]: x approx = N X i =1 u i y i v i ; .2.11 where, N 2 N isthetruncationparameterdeterminedwhensmallsingularvaluesareleft. Thesolutionforanill-posedproblembythetruncatedSVDregularizationmethoddepends thevalueof N .Ifthetruncationparameter N isequalto k ,noregularizationisapplied.If thetruncatedparameter N isequaltozero,thisleadstoneglectofallthesingularvalues,and thereisnoapproximationsolutionforthesystem.2.8.ThetruncatedSVDregularization withtoosmall N valueresultsamoreoscillatorysolution[79,80,81].Manyinvestigatorshave studiedseveralapproachestochoosetheappropriatetruncatedSVDregularizationincluding thegeneralizedcross-validationGCV[81,82],discrepancyprinciple[61,83],andL-curvecriterion[81,84,85].

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22 3.2.3TikhonovRegularization Tikhonovregularizationiswidelyusedtostabilizevariousill-posedproblems.Inthismethod, thesolutiontothelinearsystemgiveninEq..2.8isobtainedbyminimizingthecostfunctionalgivenby: min 8 x 2 X J = k Ax )]TJ/F19 10.9091 Tf 10.909 0 Td [(y k 2 + k x k 2 ; .2.12 whereaboundonthesolutionisintroducedby > 0.Astablesolutionfor x 2 X canthenbe obtainedbyleast-squareminimization.Incaseofadierentialoperator,onecanconsiderthe minimizationproblemgivenby: min 8 x 2 X J = k Ax )]TJ/F19 10.9091 Tf 10.909 0 Td [(y k 2 + k x k 2 ; .2.13 where oftenrepresentstherstorderderivative. 3.2.4OptimizationMethod Theoptimizationmethodsarewidelyusedtosolvemanyinverseproblems.Typically,oneseeks torecoveramodel x basedonobservationsofaeld u ,where u isrelatedto x byaforward problem.Theforwardprobleminpracticecanbewrittenas: A x u = y ; .2.14 where, A 2 R n n isanonsingularmatrix,and y isaright-handsidethatconsistingofsource termsandboundaryconditionvalues.Thedataontheboundaryisgivenby: Bu = g ; .2.15 wherethevector g containsthedata,andthematrix B representsthemeasurementoperator thatselectseldsatdatameasurementlocations.Asolutionto x canbeobtainedbyminimizingtheerrordierences: min u ; x J = 1 2 k Bu )]TJ/F19 10.9091 Tf 10.909 0 Td [(g k 2 ; suchthat A x u = y : .2.16

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23 Sincetheinverseproblemgenerallyisill-posed,theregularizationisneededtostabilizethesolution.ThentheconstrainedoptimizationproblemwithTikhonovregularizationisgivenby: min u ; x J = 1 2 k Bu )]TJ/F19 10.9091 Tf 10.909 0 Td [(g k 2 + 1 2 k x k 2 : .2.17 Sincethesystem.2.14islinearin u ,thenonecanarriveat: u = A x )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 y : .2.18 Thisleadstotheunconstrainedoptimizationproblem: min x J = 1 2 k B A x )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 y )]TJ/F19 10.9091 Tf 10.909 0 Td [(g k 2 + 1 2 k x k 2 : .2.19 ThisunconstrainedoptimizationproblemcanbesolvedbyGauss-Newtonmethod[86,87].The optimizationmethodsworkwellformosthyperbolicsystems,butithaspoorperformancefor ellipticandparabolicsystems.Ingeneral,theremaybemanylocalminimumsand,theconvergencecanalsobeverypoor. 3.2.5D-barMethod Nachman'suniquenessproofforthe2-DinverseconductivityproblemoutlinesadirectprocedureforreconstructingtheconductivityprolefromknowledgeoftheDirichlet-to-Neumann map[80,88].ConsiderCalderon'sproblemwhichisgivenby 5 x 5 u =0 ; x 2 R 2 ; .2.20 where, x istheconductivity,andtheDirichletboundaryconditionontheboundaryisgiven by: u x = g x ; x 2 @ : .2.21 Thefunction g x isagivenvoltage.Thegoalistorecoverthe x bymeasuringthecurrent density f ontheboundary: x u x @n = f x ; x 2 @ : .2.22

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24 Usingthefunction q = 4 p p and~ u = p u ,onecantransformtheconductivityequationtothe Schrodingerequationas )-222(4 + q z ~ u z =0 ;z 2 R 2 : .2.23 Introduceacomplexparameter k andlookforsolution z;k oftheSchrodingerequation )-222(4 + q ;k =0 ; .2.24 satisfyingtheasymptoticcondition e )]TJ/F26 7.9701 Tf 6.587 0 Td [(ikz z;k )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 W 1 ;p R 2 ; .2.25 forany2
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25 UsetheDNmap toconstructthecomplexgeometricalopticsCGOsolution z;k .Forall k 2 C nf 0 g solvetheboundaryintegralequation ;k j @ = e ikz j @ )]TJ/F25 10.9091 Tf 10.909 0 Td [(S k )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 ;k ; .2.30 where S k )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 ~ u z;k = Z @ G k z )]TJ/F25 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 ;k ds : .2.31 ;k j @ )167(! t k : Evaluatethescatteringtransform t k byusingAlessandrini'sidentity[89]as t k = Z q z e i k z z;k dA = Z @ e i k z )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 ;k ds: .2.32 t k )167(! m z;k : Foreach z 2 solvethe @ equation @ @ k m z;k = t k 4 k e )]TJ/F26 7.9701 Tf 6.587 0 Td [(k z m z;k ; .2.33 where, e )]TJ/F26 7.9701 Tf 6.587 0 Td [(k z =exp )]TJ/F25 10.9091 Tf 8.485 0 Td [(i kz + k z ,and m z; )]TJ/F15 10.9091 Tf 9.484 0 Td [(1 2 L p 0 L 1 .TheEq..2.33is generatedbydierentiatingtheLippman-Schwingerequation m =1 )]TJ/F25 10.9091 Tf 11.118 0 Td [(g k qm withrespectto k .NachmanveriedthatEq.2.33hasauniquesolutionfor m z; )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 L p 0 L 1 C forsome p 0 > 2[90]. m z;k )167(! . Examine k ! 0inEq..2.29andusing q = 4 p p implies )-222(4 m z; 0= 4 p p m z; 0 : .2.34 Thus m and p solvethesameSchrodingerequation 4 m ; 0= f m ; 0and 4 p = f p ; .2.35 with f z = 4 m z; 0 m z; 0 = 4 p z p z .Thedecayconditionin m )]TJ/F15 10.9091 Tf 10.978 0 Td [(1 2 W 1 ;p 0 and 1 on R 2 n nowimplies m ; 0= p .Recovertheconductivityby p z =lim k ! 0 m z;k = m z; 0 : .2.36

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26 3.2.6MethodofMultipleForward Consideraclosedboundeddomain 2 R 2 anda2-DHelmholtzequationgivenby[91] 4 u + k 2 g x u =0 ; x 2 R 2 ; .2.37 withDirichletboundaryconditionsandadditionalboundarymeasurementintheformofthe Neumannconditionare u x = f x ; x 2 @ ; 5 n u x = h x ; x 2 ; .2.38 where f x isthegivenfunction.Thevariable u x denotestheelectriceld,theparameter k denotesthefrequencyoftheincidentwaveandthefunction g x isaphysicalparameter.The goalistorecoverthefunction g x basedonboundarymeasurement.Thealgorithmisiterative innatureanditconsistsofthefollowingsteps. Assumeaninitialguessoftheunknownfunction g x as^ g x .Theinitialguess togetherwiththegivenDirichletconditionsgenerateabackgroundeld,^ u x , whichsatisesthesystemgivenby 4 ^ u + k 2 ^ g x ^ u =0 ; x 2 R 2 ; ^ u x = f x ; x 2 @ : .2.39 TheNeumannboundaryconditionscanbeestimatedontheboundary. 5 n ^ u x = ^ h x ; x 2 : .2.40 Subtracttheactualandbackgroundledtoobtainanequationfortheerror, accordingto e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x ,whichisgivenby 4 e + k 2 g x u )]TJ/F25 10.9091 Tf 10.91 0 Td [(k 2 ^ g x ^ u =0 ; x 2 R 2 ; e x =0 ; x 2 @ ; 5 n e x = h x )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h x ; x 2 : .2.41

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27 Itisalsopossibletowrite g x =^ g x + q x wherecorrectionterm q x is unknown.Itfollowsthat 4 e + k 2 ^ g x e + k 2 q x u =0 ; x 2 R 2 ; e x =0 ; x 2 @ ; 5 n e x = h x )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h x ; x 2 : .2.42 Thenextstepistolinearizearoundthebackgroundeldandarriveat 4 e + k 2 ^ g x e + k 2 q x ^ u =0 ; x 2 R 2 ; e x =0 ; x 2 @ ; 5 n e x = h x )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h x ; x 2 ; .2.43 whichhasbeenshowninFig.3.1.Now,thereisonlyoneunknown h x inthe FIGURE3.1:Ill-posedproblem systemneedtobecalculated.Howevertheinverseproblem.2.43isoverspeciedatboundary.Inthemultipleforwardmethod,itseparatestheill-posed problemintotwowell-posedproblemstosolvetheunknownfunction.

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28 FIGURE3.2:Well-posedproblemI FIGURE3.3:Well-posedproblemII Twowell-posedproblemshavethesameequationbutdierentboundaryconditionandmeasurements,whichareshownintheFig.3.2andFig.3.3.Combiningtwowell-posedproblemandusingleastsquareminimizationmethod,the unknowncorrectionterm q x canbeestimated.

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29 Thiscomputationalmethodisquiteversatileandcanbeappliedtovarioussystem.Ournew computationalmethodfortheinverseheatconductionproblemisbasedonit. 3.2.7ActiveSubspaceMethod Activesubspacemethodexaminesvariousdirectionsofgradientsandobtainsthedirectionof thestrongestvariability.Itthenexploitsthesedirectionstoobtaintheactivesubspacethedirectionwithmostvariability[92,93].Considerafunction f with m continuousinputs f = f x ; x 2X R m : .2.44 Let X beequippedwithaboundedprobabilitydensityfunction : R m ! R + ,where x > 0 ; x 2X and x =0 ; x = 2X : .2.45 Denethe m m matrix C by C = E [ 5 x f 5 x f T ] ; .2.46 where 5 x f x = h @f @x 1 ;:::; @f @x m i T and E [ ]istheexpectation.Thematrix C isacovariance-like matrixofthegradient,whichisusedtodeterminethedirectionsofvariabilitybyfactorization. Sincethematrix C issymmetricandpositivesemi-denite,itcanbedecomposedinto C = WW T ; .2.47 where W denoteseigenvectorsand =diag 1 ; 2 ;:::; m ,for 1 2 ::: m .With decreasingeigenvalues,theactivesubspacemethodseparatescomponentsoftherotatedcoordinatesystemintoaset y ,correspondingtogreateraveragevariationandaset z ,corresponding tosmallervariation.Then,thematrixofeigenvectors W canbepartitionedas = 2 6 4 1 2 3 7 5 ; and W =[ W 1 ; W 2 ].2.48 where 1 =diag 1 ; 2 ;:::; n with n
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30 Twonewsetsofcoordinates y and z canbedenedas y = W T 1 x ; y 2 R n ; z = W T 2 x ; y 2 R n )]TJ/F26 7.9701 Tf 6.587 0 Td [(m : .2.49 Thechoiceofeigenvectorsthatconstructthesubspaceassociatewiththepracticalconsiderationandthescaleofeigenvalues.Therealadvantageofthismethodisthatonecanconstruct responsesurfaceswithreducednumberofvariables x 2 R n insteadof f 'snaturalvariables x 2 R m ,whichlowerthedimensionofproblemfrom m to n . 3.3GeneralIdenticationAlgorithm 3.3.1IdenticationAlgorithm Inthisdissertation,ageneralidenticationalgorithmhasbeenusedtosolvevedierentillposedinverseproblems.Thepresentalgorithmisiterativeinnaturewhichcorrectstheinitial guesswithacorrectiontermineachiteration.Thealgorithmcanbedescribedasfollowing steps. Constructtheinitialguessfortheunknownfunctionand,usingthegivenboundarycondition,obtainabackgroundeld. Subtractthebackgroundeldfromactualeld,andobtaintheerroreld. Applythepropersolutionspacemethodtoobtainthecorrectiontermtothe assumedvaluefortheunknownfunction.Repeattheabovestepsuntilasignicantreductioninerrorisachieved. 3.3.2ProperSolutionSpaceMethod Thethirdstepofthealgorithminvolvestheidenticationofthecorrectiontotheassumed valueoftheunknownfunction.Wemainlyadoptthepropersolutionspacemethodinthisdissertation,whichisatypeofsubspacemethodforchoosingparticularsubspacetoreconstruct

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31 thesolutionspace.Ingeneral,thepropersolutionspacecanbethoughtofasasubspacemethod wherethesubspaceisspecicallygeneratedfortheproblemathand,i.e.,erroreld.Thismethod consistsoffollowingsteps: Choosealinearlyindependentsetoffunctions c ` x ,for ` =1 ; 2 ;:::;N over thedomain,andassumethatthecorrectiontermcanbeexpressedasalinear combinationofthislinearlyindependentsetoffunctions c ` 's.The c ` x should satisfyapartofgivenboundaryconditions. Generateasetoffunctions ` ,for ` =1 ; 2 ;:::;N ,thatspantheerroreld e x andsatisfypartofthegivenboundaryconditionstheDirichletpart. Expandtheactualerroreld e x inthespanofthespacegeneratedby ` , ` = 1 ; 2 ;:::;N . Usetheremainingboundaryconditiontoobtainthecorrection. Inthisdissertation,weassumethattheunknowncoecientcanbemeasuredontheboundary. Thisisnotgeneralnecessary.Itisonlydoneforconvenience.Therefore,thecorrectionterm fortheassumedvalueneedstosatisfythezeroDirichletboundarycondition.Fornumerical considerationsaglobalapproximationspacesuchassinefunctionsaremoreappropriate. 3.4GeneralContributionofThisDissertation Thecontributionofthisdissertationisthreefold.First,weapplythepropersolutionspace methodforsolvinginverseproblemsforparabolicandellipticsystems.Second,weuseouralgorithmtoreconstructsimultaneouslytwounknownfunctionsinvolvedinthephotondiusion equation.Finally,wetrytodotheidenticationoflocationofthedomainforaellipticsystem, whichisashapeoptimizationproblem.

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32 CHAPTERIV EVALUATIONOFTHEINITIALCONDITION Inthischapterweusethepropersolutionspacemethodfortheinverseevaluationoftheinitialconditionforaparabolicsystem.Itiswellknownthatthisproblemishighlyill-posed[94]. Insection4.1,wepresenttheiterativealgorithm.Itassumesaninitialvaluefortheunknown initialconditionandobtainscorrectionstotheassumedvalue.Thenewfeatureofthepresent algorithmistheupdatingstagewhichispresentedinsection4.2.Insection4.3,weuseseveral numericalexamplestostudytheapplicabilityofthemethod. 4.1ProblemStatementandIdenticationAlgorithm Let= f t;x ;x 2 [0 ; 1] ;t 2 [0 ;t f g andconsidera1-Dheatconductionequationgivenby u t = u xx ; t;x 2 ; u t; 0= g 0 t ;u t; 1= g 1 t ; .1.1 where u t;x isthetemperature, t f isthenialtime,thermalconductivityanddiusivityare assumedtobeequaltooneforsimplicity,andDirichletboundaryconditionsareimposed.The unknownconditionistheinitialcondition u ;x .Thisill-posedproblemissupplementedby additionalconditions,namely,theNeumannboundaryconditionsontheboundaries,i.e. u x t; 0= f 0 t ;u x t; 1= f 1 t : .1.2 Theinverseproblemofinteresthereistorecovertheunknowninitialcondition u ;x = x basedontheadditionalgivenconditionsattheboundariesEq..1.2.Thepresentalgorithm isiterativeinnatureandconsistsofthreefollowingparts.

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33 Assumeafunctionfortheunknowninitialcondition,^ u ;x = ^ x ,usingthe givenDirichletboundaryconditions g 0 t and g 1 t ,obtainabackgroundeld satisfyingthesystem ^ u t =^ u xx ; t;x 2 ; ^ u t; 0= g 0 t ; ^ u t; 1= g 1 t : .1.3 SubtractthebackgroundeldfromEq..1.1,andobtaintheerroreld, e t;x = u t;x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u t;x ,givenby e t = e xx ; t;x 2 ; e t; 0= e t; 1=0 : .1.4 Theerroreldisrequiredtosatisfyadditionalconditionsgivenby e x t; 0= f 0 t )]TJ/F15 10.9091 Tf 11.608 0 Td [(^ u x t; 0= ^ f 0 t ; e x t; 1= f 1 t )]TJ/F15 10.9091 Tf 11.608 0 Td [(^ u x t; 1= ^ f 1 t : .1.5 Torecovertheinitialcondition,thecorrectionterm e ;x needtobecomputed. UsetheadditionalboundaryconditionsinEq..1.5andobtaintheinitialconditionfortheerroreld.Updatetheassumedvaluefortheinitialconditionand gotostepone. Theabovethreestepsiterationsisabasicalgorithmthatweusedforasimilarproblems[95]. Thenovelfeatureofthepresentmethodisthethirdstepintheabovealgorithmwhichisthe evaluationoftheunknowninitialconditionusinganewmethod.Wewilldescribedthethird stepindetailinnextsection. 4.2ProperSolutionSpace Thethirdstepofthealgorithminvolvestheidenticationofthecorrectionoftheassumedvalue oftheunknowninitialvalue.Thenewmethodwasrstdevelopedfortheinverseevaluation ofaboundaryconditioninsteadyheatconductionproblems[96].Theactualvalueoftheunknownfunctionisrelatedtothebackgroundeldaccordingto u ;x =^ u ;x + e ;x .In

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34 ordertorecover e ;x ,weconsideralinearlyindependentsetoffunctions c ` x , ` =1 ; 2 ;:::;N over x 2 [0 ; 1]andassumethattheunknownfunctions e ;x canbeexpressedasalinearcombinationofthesefunctions,i.e., e ;x 2f c 1 ;c 2 ;:::;c N g .Then,weneedtogenerateasetof functions ` t;x thatsatisfytheerroreldequationwiththeknownzeroDirichletboundary condition,i.e., ` t = ` xx ; t;x ; 2 ` t; 0= ` t; 1=0 ; ;x = c ` x : .2.1 Therefore,everyfunction ` t;x satisesthezeroDirichletboundaryconditions.Itisthen possibletoexpandtheactualandunknownerroreld e t;x inthespanofthespacegeneratedby ` t;x , ` =1 ; 2 ;::;N ,accordingto e t;x = N X ` =1 ` ` t;x ; .2.2 wherethefunctions ` t;x canbeobtainedfromEq..2.1,buttheconstants ` areunknown. Wenextarguethattheerroreld e t;x mustsatisfythegradientconditionthatisfurnished bythemeasurementsandaregiveninEq..1.5.Thegradientconditionscanbeexpressedby theoperators B 0 and B 1 .Theerroreldisrequiredtosatisfytheconditionsgivenby B 0 e t; 0= B 0 N X ` =1 ` ` t; 0= N X ` =1 ` B 0 ` t; 0= ^ f 0 t .2.3 B 1 e t; 1= B 1 N X ` =1 ` ` t; 1= N X ` =1 ` B 1 ` t; 1= ^ f 1 t .2.4 Theaboveequationcanbeusedtoobtaintheunknowncoecients ` for ` =1 ; 2 ;:::;N .Once theunknowncoecientsareobtainedtheunknowninitialconditioncanbeobtainedaccording to e ;x = N X ` =1 ` ` ;x = N X ` =1 ` c ` x : .2.5 ThelastequalityinEq..2.5holdsbecauseofthechoseninitialconditiongiveninEq..2.1 whichcompletestheupdatingstage.Aftersolvingfor e ;x ,theassumedvalueoftheun-

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35 knowninitialconditioncanbeupdatedaccordingto u ;x =^ u ;x + e ;x .andthethree stepsinthealgorithmcanberepeated. 4.3NumericalExamples Inthissection,threenumericalexampleshavebeenusedtoinvestigatetheapplicabilityofthe method.Fortheinitialconditionproblemweconsider1-Ddimension,however,wewillestimate dierenttypesoftheunknownfunction.Intherstexample,asimple1-Dheatconditionequationwithuniformthermalconductivityandonehumpunknowninitialconditionareconsidered. Thentheunknowninitialconditionwillbemodiedtoatwohumpfunctioninsecondexample. Complexunknownfunctionishardertorecoverinusual.Inthethirdexample,weconsidera parabolicsystemwithnon-uniformthermalconductivity. Example1. Considertheevaluationoftheinitialconditionfora1-Dparabolicequationwith uniformthermalconductivitygiveninEq..1.1andassumethattheactualinitialcondition isgivenby u ;x =exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 0 : 03 : .3.1 withtheboundaryconditions u t; 0= u t; 1=0forall t .Torecovertheinitialcondition,the Neumannconditionat x =0and x =1areprovidedfor t 2 [0 ; 0 : 25.Theabovethreestep algorithmcanbestartedbyassumingaguessfortheinitialcondition,collectingdataatthe boundaries,thenarrivingattheerroreldinthestep3.Inordertoobtainthecorrectionof theassumedvalueinthethirdstepofthealgorithm,anappropriatesetoflinearlyindependent functionscanbeconsideredaccordingto c ` x =sin `x ;` =1 ; 2 ;:::;N; .3.2 where c ` = c ` =0for ` =1 ; 2 ;:::;N .Thesolutionsetcanbeobtainedbyusingtheindividualfunctions c ` x astheinitialconditionfortheerroreldinEq..2.1,for ` =1 ; 2 ;:::;N . Theexactsolutionforthesefunctionsaregivenby ` t;x = e )]TJ/F23 7.9701 Tf 6.586 0 Td [( ` 2 t sin `x ;` =1 ; 2 ;:::;N; .3.3

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36 and,itiseasytonotethatthesefunctionsarelinearlyindependent.Wecandenotethenodal valuesof ` t;x bythevectors " ` for ` =1 ; 2 ;:::;N .Wethenarguethattheerroreldmust satisfytheadditionalboundaryconditionsthatisprovidedbythemeasurementsgiveninEqs. .2.3and.2.4,accordingto B 0 e t; 0= B 0 N X ` =1 ` " ` t; 0= N X ` =1 ` B 0 " ` t; 0= b 0 ; .3.4 B 1 e t; 1= B 1 N X ` =1 ` " ` t; 1= N X ` =1 ` B 1 " ` t; 1= b 1 ; .3.5 wherevectors b 0 and b 1 arethemeasurementdataatboundary x =0and x =1. Groupingthesetwoequationsleadsto 2 6 4 B 0 " ` B 1 " ` 3 7 5 = 2 6 4 b 0 b 1 3 7 5 ; .3.6 where isthe N dimensionalvectorcontainingthevaluesoftheunknownparameters ` for ` =1 ; 2 ;::;N .Thevectors b 0 and b 1 containthevaluesof ^ f 0 t and ^ f 1 t atdierenttimes. Theabovecoecientmatrixisnon-squareandrank-decient.Itispossibletostabilizetheinversionbyimposingsomeformofregularizationaccording 2 6 6 6 6 6 4 B 0 " ` B 1 " ` 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 b 0 b 1 0 3 7 7 7 7 7 5 ; .3.7 where,thematrix representstherst-orderoperatorgivenby = dc ` x dx x = x k = N X ` =1 ` ` cos `x k ; .3.8 evaluatedatdiscretelocations x k alongthedomain.Theconstant > 0,andtheabovelinear systemcanbesolvedfortheunknowncoecients ` for ` =1 ; 2 ;:::;N . Dividingthedomain[0:1]into n e =100equallyintervalsleadsto n = n e +1=101nodes inthexdirection.Dividingthetimeinterval[0:0 : 25]into n t =100leadsto101timeintervals.Choosing N =20sinefunctionsinEq..3.2leadstothepropersolutionspacehaving

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37 FIGURE4.1:TherecoveredinitialconditionforExample1.Thegurecomparesthenal valuetotheactualfunction N =20vectors,witheachvectorhavingthedimension .Thecollecteddataatthe boundariesarecontaminatedwithazeromeanrandomlygeneratednoisewithnoise-to-signal ratioof4%.Thedataisthenlteredusingasimplepolynomiallter.Thealgorithmisiterative,andwearealsousingaformof under-relaxation inthesensethattheupdatingisachieved ^ u ;x =^ u ;x + e ;x where =0 : 1.Figure4.1comparestherecoveredinitialcondition withtheactualinitialconditionafter8830iterationswith =0 : 04andgure4.2showsthe reductionintheerror.Theerrorisdeneas Error= R t f t =0 h ^ f 0 t 2 + ^ f 1 t 2 i dt R t f t =0 h ^ f 0 t 2 + ^ f 1 t 2 i dt j rstiteration .3.9 where, ^ f 0 t and ^ f 1 t arethedierencebetweenthemeasurementdataandthecalculatedgradientsattheboundariesgivenbyEq.4.1.5. Example2. Wenowconsidertheevaluationof2humpoftheinitialcondition,andthe u ;x givenby u ;x =exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.91 0 Td [(0 : 26 4 0 : 007 +exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 74 4 0 : 007 : .3.10

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38 FIGURE4.2:ThereductioninerrorforExample1asafunctionofthenumberofiterations. Forthisexample,weset =25.Figure4.3comparestheactualinitialconditionwiththe recoveredvalueafter90,000iterations.Italsoshowsafewintermediatevalue.Thepresent methodreliesonTikhonovregularizationtostabilizetheinversioninEq..3.7.Theregularizationpenalizeslargevaluesoftherstderivative.Forinitialconditionswithlargegradients, alargervalueof isneededforstableandaccurateinverseevaluationoftheinitialcondition. Thefunctionisnotknownaprior,ingeneral,alargevalueof canbeusedtostarttheevaluation.ThedenitionoferrorissamesasbeforewhichshowninFig.4.4. Example3. Wenextconsideraparabolicsystemwithnonuniformthermalconductivity. Considertheevaluationoftheinitialconditionforasystemgivenby u t = k x u x x ; t;x 2 ;u t; 0= u t; 1=0 ; .3.11 with k x =0 : 5+0 : 5exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 0 : 0007 ; .3.12

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39 FIGURE4.3:TherecoveredinitialconditionforExample2.Thegurepresentstherecovered functionatafewintermediateiterationsandcomparesthenalvaluetotheactualfunction. FIGURE4.4:ThereductioninerrorforExample2asafunctionofthenumberofiterations. andthe u ;x isgivenby u ;x =exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 26 2 0 : 007 +1 : 5exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 74 2 0 : 007 : .3.13

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40 Wexedthevalueof =25.Inthegure4.5,wecomparestheactualinitialconditionwith FIGURE4.5:TherecoveredinitialconditionforExample3.Thegurepresentstherecovered functionatafewintermediateiterationsandcomparesthenalvaluetotheactualfunction. FIGURE4.6:ThereductioninerrorforExample3asafunctionofthenumberofiterations. therecoveredvalueafter90,000iterations.Italsoshowsafewintermediatevalue.Figure4.6

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41 presentsthereductionintheerrorasafunctionofthenumberofiterations. 4.4Remark Inthischapter,weusedacomputationalmethodbasedonpropersolutionspacetosolvethe inverseevaluationoftheinitialconditionforaparabolicequation.Threeexampleshavebeen implementedtostudytheapplicabilityofthismethodwhichdemonstrategoodrobustnessto noise.Nolinearizationisrequiredinthealgorithminthischapter.Thismethodalsocanbe appliedtoheatconductionproblemswithvariablematerialproperties.Evaluationofabsorption coecientforaparabolicsystembasedonpropersolutionspacemethodwillbeconsideredin thenextchapter.

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42 CHAPTERV EVALUATIONOFABSORPTIONCOEFFICIENT Inthischapter,weusepropersolutionspacemethodtorecoverabsorptioncoecientfora parabolicsystem.Weconsidertheabsorptioncoecientinoneandtwodimensionsofabounded domain.Inbothcasesweassumethattheunknownfunctioncanbemeasuredattheboundary, andforsimplicity,weassumethatitisequaltoone.InSection5.1,wepresenttheiterativealgorithm.Itassumesaninitialvaluefortheunknownfunctionandobtainscorrectionstotheassumedvalue.Thenewfeatureofthepresentalgorithmistheupdatingstagewhichispresented inSection5.2.InSection5.3,weuseanumberofnumericalexamplestostudytheapplicability ofthemethod. 5.1ProblemStatementandtheIdenticationAlgorithm Let= f t;x ;x 2 [0 ; 1] ;t 2 [0 ;t f g andconsidera1-Dparabolicequationwithabsorption coecientgivenby u t = u xx + a x u; t;x 2 ; u t; 0= g 0 t ;u t; 1= g 1 t ; u ;x = u 0 x ; .1.1 where u t;x isthetemperature, t f isthenaltime, a x istheabsorptioncoecientandDirichletboundaryconditions u t; 0and u t; 1areimposed.Theunknownfunctionistheabsorption coecient a x .Thisill-posedproblemissupplementedbyadditionaltheNeumannboundary conditionsattheboundaries,i.e. u x ;t = f 0 t ;u x ;t = f 1 t : .1.2

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43 Theinverseproblemofinteresthereistorecover a x basedontheadditionalgivenconditions attheboundariesEq..1.2.Forthepurposeofinversion,wecandene u t;x = e v t;x [97] since u t;x > 0andrewriteEq..1.1accordingto v t = v xx + v 2 x + a x ; t;x 2 ; v t; 0=ln g 0 t ;v t; 1=ln g 1 t ; v ;x =ln u 0 x : .1.3 Thegivendataaretransformedaccordingto v x = u x =u attheboundarieswith u> 0forall t 2 [0 ;t f ].ThisformulationissuitablebecausetheunknownfunctionisisolatedinEq..1.3. Thepresentalgorithmissimilaraschapter4.Duetheunknownfunctionisinthegoverning equation,thedetailofthesystemisnotexactlysameaspreviouschapter.Thedierentisthe linearizationneedtobeusedinthisalgorithm.Themethodisiterativeinnatureandconsists ofthreesteps. Assumeavaluefortheunknownabsorptioncoecient^ a x and,usingthegiven Dirichletboundaryconditions g 0 t and g 1 t ,obtainabackgroundeldsatisfyingthesystem ^ v t =^ v xx +^ v 2 x +^ a x ; t;x 2 ; ^ v t; 0=ln g 0 t ; ^ v t; 1=ln g 1 t ; ^ v ;x =ln u 0 x : .1.4 SubtractthebackgroundeldfromEq..1.3,andobtaintheerroreld, e t;x = v t;x )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ v t;x ,givenby e t = e xx + v 2 x )]TJ/F15 10.9091 Tf 11.325 0 Td [(^ v 2 x + a x )]TJ/F15 10.9091 Tf 11.065 0 Td [(^ a x ; t;x 2 ; e t; 0= e t; 1= e ;x =0 : .1.5 Theerroreldisrequiredtosatisfyadditionalconditions.Thegivendataisin theformofthegradientoftheeldattheboundaries e x ;t = f 0 t g 0 t )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ v x ;t = ^ f 0 t ; e x ;t = f 1 t g 1 t )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ v x ;t = ^ f 1 t : .1.6

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44 Assumethattheunknownfunctionisrelatedtotheassumedvalueaccordingto a x =^ a x + q x ,where q x isstillanunknownfunction.ThentheEq..1.5 changesto e t = e xx + v 2 x )]TJ/F15 10.9091 Tf 11.325 0 Td [(^ v 2 x + q x ; t;x 2 ; e t; 0= e t; 1= e ;x =0 : .1.7 UsetheadditionalmeasurementsontheboundariesinEq..1.6andobtain theunknowncorrectionterm q x ,totheassumedvalueof a x .Updatethe assumedvalue,^ a x ,andgotostep1. Thebasicstepofalgorithmissameasbefore.Thedierentpartofthismethodisthethird stepwhichistheevaluationoftheunknowncorrectionterm q x .Inthepreviouschapter,the errorintheinitialconditioncanbetreatedascorrectiontermtoupdatetheassumedvalue.In thischapter,theunknownpartisabsorptioncoecient.Therefore,thecorrectiontermshould beinthegoverningequationforerroreld.Thewaytoestimatetheunknowncoecientbase onpropersolutionspaceismoredicultthanbefore.Thedetailswillbeshowninthenext section. 5.2ProperSolutionSpace Thethirdstepofthealgorithminvolvestheidenticationofthecorrectiontotheassumed valueoftheabsorptioncoecient a x .WerstlinearizethequadratictermsinEq..1.7 accordingto e t = e xx + v 2 x )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ v 2 x + q x = e xx + v x +^ v x v x )]TJ/F15 10.9091 Tf 11.325 0 Td [(^ v x + q x = e xx +2^ v x e x + q x : .2.1 Ourgoalinthischapteristorecover q x .Sameasbefore,wecanndthelinearlyindependentsetoffunctions c ` , ` =1 ; 2 ;:::;N over x 2 [0 ; 1]andassumetheunknownfunctions q x 2f c 1 ;c 2 ;:::;c N g .Weareassumingthatitispossibletoevaluatetheunknown a x atthe

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45 boundaries,andasaresult,wecanassumethat q = q =0.Next,generateasetoffunctionsthatsatisfytheerroreldequationwiththeknownzeroDirichletboundarycondition, i.e., ` t = ` xx +2^ v x ` x + c ` x ; t;x 2 ; ` t; 0= ` t; 1= ` ;x =0 : .2.2 Therefore,everyfunction ` t;x mustsatisfythezeroDirichletboundaryconditionsat x =0 and x =1,andzeroinitialcondition.Itisthenpossibletoexpandtheactualandunknown erroreld e t;x inthespanofthespacegeneratedby ` t;x , ` =1 ; 2 ;;N; accordingto e t;x = N X ` =1 ` ` t;x ; .2.3 wherethefunctions ` t;x areknownfromEq..2.2,howevertheconstants ` arestillunknown.Wenextarguethattheerroreld e t;x mustsatisfythegradientconditionthatis furnishedbythemeasurementsandaregiveninEq..1.6.Thegradientconditionscanbe expressedbytheoperators B 0 and B 1 .Theerroreldisrequiredtosatisfytheconditionsgiven by B 0 e t; 0= B 0 N X ` =1 ` ` t; 0= N X ` =1 ` B 0 ` t; 0= ^ f 0 t .2.4 B 1 e t; 1= B 1 N X ` =1 ` ` t; 1= N X ` =1 ` B 1 ` t; 1= ^ f 1 t .2.5 Theabovetwoequationscanbeusedtoobtaintheunknownconstants ` for ` =1 ; 2 ;:::;N . Thisstepispresentedinmoredetailsinthenextnumericalexamplesection.Oncetheunknowncoecientsareobtained,theunknown q x canbecomputedbysubstitutingEq..2.3 inEq..2.1whichleadsto N X ` =1 ` [ ` t )]TJ/F25 10.9091 Tf 10.909 0 Td [( ` xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ v x ` x ]= q x : .2.6 UsingEq..2.2andassuminganexpansionfor q x = P N ` =1 ` c ` x ,andsubtractingintoEq. .2.6leadsto N X ` =1 c ` ` )]TJ/F25 10.9091 Tf 10.909 0 Td [( ` =0.2.7

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46 Duetothefunctions c ` x arelinearlyindependentandcannotbezero,wecanconcludethat ` = ` , ` =1 ; 2 ;:::;N .and q x = P N ` =1 ` c ` x .ByusingEqs..2.4and.2.5,wecanget thevalueofunknownconstants l .Thiscompletesthethirdstepofthealgorithm.Aftersolvingfor q x ,theassumedvalueoftheunknownabsorptioncoecientcanbeupdatedaccording to a x =^ a x + q x andthethreestepsinthealgorithmcanberepeated. 5.3NumericalExamples Inthissectionanumberof1-Dand2-Dexampleshavebeenusedtostudytheapplicabilityof thismethodininversecoecientevaluationproblem. Example1. First,wewillconsidertheevaluationoftheabsorptioncoecientfora1-Dparabolic equationgiveninEq..1.1andassumethattheactualfunctionisgivenby a x =1+exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 01 : .3.1 Alsoassumetheinitialconditionis u ;x =0 : 5.TwoDirichletboundaryconditionare g 0 t =0 : 5+0 : 24exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( t )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 4 2 0 : 03 ; .3.2 and g 1 t =0 : 5+0 : 24sin 2 : 4 t : .3.3 Thealgorithmwillstartfromassumingavalueofabsorptioncoecientfor^ a =1,obtaining thebackgroundeldofthetransformedsystem^ v t;x ,andprovidingtheboundaryconditions fortheerroreldinEq..1.6for t 2 [0 ; 0 : 25.Anappropriatesetoflinearlyindependent functionscanbeconsideredaccordingto c ` x =sin `x ;c ` = c ` =0 ;` =1 ; 2 ;:::;N: .3.4 The solutionset l canbesolvedbyusingtheindividualfunctions c ` x inEq..2.2,for ` = 1 ; 2 ;:::;N .Wecandenotethenodalvaluesof ` t;x bythevectors " ` for ` =1 ; 2 ;:::;N .We

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47 nextarguethattheerroreldmustsatisfytheadditionalboundaryconditionsthatisprovided bythemeasurementsgiveninEqs..2.4and.2.5,accordingto B 0 e t; 0= B 0 N X ` =1 ` " ` t; 0= N X ` =1 ` B 0 " ` t; 0= b 0 .3.5 B 1 e t; 1= B 1 N X ` =1 ` " ` t; 1= N X ` =1 ` B 1 " ` t; 1= b 1 .3.6 Groupingthesetwoequationsleadsto 2 6 4 B 0 " ` B 0 " ` 3 7 5 = 2 6 4 b 0 b 1 3 7 5 .3.7 where isthe N dimensionalvectorcontainingthevaluesoftheunknownparameters l for ` =1 ; 2 ;::;N .Thevectors b 0 and b 1 containthevaluesof ^ f 0 t and ^ f 1 t atdierenttimes. Theabovecoecientmatrixisnon-squareandrank-decient.Itispossibletostabilizetheinversionbyimposingsomeformofregularizationaccording 2 6 6 6 6 6 4 B 0 " ` B 0 " ` 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 b 0 b 1 0 3 7 7 7 7 7 5 .3.8 where,thematrix representstherst-orderoperatorgivenby dc ` x;y dx x = x k = N X ` =1 ` ` cos `x k .3.9 evaluatedatdiscretelocations x k alongthedomain.Theconstant > 0issetbythedesigner, andtheabovelinearsystemcanbesolvedfortheunknowncoecients ` for ` =1 ; 2 ;:::;N . Dividingthedomain[0:1]into n e =100equalintervalsleadsto n = n e +1=101nodesin thexdirection.Dividingthetimeinterval[0:0 : 5]into n t =100leadsto101timeintervals. Choosing N =20sinefunctionsinEq..2.3leadstothepropersolutionspacehavingN=20 vectors,witheachvectorhavingthedimension .Thecollecteddataattheboundariesarecontaminatedwithazeromeanrandomlygeneratednoisewithnoise-to-signalratioof

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48 FIGURE5.1:TherecoveredabsorptionfortheExample1.Thegurecomparesthenalvalue totheactualfunction 4%.Thedataisthenlteredusingasimple7-thorderpolynomial.Figure5.1comparestherecoveredabsorptioncoecientwiththeactualfunctionafter150iterationswith =0 : 00005, andgure5.2showsthereductionintheerror.InthisnotetheerrorisdeneasEq..3.9, where ^ f 0 t and ^ f 1 t arethedierencebetweenthegivendataandthecalculatedgradientsat theboundariesgiveninEq..1.6. Example2. Nextconsidertheevaluationofanabsorptioncoecientwithina2-Ddomain x;y 2 =[0 ; 1] [0 ; 1]givenby a x;y =1+exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 2 0 : 02 exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 35 2 0 : 02 : .3.10 whichisshowninFig.5.3.Theboundaryisaccessibleandcanbeusedtoimposetemperature accordingto u ;y;t =0 : 5+0 : 02exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( t )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 4 2 0 : 002 exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 0 : 002 : .3.11 u ;y;t =0 : 5+0 : 02exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( t )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 4 2 0 : 002 sin 2 y : .3.12

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49 FIGURE5.2:ThereductioninerrorfortheExample1asafunctionofthenumberofiterations u x; 0 ;t =0 : 5+0 : 02exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( t )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 4 2 0 : 002 exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 0 : 002 : .3.13 u ;x;t =0 : 5+0 : 02exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( t )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 4 2 0 : 002 sin 2 x : .3.14 Forthisexample,welet 4 x = 4 y =1 = 60,anddividethetimeinterval[0:0 : 5]into n t =60 equalintervalsi.e., 4 t =0 : 5 = 60.Thedicultpartofthisproblemistheunknowncorrection termisa2-Dfunction.Soweneedtondthelinearlyindependentfunctiondependson x and y .Inordertogeneratethesolutionspace,asetoflinearlyindependentfunctionscanbeconsideredaccordingto c ` x;y =sin ix sin jy ` = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j;i =1 ; 2 ;:::;M;j =1 ; 2 ;:::;M: .3.15 AnditshouldsatisfytheDirichletboundaryconditionsas c ` ;y = c ` ;y = c ` x; 0= c ` x; 1=0 : .3.16

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50 FIGURE5.3:TheactualabsorptioncoecientforExample2. For M =20,thepropersolutionspacehasthedimension N = M 2 =400.Eachvectorshould havethedimension61 61 61.Duetothesystemisstillnon-squareandrank-decient.Itis possibletostabilizetheinversionbyimposingsomeformofregularizationaccordingto 2 6 6 6 6 6 6 6 6 4 B 0 " ` B 0 " ` x y 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 b 0 b 1 0 0 3 7 7 7 7 7 7 7 7 5 .3.17 where,thematrix x and y representstherst-orderoperatordependon x and y givenby x = dc ` x;y dx x = x i ;y = y j = N X ` 1 =1 N X ` 2 =1 l 1 cos ` 1 x i sin ` 2 y j .3.18 y = dc ` x;y dy x = x i ;y = y j = N X ` 1 =1 N X ` 2 =1 ` 2 sin ` 1 x i cos ` 2 y j .3.19 where x i and y j arethediscretelocationinthedomain.Theconstant islargerthanzero, whichissetbythedesigner,andtheabovelinearsystemcanbesolvedforunknowncoecient

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51 FIGURE5.4:TherecoveredabsorptioncoecientfortheExample2,after1000iterations. ` for ` =1 ; 2 ; 3 ;:::;N .Thedataisagaincollectedintheformofthegradientattheboundariesandthesamelevelofnoise%isadded.Figure5.4showstherecoveredfunctionafter 1000iterations,andgure5.5showsthereductioninerrorasafunctionofthenumberofiterations.Theparameter issetat =0 : 00001.Fortheinverseproblem,weusuallycannot getthewholemeasurementontheboundary.Therefore,nexttwoexampleswillbeusedpartial measurementtorecovertheunknownabsorptioncoecient. Example3. Inthisexample,weconsidertheabsorptioncoecientissameastheoneinExample1andusethesameboundaryconditionsasgivenfromEqs..3.11to.3.14.However,inthisexample,partialmeasurementswillbeconsidered.Assumedatawouldbecollected ontwosides x =0and x =1. Forthisexample,thedomain[0:1]willbedividedinto n e =60equalintervalsleadsto n = n e +1=61nodesinboth x and y directions.Dividingthetimeinterval[0:0 : 5]into n t =60leadsto61timeintervals.Choosing M =20sinefunctionsinEqs..3.18and.3.19

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52 FIGURE5.5:ThereductioninerrorfortheExample2asafunctionofthenumberofiterations. FIGURE5.6:TherecoveredabsorptioncoecientfortheExample3,after150iterations.

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53 leadtothepropersolutionspacehaving N = M 2 =400vectors,witheachvectorhavingthedimension61 61 61.Figure5.6showstherecoveredresultafter150iterations.Theparameter isagainsetat =0 : 0002.Thereductioninerrorasafunctionofthenumberofiterationis presentedinFig.5.7.Evenwecollectthedataonlyontwosides,thismethodalsocanrecover theapproximateshapeandcorrectlocationoftheobjectfunction.Inordertoimprovetheresultwithpartialmeasurements,collectingdataonfoursidesofthedomainisworthatry.In thenextexample,atwohumpsobjectfunctionwillberecoveredbyusingthepartialmeasurementsonfoursides. FIGURE5.7:ThereductioninerrorfortheExample3asafunctionofthenumberofiterations. Example4. Nextconsidertheevaluationofanabsorptioncoecientgivenby a x;y =1+exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.91 0 Td [(0 : 3 2 0 : 015 exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 +exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 ; .3.20 whichisshowninFig.5.8.ImposingthesameboundaryconditionsasgiveninEqs..3.11 to.3.14andsetting =0 : 00001,gure5.9showstherecoveredfunctionafter1000itera-

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54 FIGURE5.8:TheactualabsorptioncoecientfortheExample4. tionswithfullmeasurements.Forpartialmeasurement,wearecollectingdatawiththeinterval [0 : 3:0 : 7]oneachside.Figure5.10showstherecoveredfunctionafter1000iterationsforpartial measurements.Comparewithtworesults,thismethodshowsgoodrobustnesstonoiseevenwe usepartialmeasurements. 5.4Remark Inthischapter,weusedthepropersolutionspacemethodtorecoverunknownabsorptioncoefcients.Several1-Dand2-Dexampleswereusedtostudytheapplicabilityofthismethod.The eectofnoiseandtheapplicabilityofthemethodforpartialdatawerealsostudied.

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55 FIGURE5.9:TherecoveredabsorptioncoecientfortheExample4,after1000iterationswith fullmeasurements. FIGURE5.10:TherecoveredabsorptioncoecientfortheExample4,after1000iterations withpartialmeasurements.Thedataiscollectedwithintheinterval[0.3,0.7]oneachside.

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56 CHAPTERVI EVALUATIONOFDIFFUSIONCOEFFICIENT;CALDERON'S PROBLEM Inthischapter,weconsiderinverseidenticationofthediusioncoecientforanellipticequation.Thisproblem,whichisoftenreferredtoastheCalderon'sproblem[98],appearsverynaturallyinvariousapplications.Itiswellknownthatthisproblemishighlyill-posed[99]andvariousmethodhavebeendevelopedtoovercomesuchdiculties.InSection6.1,wepresentthe iterativealgorithm.InSection6.2,wepresentupdatingstage.InSection6.3,weuseanumber ofnumericalexamplestostudytheapplicabilityofthismethod. 6.1ProblemStatementandtheIdenticationAlgorithm Letbetheboundeddomainin R 2 or R 3 withsmoothboundary @ .Intheabsenceofsinks orsources,thetemperatureeld u x isgivenby 5 x 5 u =0 ; x 2 ; u x = g x ; x 2 @ ; .1.1 where, x > 0isthepositiveunknownthermalconductivity.TheDirichletboundarycondition g x areimposed.Theextrameasurementsarethetemperaturegradientgivenby 5 n u x = @u @n = f x x 2 @ ; .1.2 where, n istheoutwardnormal.Theinverseproblemistodetermine x basedontheadditionalgivenconditionsattheboundary.1.2.Thebasicalgorithmissameasbefore,whichis iterativeandconsistsofthreesteps.However,duetotheunknownpartisdiusioncoecient, thedetailsofthealgorithmaredierent.Theiterativealgorithmhavebeenshownbelow

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57 Assumeainitialvaluefortheunknowncoecientfunction^ x andusingthe givenDirichletboundaryconditionsobtainabackgroundeldsatisfyingthesystem ^ x 4 ^ u + 5 ^ x 5 ^ u =0 ; x 2 ^ u x = g x x 2 @ .1.3 for^ u x = g x ; 8 x 2 @ . SubtractthebackgroundeldfromEq..1.1,andobtaintheerroreld, e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x ,givenby x 4 u + 5 x 5 u )]TJ/F15 10.9091 Tf 10.909 0 Td [(^ x 4 ^ u + 5 ^ x 5 ^ u =0 ; x 2 ; .1.4 for x 2 ,wheretheerroreldisrequiredtosatisfy e x =0 ; 8 x 2 @ .The erroreldisrequiredtosatisfyadditionalconditionsgivenby 5 n e x = f x )-222(5 n ^ e x = ^ f x x 2 @ : .1.5 Theassumedfunction, x ,isrelatedtotheactualvalueaccordingto x = ^ x + q x ,where q x isstillanunknownfunction.UsetheadditionalboundaryconditionsinEq..1.5toobtaintheunknowncorrectionterm q x .Then wecanusethecorrection q x toupdatetheassumedvalue,^ x ,andgoto step. wehaveappliedthisalgorithmtoanellipticproblemin[96]andaparabolicproblemin[100]. Alsowetrytorecovertheabsorptioncoecientinthechapter5.Thedetailofstep3inthe algorithmwillbeshowninthenextsection.

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58 6.2ProperSolutionSpace Thethirdstepofthealgorithmincludestheidenticationofthecorrection,i.e. q x ,totheassumedvalueofthediusioncoecient.Substitutingfor x inEq..1.4leadto ^ 4 e + 5 ^ 5 e + q x 4 u + 5 q x 5 u =0 ; x 2 ; e x =0 x 2 @ : .2.1 Notethat u x =^ u x + e x andthetermsthatarequadraticinunknownsaredropped.After linearizingtheaboveequationaroundthebackgroundelditleadsto ^ 4 e + 5 ^ 5 e + q x 4 ^ u + 5 q x 5 ^ u =0 ; x 2 e x =0 x 2 @ : .2.2 Inordertorecover q x for x 2 wecanproceedasfollow.Consideralinearlyindependent setoffunctions c ` x ;` =1 ; 2 ; 3 ;:::;N over x 2 andassumethattheunknownfunctions q x canbeexpressedasalinearcombinationoffunctions c ` x ,i.e. q x 2f c 1 ;c 2 ;c 3 ;:::;c N g .We areassumingthatitispossibletoevaluatetheunknownfunction x attheboundaries.For simplify,weassumethatitisequaltoone.Asaresultwecangetthat q x j @ =0.Thenext stepistogenerateasetoffunctionsthatsatisfytheerroreldinEq..2.2accordingto ^ 4 ` + 5 ^ 5 ` + c ` 4 ^ u + 5 c ` 5 ^ u =0 ; x 2 ; .2.3 alongwith x ; 8 x 2 @ .Therefore,everyfunction ` x satisestheDirichletboundaryconditions.Itisthenpossibletoexpandtheactualerroreld e x inthespanofthespacegenerated by ` x ;` =1 ; 2 ; 3 ;:::;N accordingto e x = N X ` =1 ` ` x ; x 2 .2.4 wherethefunctions ` x areknown,buttheconstants ` areunknown.Wenextarguethatthe erroreld e x mustsatisfythegradientconditionaregiveninEq..1.5.Thegradientconditionscanbeexpressedbytheoperator B .Theerroreldisrequiredtosatisfythecondition givenby

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59 B e x = B N X ` =1 ` ` x = N X ` =1 ` B ` x = ^ f x x 2 @ ; .2.5 Theaboveequationcanbeusedtoobtaintheunknowncoecients ` for ` =1 ; 2 ; 3 ;:::;N . Moredetailofthispartwillbeshowninthenextsection.Oncetheunknowncoecientsare obtainedtheunknownfunction q x canbycomputedbysubstitutingEq..2.4inEq..2.2 whichleadsto N X ` =1 ` ^ 4 ` + 5 ^ 5 ` + q x 4 ^ u + 5 q x 5 ^ u =0 ; x 2 : .2.6 NextweneedtouseEq..2.3andsimplifyaboveequationtoget N X ` =1 ` [ )]TJ/F15 10.9091 Tf 8.485 0 Td [( c ` 4 ^ u + 5 c ` 5 ^ u ]+ q x 4 ^ u + 5 q x 5 ^ u =0 ; x 2 : .2.7 Assumingthetheunknownfunction q x canbeexpressedaslinearcombinationoffunctions c ` x accordingto q x = P N ` =1 ` c ` x ,andsimplifyingleadsto N X ` =1 ` )]TJ/F25 10.9091 Tf 10.909 0 Td [( ` c ` 4 ^ u + 5 c ` 5 ^ u =0 ; x 2 : .2.8 Nextwearguethatingeneral^ u x 6 =0and c ` x arelinearlyindependent.Thenwecanconcludethat ` = ` forall ` =1 ; 2 ; 3 ;:::;N and q x = N X ` =1 l c ` x x 2 : .2.9 Thiscompletesthethirdstepofthealgorithm.Aftersolvingthecorrectionterm q x ,theunknowndiusioncoecientcanbeupdatedaccordingto x =^ x + q x .Thenthethree stepsinthewholealgorithmneedtoberepeated. 6.3NumericalExamples Inthissectionweuseanumberofexamplestostudytheapplicabilityofthismethod. Example1. Considertheevaluationoftheofthediusioncoecientfora2-DellipticequationgiveninEq..1.1andassumethattheactualfunctionof x;y isgivenby x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 35 2 0 : 02 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 2 0 : 02 .3.1

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60 whichisshowninFig.6.1,andtheboundaryconditionaregivenby FIGURE6.1:Thermalconductivityforexample1. g x;y =cos k x cos + y sin x;y 2 : .3.2 Inthisexampleweuse16setsofdatawith k =1 ; 2 ; 3 ;:::; 8and = = 3 : 2 ;= 4 : 2.Wecanstart thealgorithmbyassumingthevaluefor^ x;y =1,obtainingthebackgroundeld^ u x;y ,and providingthegradientboundaryconditionsfortheerroreldforEq..1.5.Inordertoobtainthecorrectionterm q x;y inthethirdstepofthealgorithm,anappropriatesetoflinearly independentfunctionsneedtobeconsideredaccordingto c ` x;y =sin ` 1 x sin ` 2 y ;c ` x =0 ; x 2 @ ; ` 1 =1 ; 2 ; 3 ;:::M;` 2 =1 ; 2 ; 3 ;:::;M; and ` = ` 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + ` 2 ;N = M 2 : .3.3

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61 Thesolutionsetcanbeobtainedbyusingtheindividualfunctions c ` x;y inEq..2.3,for ` =1 ; 2 ; 3 ;:::;N .Thenitedierenceapproximationtotheellipticsystemsimpliestolinear systemsgivenby A " ` + c ` =0 ; or ; " ` = )]TJ/F19 10.9091 Tf 8.485 0 Td [(A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 c ` ; 8 ` =1 ; 2 ; 3 ;:::;N; .3.4 where A isnitedimensionalapproximationofthe2Ddiusionoperatordependson^ with uniqueinversesincetheproblemiswell-posed, " ` and c ` arethevectorcontainingthenodal valuesofthefunction ` x;y and c ` x;y .Thematrix )]TJ/F15 10.9091 Tf 11.182 0 Td [(isthenite-dierenceapproximation oftheoperator 4 ^ u + 5 ^ u 5 thatactson c l .Sincethefunctions c ` arelinearindependent, itfollowsthat ` arealsolinearlyindependentfor ` =1 ; 2 ;:::;N .Theerroreldneedstobe expandedinthespacegeneratedby ` accordingto e x;y = N X ` =1 ` ` x;y .3.5 where ` isunknowncoecientsfor ` =1 ; 2 ; 3 ;:::;N .Inordertocomputethevalueof ` , theadditionalgradientboundaryconditionfortheerroreldneedtobeused.AccordingEq. .2.5anddenotingtheappropriatethenitedimensionalapproximationtothegradientoperatorby B leadsto Be = B N X ` =1 ` " ` = N X ` =1 ` B " ` = ^ f or,[ B " ` ] = ^ f ; .3.6 where isthe N dimensionalvectorcontainingthevaluesoftheunknowncoecients l for ` =1 ; 2 ; 3 ;:::;N 2 .InEq..3.6,thevector ^ f isknown,thevectorsinthesolutionspace, " ` for ` =1 ; 2 ; 3 ;:::;N arealsoknownfromEq..3.4.Thesquarematrix B issingularandthematrix,[ B " ` ],isnon-squareandrank-decientandthereforetheabovesystemcannotdirectlybe solvedforthecoecients ` .AsshowninEq..2.9,thecorrectiontermcanbeexpandedasa linearcombinationsof c ` x;y ,for ` =1 ; 2 ;:::;N .Inordertocomputetheunknowncoecient,

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62 itispossibletoimposesomeformofregularizationtostabilizetheinversionaccordingto 2 6 6 6 6 6 4 B " ` x y 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 ^ f 0 0 3 7 7 7 7 7 5 ; .3.7 wherethematrix x and y representtherstorderdierentialoperatorgivenby x = dc ` x;y dx x = x i ;y = y j = N X ` 1 =1 N X ` 2 =1 ` 1 cos ` 1 x i sin ` 2 y j ; y = dc ` x;y dy x = x i ;y = y j = N X ` 1 =1 N X ` 2 =1 ` 2 sin ` 1 x i cos ` 2 y j ; .3.8 where x i and y j arethediscretelocationinthedomain.Theconstant islargerthanzero, whichissetbythedesigner,andtheabovelinearsystemcanbesolvedforunknowncoecient ` for ` =1 ; 2 ; 3 ;:::;N . FIGURE6.2:Recoveredconductivityafter174iterationsforExample1

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63 Dividingtheunitsquaredomaininto n e =60equalintervalsleadsto n = n e +1=61nodes inthe x and y direction.Choosing M =20sinefunctionsinEq..3.3leadstothepropersolutionspacehaving N = M 2 =400vectors,witheachvectorhavingthedimension n n .The collecteddataattheboundariesarecontaminatedwithazeromeanrandomlygeneratednoise withnoise-to-signalratioof4%.Thedataislteredbytheseventhorderpolynomialcurvetting.Figure6.2showstherecovereddiusioncoecientafter174iterationswith =0 : 003.We alsostudythereductionintheerror,whichisdenedas Error= R [ 5 u x )-222(5 ^ u x ] 2 d x R [ 5 u x )-222(5 ^ u x ] 2 d x j rstiteration .3.9 andtheresultisshowninFig.6.3. FIGURE6.3:ThereductioninerrorforExample1asafunctionofthenumberofiteration Example2. InordertoimprovetheresultinExample1,moresetofdatahavebeenused whichmeans k =1 ; 2 ; 3 ; 4 ; 5and = = 2 : 3 ;= 3 : 2 ;= 4 : 2 ;= 5 : 2 ;= 6 : 2.Figure6.4showstherecovereddiusioncoecientafter150iterationswith =0 : 0001.Inthisexampledierentnoisy

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64 levelhavebeenconsidered.Figure6.5comparestheactualandrecoveredvaluesofthediusion coecientalongthediagonalcross-sectionforthreedierentvaluesofnoise.AndFigure6.6 presentsthereductionintheerrorasafunctionafter150iterationfordierentnoisy.Inthe nextexample,a2humpobjectfunctionwillbeconsidered. FIGURE6.4:Recoveredconductivityafter150iterationforExample2withmoresetsofdata Example3. Nextconsidertheevaluationofthediusioncoecientgivenby x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 +exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 .3.10 whichisshowninFigure6.7.ThesameboundaryconditionsasgiveninEq..3.2areimposed.Forthisexamplewesetthe =0 : 001.Therecoveredresultafter700iterationsispresentedinFig.6.8.Figure6.9showsthereductionintheerrorasafunctionofthenumberof iterations.Figure6.10comparestherecoveredandtheactualvaluesoftheconductivityfunctionalongdiagonalcross-sectionfortheexample3.Figure6.11comparestheactualandthe recoveredvaluesoftheconductivityfunctionalongthediagonalcross-sectionforthreedierent

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65 FIGURE6.5:Comparisonoftherecoveredandtheactualconductivityatthediagonalcross sectionforExample2fordierentlevelsofnoise FIGURE6.6:ComparisonthereductionerrorforExample2fordierentlevelsofnoise

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66 valuesofnoiselevel.Forincreasedlevelofnoise,theaccuracyoftherecoveredfunctiondrops, butthemethodcanstillrecoveragoodestimateofthe2humpunknownfunction. FIGURE6.7:ThermalconductivityforExample3. Example4. Considertheevaluationofthediusioncoecientgivenby x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 4 0 : 0001 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 4 0 : 0001 +exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 4 0 : 0001 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 4 0 : 0001 .3.11 whichispresentedinFig.6.12.Theunknownconductivityfunctioninthisexamplehasregions withrelativelylargegradients.Figure6.13showstherecoveredfunctionafter151iterations with =0 : 0005and4%noise.Figure38comparestheactualfunctionandtherecoveredfunctionalongthediagonalfortwodierentvaluesof .ThepresentmethodreliesonTikhonov regularizationtoinvertthematrixinEq..3.7.Itpenalizeslargegradients,andbyreducing abettervalueofthemagnitudeoftheunknownfunctioncanbeobtained.Figure38presents thereductionintheerrorfortwodierent 'sasfunctionsofnumberofiterations.

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67 FIGURE6.8:Recoveredconductivityafter700iterationforExample3. FIGURE6.9:ErrorreductionforExample3asafunctionofthenumberofiteration

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68 FIGURE6.10:Comparisonoftherecoveredandtheactualconductivityatthediagonalcross sectionforExample3. 6.4Remark Inthischapter,weusedthepropersolutionspacemethodtoevaluatetheunknowndiusion coecientof2-Dellipticproblem.Weusethree2-Dexamplestostudytheapplicabilityofthis method.Theaccuracyoftherecoveredfunctiondropsasthelevelofnoiseisincreased.

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69 FIGURE6.11:Comparisonoftherecoveredandtheactualconductivityatthediagonalcross sectionforExample3fordierentlevelsofnoise FIGURE6.12:ThermalconductivityforExample4.

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70 FIGURE6.13:ThermalconductivityforExample4. FIGURE6.14:Comparisonoftherecoveredandtheactualconductivityatthediagonalcross sectionforExample4fortwovaluesof .

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71 FIGURE6.15:ErrorreductionforExample3asafunctionofthenumberofiterationfortwo valuesof .

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72 CHAPTERVII EVALUATIONOFABSORPTIONANDDIFFUSIONCOEFFICIENT Inthischapter,weconsidertheinverseidenticationofthediusionandabsorptioncoecient foraphotondiusionequation[101].Itiswellknownthatthisproblemishighlyill-posed[36, 102].InSection7.1,wepresenttheiterativealgorithm.Itassumesinitialguessesforthetwo unknownfunctionsandobtainsthecorrectionstotheassumedvaluesateveryiteration.Section 7.2presentstheupdatingpartsofthisalgorithmindetail.Twonumericalexampleshavebeen usedtostudyapplicabilityofthismethod,whichisshowninSection7.3. 7.1ProblemStatementandtheIdenticationAlgorithm Letbeaboundeddomainin R 2 withsmoothboundary @ .InthefrequencydomainthediffusionapproximationDAequationisexpressedby )-222(5 x 5 x ;! + a x + i ! c x ;! =0 ; x 2 ; .1.1 where, x ;! isthephotondensityatposition x andmodulationfrequency ! , x denotes thediusioncoecient, a x istheabsorptioncoecient, c isthespeedoflightinthemedium andi= p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.InordertosolveEq..1.1,boundaryconditionsneedtobeimposed.Themost frequentlyusedboundaryconditionisRobinboundarycondition,whichisgivenasfollows: x ;! + 1 2 x @ x ;! @v = g x ;! ; x 2 @ ; .1.2 where g x ;! istheboundarysource, isadimension-dependentconstant 2 =1 =; 3 =1 = 4 and v istheouternormalatboundarydomain. Theunknownfunctioninthisproblemarethediusioncoecient x andabsorptioncoecient a x .Themeasurementsinthispaperconsistofthecomplexintensityintermsofphase

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73 shiftandamplitudeofreceivedopticalsignalwithadding4%Gaussiannoisetoboth,whichare measuredas x ;! = f x x 2 @ : .1.3 Theinverseproblemistodetermine x and a x basedontheadditionalmeasurementat theboundary.1.3.Twounknownfunctionsneedtobeevaluatedbasedonpropersolution space.Thealgorithmissimilartopreviouschapters,however,itshouldconcludemoresteps thanbefore.Thealgorithmisstilliterativeandconsistsofsevensteps 1.Assumeinitialvaluesforbothunknowncoecientsas^ x and^ a x .First, treat a x =^ a x andusethegivenboundaryconditionstoobtainabackgroundeldsatisfyingthesystem )]TJ/F15 10.9091 Tf 8.485 0 Td [(^ x 4 ^ + 5 ^ x 5 ^ + )]TJ/F25 10.9091 Tf 5 -8.837 Td [( a x + i ! c ^ =0 ; x 2 ; ^ x + 1 2 ^ x @ ^ x @v = g x ; x 2 @ : .1.4 2.SubtractthebackgroundeldfromEq..1.1andobtaintheerroreld, e x = x )]TJ/F15 10.9091 Tf 12.121 2.758 Td [(^ x ,givenby )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 4 + 5 x 5 +^ x 4 ^ + 5 ^ x 5 ^ + )]TJ/F25 10.9091 Tf 5 -8.836 Td [( a x + i ! c e =0 ; x 2 ; .1.5 where,theerroreldisrequiredtosatisfy e x + 1 2 ^ x @e x @v =0 ; 8 x 2 @ .The erroreldisalsorequiredtosatisfyadditionalconditionsgivenby e x = f x )]TJ/F15 10.9091 Tf 12.122 2.757 Td [(^ x = ^ f x x 2 @ : .1.6 3.Assumetheunknownfunction x isrelatedtotheassumedvalueaccordingto x =^ x + p x ,where p x isstillanunknownfunction.Usetheadditional boundaryconditionsinEq..1.6toobtaintheunknowncorrectionterm p x . Thenwecanusethecorrection p x toupdatetheassumedvalue,^ x ,andgo tostep1.Theiterationformstep1to3iscallediterationI.

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74 UsingiterationI,theunknowncoecient^ x canbeupdated.However,thevalueof^ a x is stillinitialvalue.Therefore,thepreviousalgorithmneedtobeexpandedanduseupdated^ x torecoverunknowncoecient a x . 4.Consider x =^ x ,whichisobtainedfromiterationIand,usingthegiven boundaryconditions,obtainadierentbackgroundeldsatisfyingthesystem )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 4 ^ ^ + 5 x 5 ^ ^ + )]TJ/F15 10.9091 Tf 8.057 -8.836 Td [(^ a x + i ! c ^ ^ =0 ; x 2 ; ^ ^ x + 1 2 x @ ^ ^ x @v = g x ; x 2 @ : .1.7 5.SubtractthebackgroundeldfromEq..1.1andEq..1.2,andobtaina newerroreld,^ e x = x )]TJ/F15 10.9091 Tf 12.121 5.636 Td [(^ ^ x ,givenby )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 4 ^ e + 5 x 5 ^ e + )]TJ/F25 10.9091 Tf 5 -8.836 Td [( a x + i ! c ; )]TJ/F31 10.9091 Tf 10.303 8.837 Td [()]TJ/F15 10.9091 Tf 8.058 -8.837 Td [(^ a x + i ! c ^ ^ =0 ; x 2 ; .1.8 where,theerroreldisrequiredtosatisfy^ e x + 1 2 x @ ^ e x @v =0 ; 8 x 2 @ .The newerroreldisrequiredtosatisfyadditionalconditionsgivenby ^ e x = f x )]TJ/F15 10.9091 Tf 12.121 5.636 Td [(^ ^ x = ^ ^ f x x 2 @ : .1.9 6.Assumetheunknownfunction a x isrelatedtotheassumedvalueaccording to a x =^ a x + q x ,where q x isstillanunknownfunction.UsetheadditionalboundaryconditionsinEq..1.9toobtaintheunknowncorrectionterm q x .Thenonecanusethecorrection q x toupdatetheassumedvalue,^ a x , andgotostep4.Wecalltheiterationformstep4to6istheiterationII 7.RepeatiterationII,theassumedabsorptioncoecient^ a x canbeupdated. Thenweneedtogobacktostep1anddoiterationIanditerationIIagainto improvetheresultof^ x and^ a x ,whichiscallediterationIII. Thewholealgorithminthissectionconsistsoftwoparts.TheiterationIisthebasicalgorithm usedtorecoverabsorptioncoecientinchapter5,andtheiterationIIisthealgorithmused

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75 toevaluateofdiusioncoecientinchapter6.Thedicultpartofthisproblemistwounknownsinthesystemanditrequirestorecovertheminthesametime.Therefore,wecan't recovertheunknowncoecientsbyasingleiteration.Twoassumedvaluesneedtobeusein thisalgorithm.Moredetailsaboutstep3and6willbeshowninnextsection. 7.2ProperSolutionSpace 7.2.1DetailsofStep3 Thethirdstepofthealgorithmincludesthedeterminationofthecorrectiontotheassumed valueofthediusioncoecient x ,whichisoneofthenewfeaturesofthisalgorithm.Substitutingfor x =^ x + p x inEq..1.5leadsto )]TJ/F15 10.9091 Tf 8.9 0 Td [(^ 4 e )-222(5 ^ 5 e + a x + i ! c e )]TJ/F25 10.9091 Tf 10.909 0 Td [(p x 4 )-222(5 p x 5 =0 ; x 2 ; .2.1 where,theerroreldsatises e x + 1 2 ^ x @e x @v =0 ; 8 x 2 @ .Linearizingtheequationaround thebackgroundeldleadsto )]TJ/F15 10.9091 Tf 8.9 0 Td [(^ 4 e )-222(5 ^ 5 e + a x + i ! c e )]TJ/F25 10.9091 Tf 10.909 0 Td [(p x 4 ^ )-222(5 p x 5 ^ =0 ; x 2 : .2.2 Inordertorecover p x for x 2 wecanusethesameassumptionasbefore.Consideralinearlyindependentsetoffunctions c ` x ;` =1 ; 2 ; 3 ;:::;N over x 2 andassumethatthe unknownfunctions p x canbeexpressedasalinearcombinationoffunctions c ` x ,i.e. p x 2 f c 1 ;c 2 ;c 3 ;:::;c N g .Weareassumingthatitispossibletoevaluatetheunknownfunction p x attheboundaries.Forsimplify,weassumethatitisequaltoone.Asaresultwecangetthat q x =0 8 x 2 @ .Thenextstepistogenerateasetoffunctionsthatsatisfytheerroreld accordingto )]TJ/F15 10.9091 Tf 8.901 0 Td [(^ 4 ` )-222(5 ^ 5 ` + a x + i ! c ` )]TJ/F25 10.9091 Tf 10.909 0 Td [(c ` 4 ^ )-222(5 c ` 5 ^ =0 ; x 2 ; .2.3 withtheknownRobinboundarycondition ` x + 1 2 ^ x @ ` x @v =0 ; 8 x 2 @ .Itisthen possibletoexpandtheactualerroreld e x inthespanofthespacegeneratedby ` x ;` =

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76 1 ; 2 ; 3 ;:::;N accordingto e x = N X ` =1 ` ` x ; x 2 ; .2.4 where,thefunctions ` x areknown,buttheconstants ` areunknown.Wenextarguethat theerroreld e x mustsatisfytheDirichletconditionaregiveninEq..1.6.TheDirichlet conditionscanbeexpressedbytheoperator B .Theerroreldisrequiredtosatisfytheconditiongivenby B e x = B N X ` =1 ` ` x = N X ` =1 ` B ` x = ^ f x x 2 @ : .2.5 Theaboveequationcanbeusedtoobtaintheunknowncoecients ` for ` =1 ; 2 ; 3 ;:::;N . Moredetailofthispartwillbeshowninthenextsection.Oncetheunknowncoecientsare obtainedtheunknownfunction p x canbecomputedbysubstitutingEq..2.4inEq..2.2 whichleadsto N X ` =1 ` )]TJ/F15 10.9091 Tf 8.9 0 Td [(^ 4 ` )-222(5 ^ 5 ` + a x + i ! c ` )]TJ/F25 10.9091 Tf 10.909 0 Td [(p x 4 ^ )-222(5 p x 5 ^ =0 ; .2.6 for 8 x 2 .NextweneedtouseEq..2.3andsimplifyaboveequationtoget N X ` =1 ` c ` 4 ^ + 5 c ` 5 ^ )]TJ/F25 10.9091 Tf 10.909 0 Td [(p x 4 ^ )-222(5 p x 5 ^ =0 ; x 2 : .2.7 Assumingtheunknownfunction p x canbeexpressedaslinearcombinationoffunctions c ` x accordingto p x = P N ` =1 ` c ` x ,andsimplifyingleadto N X ` =1 ` )]TJ/F25 10.9091 Tf 10.909 0 Td [( ` c ` 4 ^ + 5 c ` 5 ^ =0 ; x 2 : .2.8 Nextwearguethatingeneral ^ x 6 =0and c ` x arelinearlyindependent.Thenwecanconcludethat ` = ` forall ` =1 ; 2 ; 3 ;:::;N and p x = N X ` =1 ` c ` x x 2 : .2.9 Thiscompletesthethirdstepofthealgorithm.Aftersolvingthecorrectionterm p x ,theunknowndiusioncoecientcanbeupdatedaccordingto x =^ x + p x .Thentheiteration Ineedtorepeattoapproximatethediusioncoecient x .

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77 7.2.2DetailsofStep6 Inthesixthstep,thesamemethodneedtobeusedtoexpresstheunknownfunction q x by thesamelinearcombinationoffunctions c ` x .Substitutingfor a x =^ a x + q x inEq. .1.8andlinearizingthenewerroreldequationleadto )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 4 ^ e + 5 x 5 ^ e + ^ a x + i ! c ^ e + q x ^ ^ =0 ; x 2 ; .2.10 wherethenewerrorledalsoneedtosatisfy^ e x + 1 2 x @ ^ e x @v =0 ; x 2 @ .Usethesameset offunctions c ` x andassumethat q x canbeexpressedasalinearcombinationoffunctions c ` x .Inordertoevaluatethecorrectionterm q x ,adierentsetoffunctions^ ` whichsatisesthenewerroreldwiththeknownRobinboundaryconditionneedtobegenerated,i.e., )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 4 ^ ` + 5 x 5 ^ ` + ^ a x + i ! c ^ ` + c ` x ^ ^ =0 ; x 2 ; .2.11 with^ ` x + 1 2 x @ ^ ` x @v =0 ; 8 x 2 @ .Thenitisalsopossibletoexpandtheerroreld^ e x inthespanofthespacegeneratedby^ ` x , ` =1 ; 2 ; 3 ;:::;N accordingto ^ e x = N X ` =1 ^ ` ^ ` x ; x 2 ; .2.12 where^ ` x canbeobtainedfromEq..2.11andtheconstant^ ` arestillunknown.Inorder tocalculate^ ` ,thesamemeasurement f x needtobeusedaccordingto B ^ e x = B N X ` =1 ^ ` ^ ` x = N X ` =1 ^ ` B ^ ` x = ^ ^ f x x 2 @ ; .2.13 where,operator B isthesameasbefore.Usingthesameanalyticalmethod,wecanconclude that q x = N X ` =1 ^ ` c ` x x 2 : .2.14 Thiscompletesthesixthstepofthealgorithm.Solvingforthecorrectionterm q x ,theunknownabsorptioncoecientcanbeupdatedaccordingto a x =^ a x + q x .ThentheII needtoberepeatedtoupdatetheunknowncoecient a x .Obtainingupdatedvalueofabsorptioncoecient,thealgorithmshouldgobacktostep1anddoiterationIIItoimproveboth x and a x .

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78 7.3NumericalExamples Inthissectionweusetwoexamplestostudytheapplicabilityofthismethodinthisproblem. Example1. Considertheevaluationoftheofabsorptioncoecientandthediusioncoefcientfora2-DellipticequationgiveninEq..1.1,andassumethattheactualfunctionof x;y and a x;y aregivenby x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 35 2 0 : 02 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 2 0 : 02 ; a x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 2 0 : 02 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 35 2 0 : 02 ; .3.1 whichareshowninFig.7.1andFig.7.2,andtheboundaryconditionaregivenby g x;y =cos k x cos + y sin : .3.2 FIGURE7.1:Theactualdiusioncoecient forExample1. Recover x

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79 FIGURE7.2:Theactualabsorptioncoecient a forExample1. Inthisexampleweusedatawith k =5 ; 7 ; 10 ; 13 ; 15and = = 5 : 2 ;= 7 : 2 ;= 9 : 2 ;= 11 : 2, = 13 : 2 ;= 15 : 2.Wecanstartthealgorithmbyassumingthevaluefor^ x;y =1and^ a x;y = 1,obtainingthebackgroundeld ^ x;y ,andprovidingtheboundaryconditionsfortheerror eldforEq..1.6.Inordertoobtainthecorrectionterm p x;y instep3,anappropriateset oflinearlyindependentfunctionsneedtobeconsideredaccordingto c ` x;y =sin ` 1 x sin ` 2 y ; c ` =0 ; 8 x 2 @ ; ` 1 =1 ; 2 ; 3 ;:::M;` 2 =1 ; 2 ; 3 ;:::;M; ` = ` 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 N + l 2 ; and M 2 = N: .3.3 Thesolutionset ` canbeobtainedbyusingtheindividualfunctions c ` x;y inEq..2.3,for ` =1 ; 2 ; 3 ;:::;N .Thenitedierenceapproximationtotheellipticsystemsimpliestolinear systemsgivenby A ^ " ` + A c ` ^ =0 ; or ; " ` = A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ^ A c ` ^ ; for ` =1 ; 2 ; 3 ;:::;N; .3.4

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80 where A ^ and A c ` arenitedimensionalapproximationofthe2Ddiusionoperatordepends on^ and c ` individually, " ` and ^ arethevectorcontainingthenodalvaluesofthefunction ` x;y and ^ x;y .Sincethefunctions c ` arelinearindependent,itfollowsthat ` arealsolinearindependent.Inordertocomputetheunknownvalue ` inEq..2.4,themeasurement boundaryconditionfortheerroreldneedtobeused.AccordingEq..2.5anddenotingthe appropriatethenitedimensionalapproximationtothegradientoperatorby B leadsto Be = B N X ` =1 ` " ` = N X ` =1 ` B " ` = ^ f ; or,[ B " ` ] =[ ^ f ] ; .3.5 where, isthe N dimensionalvectorcontainingthevaluesoftheunknowncoecients ` for ` =1 ; 2 ; 3 ;:::;N .Intheaboveequation,thevector ^ f isknown,thesolutionspace " ` for ` = 1 ; 2 ; 3 ;:::;N 2 arealsoknownfromEq..3.4.Theabovecoecientmatrix,[ B " ` ],isnon-square andrank-decient.AsshowninEq..2.9,thecorrectiontermcanbeexpandedby c ` x;y .It ispossibletostabilizetheinversionbyimposingsomeformofregularizationaccordingto 2 6 6 6 6 6 4 B " ` 1 x 1 y 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 ^ f 0 0 3 7 7 7 7 7 5 ; .3.6 wherethematrix x and y representtherstorderdierentialoperatorgivenby x = dc ` x;y dx x = x i ;y = y j = N X ` 1 =1 N X ` 2 =1 ` 1 cos ` 1 x i sin ` 2 y j ; y = dc ` x;y dy x = x i ;y = y j = N X ` 1 =1 N X ` 2 =1 ` 2 sin ` 1 x i cos ` 2 y j ; .3.7 where x i and y j arethediscretelocationinthedomain.Theconstant 1 islargerthanzero, whichissetbythedesigner,andtheabovelinearsystemcanbesolvedforunknowncoecient ` for ` =1 ; 2 ; 3 ;:::;N .Substituting ` inEq..2.9,wecangetthecorrectionterm p x and updatethediusioncoecientaccordingto^ a x =^ a x + q x . Recover a x

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81 Similaraspreviousiteration,theupdatedabsorptioncoecientneedtobeusedinthispart. ThenitedierenceapproximationtoEq..2.11tolinearsystemsgivenby A ^ a ^ " ` + c ` ^ ^ =0 ; or ; ^ " ` = A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ^ a c ` ^ ^ ; for ` =1 ; 2 ; 3 ;:::;N; .3.8 where A ^ a isnitedimensionalapproximationofthe2Ddiusionoperatordependson^ a with uniqueinverse, ^ " ` and ^ ^ arethevectorcontainingthenodalvaluesofthefunction^ ` x;y and ^ ^ x;y ,and c ` isavectordependsonthefunctions c ` x;y =sin ` 1 x sin ` 2 y .Since c ` are linearindependent,itshowsthat^ ` arealsolinearindependent.Accordingtoequation.2.13 andaddingregularizationterm,thenewsystemisgivenby 2 6 6 6 6 6 4 B ^ " ` 2 x 2 y 3 7 7 7 7 7 5 ^ = 2 6 6 6 6 6 4 ^ ^ f 0 0 3 7 7 7 7 7 5 : .3.9 Bysolvingtheabovesystem,theunknownconstants^ ` canbeobtained.Substituting^ ` inEq. .2.14,wecangetthecorrectionterm q x andupdatetheabsorptioncoecientaccordingto a x =^ a x + q x . Consideringatwo-dimensionaldomain x;y 2 =[0 ; 1] [0 ; 1]anddividingthedomaininto n e =60equalintervalsleadsto n = n e +1=61nodesinthe x and y direction.Choosing M =20sinefunctionsinEqs..2.4and.2.12leadtothepropersolutionspacehaving N = M 2 =400vectors,witheachvectorhavingthedimension n n .Thecollecteddataatthe boundariesarecontaminatedwithazeromeanrandomlygeneratednoisewithnoise-to-signal ratioof4%.Thedataislteredbytheseventhorderpolynomialcurvetting.Figure7.3and Figure7.4showtherecovereddiusioncoecientandabsorptioncoecient.TheiterationIfor updating^ x with 1 =8 e )]TJ/F15 10.9091 Tf 11.494 0 Td [(7is5,theiterationIIforrecovering a x with 2 =1 e )]TJ/F15 10.9091 Tf 11.494 0 Td [(5is 40,andtheiterationIIIis10.Wealsostudythereductionintheerror,whichisdenedasEq. .3.9.TheerrorresultofdiusioncoecientisshowninFig.7.5andresultofabsorptionis showninFig.7.6.

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82 FIGURE7.3:Recovereddiusioncoecient forExample1 FIGURE7.4:Recoveredabsorptioncoecient a forExample1

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83 FIGURE7.5:Errorreductionfordiusioncoecient asafunctionofthenumberofiteration FIGURE7.6:Errorreductionabsorptioncoecient a asafunctionofthenumberofiteration

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84 Example2. Nextconsidertheevaluationofthediusioncoecientgivenby x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 +exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 ; .3.10 andabsorptioncoecientgivenby a x;y =1+exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 015 +exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 2 0 : 015 ; .3.11 whichareshowninFig.7.7andFig.7.8,andthesameboundaryconditiongiveninEq..3.2. Forthisexamplewesetthe 1 =5 e )]TJ/F15 10.9091 Tf 11.483 0 Td [(7, 2 =5 e )]TJ/F15 10.9091 Tf 11.483 0 Td [(5and M =20.TheiterationIis10,the iterationIIis40,andtheiterationIIIis10.Therecoveredresultsfor x and a x arepresentedinFig.7.9andFig.7.10.Figure7.11andFigure7.12showthereductionintheerroras afunctionofthenumberofiterationsfor x and a x . FIGURE7.7:Theactualdiusioncoecient forExample2.

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85 FIGURE7.8:Theactualabsorptioncoecient a forExample2. FIGURE7.9:Recovereddiusioncoecient forExample2.

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86 FIGURE7.10:Recoveredabsorptioncoecient a forExample2. FIGURE7.11:Errorreductionfordiusioncoecient asafunctionofthenumberofiteration.

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87 FIGURE7.12:Errorreductionfordiusioncoecient a asafunctionofthenumberofiteration. 7.4Remark Inthischapter,westudiedtheinverseevaluationofthediusionandabsorptioncoecientin a2-Dellipticproblembyusingpropersolutionspacemethod.Weusedtwo2-Dexamplesto studytheapplicabilityofthismethod.

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88 CHAPTERVIII EVALUATIONOFTHEBOUNDARYOFTHEVACUUMINA TOKAMAK Inthischapter,westudytheinverseevaluationoftheinternalboundaryofthetoresupraina tokamak.Thisisessentialtothestableoperationofamagneticconnementdevicewithtoroidal geometryasshowninFig8.1.Theexactlocationoftheplasmawithintheconnementisessentialforthedesignoffeedbackcontrollawswhicharenecessaryforthestableoperationofa fusionbasedenergysource.Fusionbasedenergyproductionhasapotentialtoprovideunlimitedenergywithlittlenegativebyproductssuchaswaste,orpollution[103].Inthischapter,we investigateaninverseproblemforthepoloidaluxwhichactsasabuerbetweentheplasma andtheboundary.Analyticalissuesonthisproblemhavebeenpresentedin[104],andareview ofrecentworksonthisinverseproblemcanalsobefoundin[105]. Twocomputationalmethodshavebeenusedforevaluatingoftheinteriorboundaryofthetore supraorplasmaboundaryinsideaTokamak.By interiorboundary wearereferringtothe interiorboundaryofthevacuumwhichisthesameastheouterboundaryoftheplasma.The rstoneisbasedontheadjointmethod,whichseekstominimizeacostfunctional.Thismethod iscomputationallyfeasible,andfast.However,wehavenoresultontheconvexityoftheminimization.Thesecondmethodissomewhatindirect.Inthesensethatwerstobtainthevalue ofthepoloidaluxwithinaringwhichincludesthetoresupraorthevacuum.Inotherwords, weextendthedomainandletthedomainofinterestbewithintwoconcentriccircles.Thenusingthefactthatthepoloidaluxhasaconstantvalueofsay c attheinteriorboundary,wecan interpolatethelocationoftheboundary.Basically,thesecondmethodinvolvesthesolutionof aCauchyproblemforanannulardomain.Thisproblemwhichisstillill-posedhasbeencon-

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89 sideredforaLaplaceequationin[106].However,inthiswork,weapplyourpreviouscomputationalmethod.Itisalsoveryaccurateandissuitablefortheproblemathand.InSection8.2, wepresenttheadjointmethod,whichseekstominimizeacostfunctional.InSection8.3,we presenttheinterpolationmethodwhichincludesthemethodbasedonpropersolutionspace.In Section8.4,weuseanumberofnumericalexamplestostudytheapplicabilityofthesemethods. FIGURE8.1:ToroidalgeometryofthetokamakToreSupra 8.1ProblemStatement Considergure8.1andlet= f ;z g betheboundeddomainin R 2 with C 1 boundary.The startingpointistheellipticequationforthepoloidalux u ;z invacuumgivenby 5 1 5 u =0 ; ;z 2 ; .1.1 wherethedierentialoperator 5 isdened 5 u = @u @ ; @u @z .Wearealsousingthephysical dimensionsclosetowhatisreportedin[104],i.e. R =2,andradiusofthecross-section1.Ex-

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90 pandingtheequationleadsto u + u zz )]TJ/F15 10.9091 Tf 12.197 7.38 Td [(1 u =0 ; ;z 2 : .1.2 AsshowninFig.8.2,itisconvenienttoperformtwocoordinatetransformations.Firstwecan FIGURE8.2:Acrosssection deneanewcoordinateintheradialdirectionaccordingto = )]TJ/F25 10.9091 Tf 11.207 0 Td [(R where R istheradiusof theTokamakwhichleadsto u + u zz )]TJ/F15 10.9091 Tf 23.132 7.38 Td [(1 + R u =0 ; ;z 2 : .1.3 Then,wecanswitchtopolarcoordinatesaccordingto = r cos and z = r sin ,whichleads to r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r u =0 : .1.4 Theouterboundaryofthetokamak[104]ortheouterboundaryofthevacuumisaccessible andbothDirichletandNeumannconditionscanbeprovidedaccordingto u ; = d ;u r ; = ^ d ; 2 [0 ; 2 ] : .1.5

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91 Dirichletconditionattheinnerboundaryofthevacuumisknown,howeverthelocationofthe innerboundaryisnotknown,i.e., u h ; = c; 8 2 [0 ; 2 ] ; .1.6 where,thefunction h isunknown.Theinverseproblemofinterestistorecovertheinner boundary h basedontheadditionalconditionthatcanbeobtainedattheouterboundary. 8.2AdjointMethod Weseektominimizethecostfunctionalgivenby J = 1 2 Z 2 0 h ^ d )]TJ/F25 10.9091 Tf 10.909 0 Td [(u r ; i 2 d; .2.1 subjecttoEq..1.4.Usingtheadjointvariable r; onecanformulateanunconstrained minimizationproblemgiven J = 1 2 Z 2 0 h ^ d )]TJ/F25 10.9091 Tf 10.909 0 Td [(u r ; i 2 d + Z 2 0 Z 1 h " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r u # drd: .2.2 theunknownfunction h appearsatthelowerboundary.Beforetakingtherstvariationsof theabovecostfunctional,itispossibletoconsidertherstordervariationsatthelowerboundaryaccordingto J = 1 2 Z 2 0 h ^ d )]TJ/F25 10.9091 Tf 10.909 0 Td [(u r ; i 2 d + Z 2 0 Z 1 ^ h + h " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u # drd: .2.3 where, h istherstordervariationsatthelowerboundary.Rewritingtheaboveequation leadsto J = 1 2 Z 2 0 h ^ d )]TJ/F25 10.9091 Tf 10.909 0 Td [(u r ; i 2 d + Z 2 0 Z 1 ^ h " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u # drd )]TJ/F31 10.9091 Tf 10.303 14.849 Td [(Z 2 0 Z ^ h + h ^ h " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u # drd: .2.4

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92 Takingtherstvariationsoftheabovecostfunctional,integratingvarioustermsbyparts,and usingthefactthatthevariation u isarbitraryonecanarriveattheadjointequationgivenby r 2 rr + 4 r )]TJ/F25 10.9091 Tf 34.921 7.38 Td [(R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r ! r + 2 )]TJ/F25 10.9091 Tf 37.459 7.38 Td [(R 2 r cos + R 2 )]TJ/F15 10.9091 Tf 15.834 8.073 Td [(1+ )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r cos )]TJ/F15 10.9091 Tf 5 -8.836 Td [(cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r 2 ! )]TJ/F15 10.9091 Tf 26.363 7.38 Td [(sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r + =0 ; .2.5 withtheappropriateboundaryconditionsgivenby ; = ^ d )]TJ/F25 10.9091 Tf 10.909 0 Td [(u r ; ; ^ h ; =0 : .2.6 Aftereliminatingvarioustermswearriveatthevariationsofthecostfunctionalgivenby J = )]TJ/F31 10.9091 Tf 10.303 14.849 Td [(Z 2 0 Z ^ h + h ^ h " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r u # drd; 2 [0 ; 2 ] : .2.7 Notethat ^ h ; =0.However,assumingalinearvariationoftheintegrandoverthesmall regionof h leadsto J = )]TJ/F31 10.9091 Tf 10.303 14.849 Td [(Z 2 0 " " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.541 Td [(R r u ## r = ^ h + 4 r dh; 2 [0 ; 2 ] ; afterwhich, J h = )]TJ/F31 10.9091 Tf 10.303 18.655 Td [(" " r 2 u rr + R cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r u r + u + sin cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r u ## r = ^ h + 4 r d; 2 [0 ; 2 ] ; .2.8 wheretheterminthesquarebracketisevaluatedat ^ h + 4 r where 4 r isthelength ofthenumericalmeshin r .Inaniterativealgorithm,theabovegradientfurnishesthedirectionforthenextvalue.Theparameter isintroducedtoscalethedistancetakenalongthe computeddirection.Therefore,aniterativealgorithmbasedontheminimizationofacostfunctionalcanbeformulatedaccordingto Assumeavalueforthelocationoftheboundary ^ h . UsetheDirichletconditionsin.1.5and.1.6andsolvetheforwardproblem inEq..1.4.Obtaintheerrordierencebetweenthegivengradientat r =1

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93 andcomputedvaluefortheguessedvalue.Thisfurnishestheboundaryconditionfortheadjointvariable r; giveninEq..2.6 Solvetheadjointequationin.2.5,andobtainthegradientdirectionin.2.8. Updatetheassumedvalueaccordingto ^ h = ^ h + J h . Thiscompletestherstmethodwhichisbasedontheminimizationofacostfunctional.We nextusethespecialpropertyoftheproblemathandandobtainadierentmethodtorecover theinteriorboundaryofthevacuuminaTokamak. 8.3Interpolation Withinthevacuumsurroundingtheplasma,thepoloidaluxreachesitshighestvalueatthe interiorboundary h whichisaconstantvalue.Thelocationoftheboundaryisunknown. Foranygiven ,thelowestvalueisattainedattheouterboundary r =1whichisknown,and thehighestvalueisreachedattheinteriorboundary, h [104].Theinverseproblemthat canbeconsideredistotrytorecovertheuxdistributionata xed ,andctitiousandknown boundary.Inotherwords,recovertheuxdistributionwithintheregionenclosedbytwoconcentriccircles,asshowninFig.8.3.Thenwiththecomputedvalueoftheux,andknowing thatattheboundary u h ; = c ,thelocationoftheboundary h canbecomputed.ConsideracrosssectionofthetoroidalconnementshowninFig.8.3. Theouterboundaryofthevacuumisknown.Wecanconsideraninteriorandctitiousboundary r = w asshowninFig.8.3andcomputetheuxdistributionwithintwoconcentriccircles.Noconditionisgivenonthectitiousinteriorboundary,andbothDirichletandNeumannconditionsaregivenontheouterboundary.Thisproblemisstillhighlyill-posed. However,wecancomputethepoloidaluxattheinteriorboundary,andthen,interpolatefor thelocationoftheactualboundary.Inthissection,weapplyaniterativealgorithmtorecover theinteriorboundaryconditioni.e., u !; = g ,for 2 [0:2 ].Itconsistsofthreesteps.

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94 FIGURE8.3:Acrosssection Assumeaninitialvaluefortheunknownboundarycondition,i.e., u !; =^ g and,usingthegivenDirichletboundaryconditionon u ; = d ,obtaina backgroundeldsatisfyingthesystem r 2 ^ u rr + R cos + )]TJ/F26 7.9701 Tf 6.195 -4.542 Td [(R r ^ u r +^ u + sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.542 Td [(R r ^ u =0 ; r; 2 ^ ; .3.1 where ^ istheannulusregionbetweenthetwoconcentriccircles r = ! and r = 1. SubtractthebackgroundeldfromEq..1.4,andobtaintheerroreld, e r; = u r; )]TJ/F15 10.9091 Tf 11.608 0 Td [(^ u r; ,givenby r 2 e rr + R cos + )]TJ/F26 7.9701 Tf 6.196 -4.542 Td [(R r e r + e + sin cos + )]TJ/F26 7.9701 Tf 6.196 -4.542 Td [(R r e =0 ; r; 2 ^ : .3.2 SincethebackgroundeldsatisestheDirichletboundarycondition,thebound-

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95 aryconditionsfortheerroreldaregivenby e ; =0 ;e r ; = ^ d )]TJ/F15 10.9091 Tf 11.608 0 Td [(^ u r ; ; e !; = g )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g = ; 2 [0 ; 2 ] .3.3 where, isstillunknown. Obtaincorrectionstotheassumedvaluefortheunknownfunction.Theassumed valueofthefunctionisrelatedtotheactualvalueaccordingto g x =^ g + . Theabovethreestepsformastandarditerativealgorithm.Thethirdstepcanbeachievedusinganewmethodwhichwerefertoaspropersolutionspaceinpreviouschapters. 8.4ProperSolutionSpace Torecover for 2 [0 ; 2 ]wecanproceedasfollow.Consideralinearlyindependentset offunctions c ` ;` =1 ; 2 ;:::;N over 2 [0 ; 2 ]andassumethattheunknownfunction canbeexpressedasalinearcombinationofthesefunctions,i.e. 2f c 1 ;c 2 ;:::;c N g .Next, generateasetoffunctionsthatsatisfytheerroreldequation.3.2withtheknownzero Dirichletboundarycondition,i.e., r 2 ` rr + R cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r ` r + ` + sin cos + )]TJ/F26 7.9701 Tf 6.195 -4.541 Td [(R r ` =0 ; .4.1 with ` ; =0 ; ` !; = c ` .Therefore,everyfunction ` r; for r; 2 ^ satisesthe Dirichletboundaryconditionsfor r =1.Itisthenpossibletoexpandtheactualandunknown erroreld e r; inthespanofthespacegeneratedby ` r; ;` =1 ; 2 ;:::;N accordingto e r; = N X ` =1 ` ` r; ; r; 2 ^ ; .4.2 wherethefunctions ` r; areknown,buttheconstants ` areunknown.Wenextarguethat theerroreld e r; mustsatisfythegradientconditionthatisfurnishedbythemeasurements, i.e., e r ; = ^ d )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u r ; ; 2 [0 ; 2 ] : .4.3

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96 Thegradientconditioncanbeexpressedbytheoperator B andtheerroreldisrequiredto satisfytheconditiongivenby B e r; = B N X ` =1 ` ` r; = N X ` =1 ` B ` r; = ^ d )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u r ; ; 2 [0 ; 2 ] : .4.4 Theaboveequationcanbeusedtoobtaintheunknowncoecients.Oncetheunknowncoecientsareobtainedtheunknownboundaryconditioncanbeobtainedaccordingto = e r; j r = ! = N X ` =1 ` ` r; j r = ! = N X ` =1 ` c ` : .4.5 ThelastequalityinEq..4.5holdsbecause ` !; = c ` ,for ` =1 ; 2 ;:::;N byconstruction.Thiscompletesthethirdstepoftheiterativealgorithm.Oncethepoloidaluxisevaluatedwithintheconcentriccircles,theactualvalueoftheux u h ; = c canbeusedto tracethelocationoftheinteriorboundary. 8.5NumericalResults Inthissectionweuseanumberofnumericalexamplestostudytheapplicabilityofthetwo methodspresented.Bothmethodsrequireaccuratenumericalsolutionoftheellipticsystem withinanirregulardomain.Anappropriatecoordinatetransformationisgivenby = ; = r )]TJ/F25 10.9091 Tf 10.909 0 Td [(h 1 )]TJ/F25 10.9091 Tf 10.909 0 Td [(h ; 2 [0 ; 2 ] ; 2 [0 ; 1] : .5.1 Accurateresultscanbeobtainedbyusing50equalintervalsin and200equalintervalsin .Werstgeneratethedataandaddnoisetoprovidearealisticdatafortheinversion.The noiseisazero-meanrandomgeneratedGaussiannoise.WeareusingFourierseriestolterout thehigh-frequencyuctuations.Fortheexamplesinthisnote,weareusing100termsinthe Fourierseriestoexpandthedataandkeeptherst50terms. 8.5.1AdjointMethod Fortheadjointmethodthealgorithmgiveninsection8.3canbefollowedStartingwithaguess fortheinternalboundary ^ h ,Eq..1.4withDirichletconditionsgiveninEqs..1.5and

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97 .1.6canbesolved.Theassumedvalueiscertainlydierentthantheactualvalue.Theerror attheboundarycanbecomputedusingEq..2.6.Thisisalsotheboundaryconditionforthe adjointequationgivenin.2.5.Aftersolvingtheadjointequation,thedirectionofgradient canbecomputedwhichisgiveninEq..2.8.Itisalsonecessarytointroducesomeformof regularization.Ateveryiterations,onceagradientdirectionisobtained,regularizationisobtainedaftersolvingthefollowingover-determinedlinearsystem 2 6 4 I )]TJ/F31 10.9091 Tf 7.545 39.27 Td [(3 7 5 q = 2 6 4 J h 0 3 7 5 ; .5.2 where, I istheidentitymatrixand)-333(istheTikhonovregularizationterm[107]page14which isarst-orderdierentialoperator @ @ .Thesolution q istheregularizeddirectionofgradient.Theassumedvalueoftheinternalboundaryisupdatedaccordingto ^ h = ^ h + q .5.3 where, isascalarparameterwhichcanbeobtainedafteraone-dimensionalsearch[108].For allthenumericalresultsinthisnoteweareusing =0 : 7.Theouterboundaryisaccessibleand bothDirichletandNeumanndatacanbeobtain.ItispossibletoimposeaDirichletcondition ontheexteriorboundarygivenby u ; =0 : 5+0 : 01exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F25 10.9091 Tf 10.909 0 Td [( 2 1 : 5 : .5.4 Thevalueoftheuxattheinteriorboundaryissetat u h ; =5 : .5.5 Example1. Considertheevaluationofaninternalboundarythatisgivenby h = s +0 : 1exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 4 : 025 +0 : 1exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 : 1 4 : 02 .5.6 Figure8.4showsthenumericallygenerateddataattheboundary.Figure8.5showstherecoveredinternalboundaryforthreedierentvalueofs.Astheinternalboundarygetsclosertothe

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98 FIGURE8.4:Normalderivativeattheboundarydata,andnoisydata. externalboundarythequalityoftherecoveredinternalboundaryimproves.Figure8.6shows thereductionintheerrorasanumberofiterations.Theerrorisgivenby Error= R 2 0 h ^ d )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u r ; i 2 d R 2 0 h ^ d )]TJ/F15 10.9091 Tf 11.608 0 Td [(^ u r ; i d Firstiteration .5.7 Figure8.7showsthenormalizederrorintherecoveredinteriorboundarygivenby NormalizedError= h )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h h : .5.8 Thequalityoftherecoveredinteriorboundaryimprovesastheinteriorboundaryiscloserto theouterboundarywherethedataiscollected.Forthesecasesthelevelofnoiseissetat2%. Wenextlets=0.7andstudytheeectofnoiseontherecoveredinteriorboundary.Figure8.8 showstherecoveredboundaryforthreelevelsofnoise.Namely,0.02%,0.05%and0.08%.Figure8.9showsthereductionintheerrorforthesethreelevelsofnoiseasfunctionsofthenumberofiterations,andgure8.10showstherelativeerrorintherecoveredfunctions.

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99 FIGURE8.5:Recoverunknowninteriorboundaryfortheexample1. 8.5.2InterpolationandProperSolutionSpace Wenextapplythemethoddevelopedinsection8.3.Inthismethod,werstcomputetheux withinaringthatincludesthevacuum.Theringistheareabetweentwoconcentriccircles. Oneistheouterboundaryr=1whichisaccessibleanddatacanbecollected.Theotheris r = ! wherethevalueofthepoloidaluxisunknown.Oncetheuxiscomputedwithinthis region,thenusingtheconditionthat u h ; = c here c =5thelocationoftheinterior boundary h canbecomputed.Theproblemofcomputingtheuxwithinthexedringis

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100 FIGURE8.6:Reductioninerrorfortheexample1. FIGURE8.7:Normalizederrorintherecoveredinteriorboundaryfortheexample1.

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101 FIGURE8.8:Recoveredunknowninteriorboundary:example1,eectofnoise. stillhighlyill-posed. Example2. Considertheevaluationofaninternalboundarythatisgivenby h = s +0 : 1exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 : 5 4 : 015 +0 : 1exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 : 51 4 : 015 .5.9 ForthisexamplewearealsousingtheDirichletconditiongivenas u ; =0 : 5+0 : 01exp )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F25 10.9091 Tf 10.909 0 Td [( 2 1 : 5 .5.10 Thedataisalsocollectedinthesameway.Assumingavalueforthepotentialattheinterior

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102 FIGURE8.9:Reductioninerror,example1,eectofnoise FIGURE8.10:Normalizederrorintherecoveredinteriorboundary,example1,eectofnoise

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103 boundary r = ! ,i.e.,^ u !; =^ g thethreestepalgorithminsection8.3canbeinitiated. AfterusingthegivenDirichletconditionsandcomputingthebackgroundeld,itleadstothe erroreldgiveninEq..3.2.TheboundaryconditionsfortheerroreldaregiveninEq. .3.3,wherethefunction isstillunknown.Thethirdstepofthisalgorithmisachieved withtheapplicationofpropersolutionspace.Thismethodcallsforthegenerationofasetof linearlyindependentfunctionseachofwhichsatisestheDirichletconditionat r =1given inEq..3.3.Inordertogeneratethesolutionsetanappropriatesetoflinearlyindependent functions c ` over 2 [0 ; 2 ]isgivenby c 1 =0 : 1 ;c j =0 : 1sin j ;c ` =0 : 1cos ` .5.11 for j =1 ; 2 ;:::;J and ` =1 ; 2 ;:::;J ,where, N =1+ J + J andthefunctionsaresimplythe Fouriesfunctionswhichformalinearlyindependentsetover[0 ; 2 ].Withthefunctions c ` specied,thepropersolutionspacecanbeobtained.ThesefunctionsarethesolutionsofEq. .4.1.For j =1 ; 2 ;:::;N theseboundaryvalueproblemsarewell-posedandthepropersolutionspace ` r; ; 8 ` =1 ; 2 ;:::;N canbecomputed.Thenitedierenceapproximationtothese ellipticproblemssimplifytolinearsystemsgivenby A " ` = c ` or ; " ` = A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 c ` ; for ` =1 ; 2 ;:::;N; .5.12 where A isthenite-dimensionalapproximationoftheellipticoperatorinEq..4.1witha uniqueinversesincetheproblemiswell-posed, " ` isthevectorcontainingthenodalvaluesof thefunction ` r; and c ` isthevectorthatcontainsthenodalvaluesoftheboundarycondition c ` .Itisalsoeasytoshowthatthevectors " ` ,for ` =1 ; 2 ;:::;N ,arelinearlyindependent A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 existsand c ` ,for ` =1 ; 2 ;:::;N ,arelinearlyindependent.Next,istoexpandthe erroreldinthespacegeneratedby " ` accordingtoEq..4.2or, e = N X l =1 ` " ` .5.13 where e isthevectorcontainingthenodalvaluesoftheerroreld e r; .ThegradientboundaryconditionfortheerroreldinEq..4.3canbeusedtocomputetheunknowncoecients

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104 l .Denotingtheappropriatenitedimensionalapproximationtothegradientoperatorat r =1 by B leadsto Be = b ,where b isthevectorthatcontainsthevaluesofthegradientboundary conditiongiveninEq..4.3.Substitutingfor e fromEq..5.13leadsto Be = B N X ` =1 ` " ` = N X ` =1 ` B " ` = b ; or ; [ B " ` ] = b ; .5.14 where 2 R N istheunknownvectorcontainingthecoecients ` .Intheaboveequationthe vector b isknown,thesolutionspace " ` for ` =1 ; 2 ;:::;N arealsoknown.Thecoecients ` areunknown.Theabovesystemcanbetreatedasalinearsystemfortheunknowncoecients ` for ` =1 ; 2 ;:::;N thatareplacedinthevector .Theabovecoecientmatrix,[ B " ` ],isnot square.Thesquarematrix B issingular,andasexpectedtheabovesystemcannotbereadily solvedforthecoecients ` .AspointedoutinEq..4.5,theunknownfunctionbeexpanded accordingto x = P N ` =1 ` c ` .Itisthenpossibletostabilizetheinversionbyimposingsome formofTikhonovregularization[107]or[109]page68,accordingto 2 6 4 B " ` 3 7 5 = 2 6 4 b 0 3 7 5 ; .5.15 where,thematrix representtherst-orderoperatorgivenby d d = k = J X j =1 j ` cos j k )]TJ/F26 7.9701 Tf 16.1 13.636 Td [(J X ` =1 ` ` sin ` k .5.16 evaluatedatdiscretelocations k alongthedomain.Theconstant > 0issetbythedesigner, andtheabovelinearsystemcanbesolvedfortheunknowncoecients ` for ` =1 ; 2 ;:::;N . Thedimensionofthevectorsinthesolutionsetis51 200.Using J =15,itfollowsthat thereare N =31linearlyindependentvectorsinthesolutionspace.Forthisexamplesweset = : 04.Figure8.11comparesthegenerateddatawiththedatausedinthecalculationwith 0.09%noise.Figure8.12presentstherecoveredinteriorboundariesforthreedierentvalueof s ,i.e., s =0 : 4 ; 0 : 6 ; 0 : 7andcomparesthemtotheiractualvalue.Thelevelofnoiseissetat 0.03%.Theaccuracyoftherecoveredboundariesimproveastheinteriorboundarygetscloser totheexternalboundary.Figure8.13showsthereductionintheerrorasfunctionsofthenum-

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105 berofiterationsforthethreeinteriorboundaries,andgure8.14showtherelativeerrorforthe recoveredfunctions. FIGURE8.11:Normalderivativeattheboundarydata,andnoisydata. Wenextstudytheeectofnoiseinthedata.Figure8.15showstherecoveredinteriorboundariesforthreedierentlevelsofnoise,andgure8.16showsthereductionsintheerrorasfunctionsofthenumberofiterations.Figure8.17presentsthenormalizederrorfortherecovered interiorboundaries. 8.6Remark Inthischapter,wepresentedtwocomputationalmethodstorecovertheinteriorboundaryof thevacuumthatenclosestheplasma.Thetwoalgorithmsareconceptuallydierentandhave dierentadvantages/disadvantages.Theadjointmethodissimpletosetup,andcomputationallyisabitfaster.However,itisbasedonthelinearizedrst-ordernecessaryconditionsonly. Thesecondmethodisbasedontheideaofsolvinganill-posedinverseboundaryproblemrst,

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106 FIGURE8.12:Recoveredunknowninteriorboundaryfordierentvalueof s . andtheninterpolatefortheinteriorboundary.Bothmethodshavebetterresults,whentheunknownboundaryisclosertotheexternalboundary.Ournumericalresultsalsoindicatedthat theybothhavegoodrobustnesstonoise.

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107 FIGURE8.13:Reductioninerrorfordierentvalueof s . FIGURE8.14:Therelativeerrorfortherecoveredfunctionsfordierentvalueof s .

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108 FIGURE8.15:Recoveredunknowninteriorboundaryfordierentvalueofnoisy.

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109 FIGURE8.16:Reductioninerrorfordierentvalueofnoisy. FIGURE8.17:Therelativeerrorfortherecoveredfunctionsfordierentvalueofnoisy.

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110 CHAPTERIX CONCLUSION Inthisdissertation,wepresenttwocomputationalmethodsforevaluatingvedierentinverse heatconductionproblems.Therstproblemistheinverseevaluationoftheunknowninitial conditionforaparabolicsystem.Secondandthirdproblemsareinverseevaluationofabsorptionanddiusioncoecients.Thefourthproblemistheinverseevaluationofcoecientsfor aphotondiusionequation,andthefthproblemistheinverseevaluationoftheboundaryof toresuprainaTokamakthatariseinmanyapplications.Itiswell-knownthattheseproblems arehighlyill-posed,andvariousmethodshavebeendevelopedtodealwiththeill-posedness. Therstmethodisbasedonpropersolutionspace.Thealgorithmisiterativemethodinnature,whichstartswithaninitialguessfortheunknownfunctionandobtainscorrectionsofthe assumedvalueineachiterationstep.Theupdatingstepisthenewfeatureofthepresentalgorithm.Thismethodgeneratesasetoffunctionsthatsatisfysomeoftheboundarycondition.It thenusestheremainingboundaryconditiontoupdatetheassumedvalue.Thesecondmethod isbasedonadjointmethod,whichseekstominimizeacostfunctional.Thismethodisalsoan iterativemethodandcomputationallyfeasible. Therstmethodhasbeenusedtosolveallveproblems.ThesecondmethodisusedinevaluatingtheboundaryshapeofthevacuuminaTokamak,whichispresentedinchapterVIII.The twoalgorithmsareconceptuallydierentandhavedierentadvantages/disadvantages.Theadjointmethodshowstheadvantageoncomputationallyecientthantherstmethod.However, itisbasedonthelinearizedrst-ordernecessaryconditionsonly.Therstmethodisbasedon theideaofsolvinganill-posedinverseboundaryproblemrst,andtheninterpolatefortheinteriorboundary.Themethod,whichisbasedonpropersolutionspace,hasawidlyuseinsolv-

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111 ingdierentkindsofinverseproblem,however,theiscomputationallyinecientthanadjoint method.Bothalgorithmsshowgoodrobustnesswhichcanbeusedtoobtainagoodestimation oftheunknownfunction.Numberofnumericalexamplesfordierentinverseproblemsareused toshowtheapplicabilityofthemethods.

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