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Enhanced surrogate modeling with adjoints for goal-oriented measure-theoretic inversion

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Enhanced surrogate modeling with adjoints for goal-oriented measure-theoretic inversion
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Butler, Troy
Mattis, S.
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Many approaches for solving stochastic inverse problems use some form of finite sampling, which leads to a type of stochastic error in solutions. When computational models are expensive to evaluate, surrogate response surfaces are often employed to increase the number of samples used in approximating the solution. The result is generally a trade off in errors where the stochastic error is reduced at the cost of an increase in deterministic/discretization errors in the evaluation of the surrogate. In this work, we reformulate a recently developed measure-theoretic approach for solving stochastic inverse problems in terms of a special class of surrogate response surfaces. We prove that the computational algorithm developed in a separate work produces an exact answer under certain conditions of an implicitly defined surrogate response surface and the exact response surface. This motivates the use of adjoint techniques to enhance the local approximation properties of the surrogate. Several examples demonstrate how this local enhancement of surrogates produces more accurate approximations of probabilities of input events at a decreased computational cost.
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DRAFT c r xxxxSocietyforIndustrialandAppliedMathematics Vol.xx,pp.x x{x Enhancedsurrogatemodelingwithadjointsforgoal-orient edmeasure-theoretic inversion T.Butler and S.Mattis y Abstract. Manyapproachesforsolvingstochasticinverseproblemsus esomeformofnitesampling,which leadstoatypeofstochasticerrorinsolutions.Whencomput ationalmodelsareexpensiveto evaluate,surrogateresponsesurfacesareoftenemployedt oincreasethenumberofsamplesusedin approximatingthesolution.Theresultisgenerallyatrade oinerrorswherethestochasticerror isreducedatthecostofanincreaseindeterministic/discr etizationerrorsintheevaluationofthe surrogate.Inthiswork,wereformulatearecentlydevelope dmeasure-theoreticapproachforsolving stochasticinverseproblemsintermsofaspecialclassofsu rrogateresponsesurfaces.Weprove thatthecomputationalalgorithmdevelopedinaseparatewo rkproducesanexactanswerunder certainconditionsofanimplicitlydenedsurrogaterespo nsesurfaceandtheexactresponsesurface. Thismotivatestheuseofadjointtechniquestoenhancethel ocalapproximationpropertiesofthe surrogate.Severalexamplesdemonstratehowthislocalenh ancementofsurrogatesproducesmore accurateapproximationsofprobabilitiesofinputeventsa tadecreasedcomputationalcost. Keywords. uncertaintyquantication,aposteriorierrorestimate,M onteCarlomethods,inverseproblems, surrogatemodeling,adjointproblems 1.Introduction. Inthiswork,weconsiderarecentlydevelopedmeasure-theo reticapproachforsolvingstochasticinverseproblemsoriginally introducedandanalyzedin[ 8 13 14 15 ].Theapproachhassincebeenemployedtoquantifyuncertai ntiesinstormsurgemodels[ 15 16 33 ],subsurfacecontaminanttransport[ 43 ],andinstructuraldamageofvibrating beams[ 17 ].Were-interpretthesamplebased,non-intrusive,comput ationalalgorithmintroducedin[ 15 ]intermsofsolvingthestochasticinverseproblemusingim plicitlydenedsurrogateresponsesurfaces.Theendresultisanon-parametricp robabilitymeasurethatapproximatelysolvesthestochasticinverseproblemfortheexactr esponsesurface.Wesubsequently provethatthealgorithmproducesexactsolutionstothesto chasticinverseproblemunder certainconditionsoftheimplicitlydenedsurrogateandt heexactresponsesurfaces.This motivatesthemajorcontributionsofthiswork,whichisthe useofadjointbasedtechniques toestimateandcorrectfornumericalerrorinthesurrogate whilesimultaneouslyincreasing thelocalorderofthesurrogateresponsesurface.Theuseof theresultingenhancedsurrogatesaretwo-foldwhereweobserveanincreaseinaccuracya nddecreaseincomputational complexityinthecomputationofprobabilitiesofspecied events.Perhapsthemostclosely relatedworkcanbefoundin[ 13 ],whichusedaposteriorierrorestimatestoderivebounds oncumulativedistributionfunctionsdenedbymeasure-th eoreticinversionrestrictedtoa classofinitialvalueproblems.However,thedierencestot hisworkaresubstantialincluding dierentclassesofdierentialequationsstudied,thealgor ithmicproceduresforcomputing bothQoIsurrogatesandthemeasure-theoreticsolution,an dthegoal-orientedapproachof measure-theoreticinversionconsideredherethatfocuses onthecomputationoftheprobability DepartmentofMathematicalandStatisticalSciences,Univ ersityofColoradoDenver,Denver,CO80202 ( Troy.Butler@ucdenver.edu ). y ZentrumMathematik,TechnischeUniversitatMunchen,Ga rching,Germany( mattis@ma.tum.de ). 1

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DRAFT 2 T.ButlerandS.Mattis ofaspeciedevent.Below,weprovideabriefreviewofrecen tworkdoneinsolvingstochastic inverseproblemsandsurrogatemodelingthatframesthiswo rkandlocatesitscontribution withintherapidlyadvancingeldofuncertaintyquantica tion(UQ)moreclearly. 1.1.RelationshiptoPriorandOtherWork. TheeldofUQhasdevelopedintoasophisticatedareaofresearchwithmanycontributorsoverth elastfewdecades.Considerable attentionandeorthasbeenfocusedonthedevelopmentofec ientandaccurateUQmethods.WhiletherearemanyUQmethodsthatsolveavarietyofre lated,butoftendistinct, stochasticinverseproblems,themostcommonlyusedmethod s(e.g.,basedonBayesianinference[ 21 49 48 ]orensembleKalmanltermethodsfordataassimilation[ 35 27 26 ])use (MonteCarlo[ 47 28 ]orMarkovChainMonteCarlo[ 32 29 ])samplingtechniquesintheparameterspacetointerrogatetheQoImap,whichintroducese rrorduetonitesampling.This errorisoftenexacerbatedwhennumericaltechniquesarere quiredtosolvethemodelsince nitecomputationalbudgetsoftenmakeitimpossibletosu cientlysampletheQoIresponse surface. Onewaytoreducetheeectofnitesamplingerroristoconstr uctasurrogatetotheQoI responsesurfacesuchthatsamplinghasalowercomputation alcost.Thepastseveraldecades haswitnessedtremendousadvancementinthedevelopmentan duseofsurrogatestopropagate uncertaintiesmakingafullreviewofsuchtechniquesvirtu allyimpossible.Whileitispossible totracebacktherootsofmuchoftheworkinvolvingstochast icniteelementapproaches [ 31 30 ]forbuildingsurrogatestotheseminalpapersofWiener[ 51 ]andCameronandMartin [ 46 ],therearenowmanyaccessiblestartingpointsthatusesim ilarideasforbuildingglobal polynomialapproximationsbasedonstochasticspectralme thods[ 53 50 37 38 42 11 10 12 45 9 ].Tensorgridandsparsegridstochasticcollocationmetho dsforbuildingsurrogateshave alsogainedinpopularity[ 2 44 ].Thereisalsointerestingnewresearchonusingdimension reductiontechniquesandreducedordermodelsforbuilding surrogates[ 20 ].Thesurrogate modelingapproachconsideredinthisworkmostcloselyrese mblestechniquesthatexploit derivativeinformationforbuildingpiecewiselow-orders urrogateapproximationstoimprove pointwiseaccuracyinpropagationsofuncertainties[ 42 24 11 8 ]. Asshownin[ 11 10 12 45 9 ],thesurrogateresponsesurfaceispollutedbytwosources oferroraectingtheaccuracyofallsamples:(1)theapproxi mationerrorofthechoiceof surrogate;(2)thenumericalerrorinevaluationofthesamp lesusedtoconstructthesurrogate. Werefertoboththesetypesoferrorasdiscretizationerror s.Thus,usingasurrogatecan representatrade-obetweenreductioninnitesamplinger rorattheexpenseofanoverall increaseindiscretizationerror.Theendresultisthatour abilitytoaccuratelyquantify uncertaintiesbysolutionofastochasticinverseproblemm aybecompromisedbytheuseof surrogatesunlessadditionalstepsaretakentoreducethed iscretizationerrors. Thederivationofcomputableandaccurateaposterioriesti matesofdiscretizationerrors basedonvariationaltechniquesandadjointsdatesbacksev eraldecades[ 4 23 1 25 19 52 ].Suchtechniquesservedasthebasisoftheerrorestimates forpolynomialchaosand pseudospectralbasedsurrogatesderivedin[ 11 10 ].Subsequently,in[ 12 45 ],sucherror estimateswereusedaspartofaBayesianinferencetoquanti fyuncertaintiesonparameters toevolutionarypartialdierentialequationswhereQoIres ponsesurfaceswereapproximated withpolynomialchaostechniquesandenhancedbytheerrore stimates.

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 3 TheUQapproachconsideredinthisworkisdistinctfromtheo therUQmethodsmentioned aboveinavarietyofways.Weinvitetheinterestedreaderto review[ 8 14 16 43 ]formore detailsonthecomparisonsofthemeasure-theoreticapproa chtoothermethodsforformulating andsolvingstochasticinverseproblemssuchasBayesianin ference,regularization,anddata assimilationtechniques.Themaindierencesarethattheme asure-theoreticapproachhas minimalassumptionsandworkswiththedirectinversionoft heQoImapevenwhenitis denedbyaset-valuedfunction.Theminimalassumptionsar e(1)themeasurabilityofthe QoImap,and(2)specicationofmeasure/probabilityspace sontheparametersandoutput datadenedbytherangeoftheQoImap.Practicalcomputatio nalconsiderationsgenerally requirethatwefurtherassumetheQoImapisatleastpiecewi sesmooth,whichisconsistent withwhatisassumed,atleastimplicitly,bymostothermeth ods.Exploitingthegeometric informationintheset-valuedinversesoftheQoImaptocons tructnon-parametericprobability measuresontheparameterspaceisadistinguishingcharact eristicofthisapproachfromothers, andweprovidemoredetailsinSection 3 .Moreover,thereiscurrentlynootherworkstudying theeectofusingadjointbasedtechniquestoenhancesurrog atemodelsdesignedspecically forreducing,simultaneously,thevarioussourcesoferror inthesamplebasedapproximation tothesolutionofthemeasure-theoreticinverseproblem. 1.2.Outline. Thispaperisorganizedasfollows.Weprovidesomegeneraln otation, terminology,andassumptionsusedinthisworkinSection 2 .InSection 3 weprovideabrief summaryofthetheorybehindthemeasure-theoreticformula tionandsolutionofastochastic inverseproblem,andinSection 4 wesummarizetherandomsamplebasedapproximationto measure-theoreticsolutionsanderroranalysis.InSectio n 5 ,wedescribetheabstractprocess ofconstructingsurrogateapproximations,identifytheva rioussourcesoferrorinthesurrogate, anddescribetheimplicitconstructionofapiecewiselow-o rdersurrogatebasedontherandom samplemeasure-theoreticsolutiontoastochasticinverse problem.Wesubsequentlyprovide theconditionsrelatingtheexactandsurrogateresponsesu rfacesforwhichtheapproximation solutiontothestochasticinverseproblemisinfactexact. Abriefreviewofadjointbased aposteriorierrorandderivativeestimatesalongwithalis tofusefulreferencesareprovided inSection 6 .InSection 6 ,wealsodescribehowweusesucherrorestimatestoenhance surrogatesbycorrectingforpersistentlocalbiasesdueto discretizationerrors.Weprovide somenumericalexamplesinSection 7 ,andconcludingremarksfollowinSection 8 2.Notation,Terminology,andAssumptions. Weassumethata(deterministic)model M ( u ; )=0ofaphysicalsystemisgivenwhere u denotesavectorofstatevariablesdeterminedbythesolutionofthemodelforaspeciedvectorof (model)parameters ,which mayincludeinitialconditions,boundarydata,orsourcete rms.Inotherwords,specication of determinesthesetupofthesystembeingmodelled.Weassume thespaceofpossible parameters,denotedby ,isalsoknown.Then,foreach 2 ,thesolutionoperatorof M ( u ; )=0determinesaparticular u ( ),wherewemakethedependenceofthesolution onthemodelparametersexplicit.InFigure 2.1 ,weillustratethemappingbyconnectinga particularsample,labeled ,intheparameterspacetoaparticularsolution, u ( ),bythe arrowlabeledwith M ( u ; ). Thevectorof(linear)functionalscorrespondingtomodelo bservablesdenesaQoImap, denotedby Q ,and Q ( u ( ))representsaparticularoutputdatumassociatedtoapart icular

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DRAFT 4 T.ButlerandS.Mattis Figure2.1:Schematicofmappingsbetweenrelevantspaces. choiceofparametersdenedby .Wethenwrite Q ( ):= Q ( u ( ))toemphasizeboththe dependenceoftheoutputdataonthemodelparametersandthe factthatinanexperimental settingwemaybeabletocontrol toobserve Q ( )withoutfullyobserving u ( ).Welet D := Q ( )denotethespaceofmodelobservables.Thisisillustrated inFigure 2.1 wherea particularsolution u ( )inthemiddleismappedbythearrowlabeledbytheQoImaptoa modelobservableontherightdenotedby Q ( u ( )). Exceptinrarecases,wegenerallycanonlyobserveaselectf ewQoI,i.e.,themodel observablespaceistypicallyalow-dimensionalspacewher easthestatespaceofthemodel maynominallybeininnitedimensions.Inthiscase,whenth ephysicalsystemisbeing observed,wecanoftenonlyrecordinformationofthemapbet ween and D denedby Q ( ) asillustratedbythelongerarrowlabeledby Q ( )inFigure 2.1 .Thisisalsooftentruein caseswherethemodeliscomputationallycomplex,anditisn otpossibletostoretheentire numericalstatespaceofthemodel. Weassumethat( ; B ; )and( D ; B D ; D )aremeasurespaceswhere and D are (volume)measuresusedtodescribesizesofeventsinthe -algebras B and B D ,respectively. Wealsoassumethat Q isatleastpiecewisedierentiable,whichimpliesmeasurab ilityofthe QoImap.Unlessotherwisestated,weassumethat R n and D R d Inthecasewherethemodelissolvednumerically,wegeneral lycompute Q h ( ):= Q ( u h ( ))) Q ( ).Here, u h ( )denotesanumericalapproximationto u ( ),and h denotesa discretizationparameter(e.g.,determinedbythemeshsiz eornumberofmaximumiterations inthesolutionmethod).Then, u h ( )= u ( )+ u;h ( ) ; where u;h ( )isthenumericalerrorin u h ( ).Wealsowrite Q h ( )= Q ( )+ Q;h ( ) ; where Q;h ( )isthenumericalerrorin Q h ( ).Wemayviewtheseerrorsasatypeofperturbationfromtheexactsolutionordata.Anapriorierrora nalysisoftencanbeusedto

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 5 determineboundson u;h ( )and Q;h ( ),whichdenethemaximumperturbationstotheexactsolutionsanddatawecanexpectfromnumericallysolvin gthemodel.Thisisillustrated bytheshadedareasaroundtheexactsolutionsandQoIdatain themiddleandrightspaces ofFigure 2.1 whichindicatesthemagnitudeofpossibleperturbationsde nedbysuchana priorierroranalysis. 3.Measure-TheoreticInversion. Wehighlightonlythemainideasbehindtheformulationandsolutionofameasure-theoreticstochasticinvers eproblem,anddirecttheinterested readerto[ 8 13 14 15 ]formoredetails.Thespecicationofthevariousmeasure/ probability spacesinvolvedinthisprocess,aswellastheordertheyare introduced,iscriticaltothe properdescriptionofboththeproblemandthedesiredsolut ion.Wethereforerefertothe schematicofFigure 3.1 toillustratethemainstepsandmeasure/probabilityspace sinthe formulationandsolutionofthemeasure-theoreticstochas ticinverseproblemconsideredhere. (S1):StochasticInverseProblem(SIP) z }| { ( ; B ; ) Q 7! ( D ; B D ; D ) P D 7! ( D ; B D ;P D ) Q 1 7! ( ; C ;P ) | {z } (S2):SolutiontoSIPSatisfyingEq.( 3.1 ) f P ` g ` 2L 7! ( ; B ;P ) | {z } (S3):UniqueSolutiontoSIPbyEq.( 3.2 )andAnsatz Figure3.1:Therststep(S1)givethenecessaryingredient sfortheformulationoftheSIPby specicationofthemodel,measurespacesofmodelparamete rsandmodelobservables,anda probabilitymeasureonmodelobservables.Then,thesecond step(S2)denesauniquesolutionon( ; C )usingthedenitionofapush-forwardmeasure.Finally,in (S3),anapplication oftheDisintegrationTheoremandanAnsatzuniquelydenes asolutionon( ; B ). Specicationofaprobabilitymeasure P D on( D ; B D )(e.g.,modelinguncertaintyinthe observeddata),denesaparticularstochasticinversepro blemofdeterminingaprobability measure P on( ; B )suchthatthepushforwardmeasureof P matches P D ,i.e., P ( Q 1 ( E ))= P D ( E ) ; 8 E 2B D : (3.1) When P D isabsolutelycontinuouswithrespectto D ,then,byanapplicationoftheRadonNikodymtheorem,wecanrewriteEq.( 3.1 )intermsofprobabilitydensityfunctionsand integrals.Wecallanysuch P satisfyingEq.( 3.1 )a(measure-theoretic)solutiontothe stochasticinverseproblem.NotethattheEq.( 3.1 )impliesthatanysolutionisuniquely determinedontheinduced -algebra C = Q 1 ( E ): E 2B D B : However,for A 2B nC ,werequiremoreinformationthanEq.( 3.1 )providesforcomputing P ( A ).Seethersttwosteps,labeledas(S1)and(S2)inFigure 3.1 ,foranillustrationof thispartoftheformulationandsolutionofthestochastici nverseproblem.

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DRAFT 6 T.ButlerandS.Mattis Forcontinuouslydierentiable Q andxeddatum q 2D Q 1 ( q )existsasa(possibly piecewisedened)( n d )-dimensionalmanifoldreferredtoasa generalizedcontour .There existsanindexing(possiblypiecewisedened)manifoldof dimension d calleda transverse parameterization thatintersectseachgeneralizedcontouronceandonlyonce .Welet L denoteanyparticulartransverseparameterization.Sinceea chpoint ` 2L correspondstoa uniquegeneralizedcontour,denoted C ` ,andviceversa,wehaveabijectionbetweenthe partitioningof intogeneralizedcontoursandthemanifold L .Theinduced -algebra C andthisbijectioncanbeusedtodeneameasurablespace( L ; B L ).Letting f C ` g ` 2L denote thefamilyofgeneralizedcontoursindexedby L ,wecandeneafamilyofmeasurablespaces f ( C ` ; B C ` ) g ` 2L .Then,anapplicationofaDisintegrationTheorem[ 14 22 ]impliesaunique decompositionofany P denedon( ; B )asamarginalprobabilitymeasureon( L ; B L ) andafamilyofconditionalprobabilitymeasures f P ` g ` 2L on f ( C ` ; B C ` ) g ` 2L suchthat P ( A )= Z L ( A ) Z 1 L ( ` ) \ A dP ` ( ) dP L ( ` ) ; 8 A 2B : (3.2) Theuniquenessofaprobabilitymeasureon( ; C )satisfyingEq.( 3.1 )impliesthatspecifying P D on( D ; B D )uniquelydenesthemarginal P L .Accordingtothedinsintegrationof Eq.( 3.2 ),specicationofafamilyofconditionalprobabilitymeas uresthenuniquelydenesa solutiontothestochasticinverseproblem.Following[ 14 ],weusethestandardansatzbased ontheuniquedisintegrationofvolumemeasuresthateectiv elyproportionsprobabilityalong contoureventsaccordingtotheirrelativesize,andresult sinauniquesolution P forthis choiceofansatz.Thisdenesthethirdsteplabeledas(S3)s howninFigure 3.1 4.NumericalAnalysisofComputationalErrorsinMeasure-TheoreticInver sion. 4.1.ANon-intrusiveComputationalAlgorithm. In[ 15 ],anon-intrusivesamplebased algorithmforproducingacountingmeasureapproximationt o P satisfyingEq.( 3.1 )using asamplesetofindependentidenticallydistributed(i.i.d .)uniformrandomsamplesin was developedandanalyzed.Anecessaryconditionforthecount ingmeasuretoproduceanontrivialprobabilityestimateofanevent A 2B wasthattheintersectionof A withthesample setwasnon-empty. Here,weuseAlgorithm 1 ,whichisavariantoftheoriginalsamplebasedalgorithm including -measureapproximationsoftheimplicitlydenedVoronoic ellsassociatedwith thesetofsamplesin .Thealgorithmproceedsinfourstageswrittenasfourdisti nctnonnestedfor-loopsthatcanbelinkedtothestepsdescribedin Figure 3.1 Thersttwostagestakentogethercorrespondtoadiscretes tatementofthestochastic inverseproblemillustratedbystep(S1)inFigure 3.1 .Intherststage,wediscretizetheprobabilityspace( D ; B D ;P D ).Then,thesecondstagediscretizesthemeasurespace( ; B ; ) andconstructsasimplefunctionapproximationtothemap Q ThethirdstageidentiesthefamilyofVoronoicellsin thatapproximatethecontour eventsdenedby Q 1 ( D i )for i =1 ;:::;M .Thisstagecanbeusedtoformulatetheconsistent solutionon( ; C ;P )shownasstep(S2)inFigure 3.1 Thefourthstagecorrespondstoadiscretizedapproximatio ntostep(S3)inFigure 3.1 ,and usesadiscreteformoftheansatztoapproximatetheprobabi lityof V j for j =1 ;:::;N .The

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 7 resultingapproximatemeasure,denotedby P ;M;N;h ,producesexactlythesameprobability estimatefortheevents A and A n ( j ) 1 j N ,whichareidentical -a.e.Wenotethat Algorithm 1 appliestoanydiscretizingsetofsamplesin nomatterhowthesamplesare generated.TheauthorsaredevelopersofBET[ 34 ],anopen-sourcePythonpackagefor stochasticmeasure-theoreticproblemswhichincludesimp lementationsofAlgorithm 1 Algorithm1: NumericalApproximationoftheInverseDensity Chooseadiscretizationpartition f D i g 1 i M of D for i =1 ;:::;M do Compute p D ;i = P D ( D i ). endChoosesamples ( j ) Nj =1 (implicitly)deningaVoronoipartition fV j g Nj =1 of for j =1 ;:::;N do Computeapproximations Q j = Q h ( ( j ) ) Q ( ( j ) ). Bin Q j inthepartition f D i g 1 i M andlet O j = f i : Q j 2 D i g Computeapproximations V j ( V j ). endfor i =1 ;:::;M do Compute C i = f j : Q j 2 D i g endfor j =1 ;:::;N do Compute p ;j = V j P k 2C O j V k p D ; O j endForany A 2B ,compute P ;M;N;h ( A )= X 1 j N p ;j V j ( A ) : (4.1) 4.2.IdenticationofErrors. Weassumethat P D isabsolutelycontinuouswithrespect to D ,whichimplies P D candescribedbyadensity D .Then, P D ( D i )= Z D i D D ; for i =1 ;:::;M: Inthiscase,wetypicallycomputetheapproximations p i givenintherstfor-loopofAlgorithm 1 usingMonteCarloapproximationsbysampling P D .Sincethesamplesaregenerated directlyon D ,approximating p i doesnotrequirenumericalsolutiontothemodel.Wethereforeassumethattheseapproximationscanbemadesucientl yaccurateforanydiscretization of D ,andneglecttheerrorinthediscretizationoftheprobabil ityspace( D ; B D ;P D ).Fora xed M associatedtoadiscretizationof( D ; B D ;P D ),welet P ;M denotethe exact solution tothestochasticinverseproblemassociatedwiththisdisc retizationandthediscreteformof theansatz.

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DRAFT 8 T.ButlerandS.Mattis Foreach N 2 N assume ( j ) 1 j N isasetofi.i.d.uniformrandomsamplesin ,and that V j = ( ) =N inAlgorithm 1 isusedforcomputing P ;M;N;h ( A ).Then,accordingto theconvergenceresultsfrom[ 15 ], P ;M ( A )=lim N !1 lim h # 0 P ;M;N;h ( A ) ; (4.2) wherebylim h # 0 P ;M;N;h ( A )wemeantheprobabilityof A computedusingAlgorithm 1 with Q j = Q ( ( j ) )foreach j =1 ;:::;N .Inpractice,weapplyAlgorithm 1 foronlyanite N (possiblyafterseveraliterationsofthealgorithm)andth enumericalerrorscannotbeneglected intheestimationof Q j .Theresultisanapproximationof P ;M ( A )by P ;M;N;h ( A ),andthe signederror e P ;M;N;h ( A )= P ;M;N;h ( A ) P ;M ( A ) ; (4.3) isingeneraluncomputable.Foreach A 2B ,let P ;M;N ( A ):=lim h # 0 P ;M;N;h ( A ) : Withthisnotation,wedecomposethesignederroras e P ;M;N;h ( A )=( P ;M;N;h ( A ) P ;M;N ( A )) | {z } e P ;M;h ( A ) ( P ;M;N ( A ) P ;M ( A )) | {z } e P ;M;N ( A ) : (4.4) Wereferto e P ;M;h ( A )asthe deterministicerror duetonumericalapproximationofthe solutionto M ( u ; )=0leadingtoamiss-classicationofVoronoicells.Seeth eleft-handplot inFigure 4.1 foranillustrationofthefollowingdescriptionofthiserr orterm.Inthecontext ofAlgorithm 1 ,foragiven1 l N ,suppose Q l = Q h ( ( l ) ) 2 D k forsome1 k M ,and Q l + Q;h ( ( l ) ) 2 D k 0 forsome k 0 6 = k .Then,forall j 2C k (and j 2C k 0 ), p ;j iscomputedusinganincorrectset ofindicesfortheVoronoicellsdescribingtheassociatedc ontourevents.Thus,foranyevent A 2B suchthat A \ Q 1 ( D k )or A \ Q 1 ( D k 0 ),theincorrectbinningof Q j contributes(a signed)errortotheapproximationoftheprobabilityof A describedby e P ;M;h ( A ). Wereferto e P ;M;N ( A )asthe nitesamplingerror fromusing N nitesamplesin toimplicitlydenetheVoronoicellsapproximatingthe M contourevents Q 1 ( D i )for i = 1 ;:::;M leadingtoamiss-classicationofVoronoimeasures.Seeth eright-handplotin Figure 4.1 foranillustrationofthefollowingdescriptionofthiserr orterm.Inthecontextof Algorithm 1 ,considerall1 k M suchthat p D ;k 6 =0.Ifforanyofthese k Q 1 ( D k ) 6 = f [ j 2C k V j g a.e. ; (4.5) thenthereexists A 2B D suchthat ( A \ Q 1 ( D k )) 6 = ( A \ [ [ j 2C k V j ]) :

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 9 Thiscanberephrasedintermsofthelocalapproximationerr orin -measureofsubsetsof anyVoronoicell V j whereif ( 2V j : Q ( ) = 2 D O j ) > 0 ; thentheeectofmiss-classicationofthissubsetof V j ontheapproximateprobabilityof A isdescribedbytheerrorterm e P ;M;N ( A ). Figure4.1:Acontourevent Q 1 ( D k ),withexactboundariesgivenbythesolidcurves,is approximatedbyimplicitlydenedVoronoicells.Onthelef t,wehighlighttheVoronoicells ofmiss-classicationdenedasanincorrectbinningof Q j duetodiscretizationerrors.The lightershadecorrespondstoacellthatshouldbelongto Q 1 ( D k )beingincorrectlyclassied, andtheoppositeistrueforthedarkershade.Ontheright,we highlighttheVoronoicells withmiss-classiedsubsetsofpositive -measureduetonitesampling.Thelightershadeof graycorrespondstoalocalover-approximationof Q 1 ( D k ),andthedarkershadecorresponds toalocalunder-approximationof Q 1 ( D k ).Theerrorsarenotmutuallyexclusive,anda cancellationoferrorsispossible. 5.SurrogateModelingforMeasure-TheoreticInversion.5.1.IdentifyingandDecomposingErrorsinGeneralSurrogateModeling. Let Q s ( ) denoteacomputationallyinexpensivesurrogateapproxima tionto Q ( ),so Q s ( )= Q ( )+ s ( ) ; where s ( )istheerrorinthesurrogate.Weuse Q s ( )tomoreecientlymaplargenumbers ofsamplesbetween and D asindicatedbythedottedarrowmappingbetweenthesespace sin Figure 2.1 .However,constructing Q s ( )oftenrequiresusingsomeparticularsetofsamples of Q h ( )basedonaspecictypeofsamplingin ,e.g.,usingapossiblydierentsetof randomsamplesorusingdeterministicsamplingapproaches suchassparsegrids[ 2 44 ].We let Q s;h ( )denotethisnumericallyconstructedsurrogateand s;h ( )denoteitserror.We decomposetheerroras s;h ( ):= s ( )+ h ( ) ;

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DRAFT 10 T.ButlerandS.Mattis where s ( )istheerrorinthechoiceofsurrogateduetolimitedapprox imationpropertiesof thesurrogate,and h ( )istheerrorinthesurrogatefromnumericalsolutionofthe model. Inotherwords,inpractice,thedottedarrowofFigure 2.1 isreplacedbyamappingthatis pollutedbymultiplesourcesoferrorthataectallsetsofpr opagatedsamplesbetweenthe spaces. 5.2.APiecewiseSurrogateInterpretationofMeasure-TheoreticInversion. Consider thesurrogate Q s; 0 ( )denedasasimplefunctiononasetofVoronoicells fV k g 1 k N s 0 dened byasampleset ( k ) 1 k N s 0 ,i.e., Q s; 0 ( )= X 1 k N s 0 Q h ( ( k ) ) V k ( ) : (5.1) ThefollowingisacorollarytoAlgorithm 1 thatprovidesausefulinterpretationofapproximate solutionstothestochasticinverseproblemobtainedbythe algorithm. Corollary5.1. Let ( j ) 1 j N beanysetofsamplesusedinAlgorithm 1 anddene Q s; 0 inEq. ( 5.1 ) fromthissampleset.If Q ( )= Q s; 0 ( ) ,and V j = ( V j ) for 1 j N in Algorithm 1 ,then P ;M;N;h ( A )= P ;M ( A ) ; 8 A 2B : Proof TheproofisalmostimmediatefromAlgorithm 1 wherewemakethesubstitutions oftheexactvaluesof V j = ( V j )and Q h ( ( j ) )= Q s; 0 ( ( j ) )inthesecondfor-loop.Since both s ( )=0and h ( )=0byassumption,theconclusionfollows. Below,weprovideavariantofthecorollarywherewereplace thepointwiseeverywhere equalityofthemap Q andthesurrogate Q s; 0 withequalityoncontourevents. Corollary5.2. Let ( j ) 1 j N beanysetofsamplesusedinAlgorithm 1 anddene Q s; 0 inEq. ( 5.1 ) fromthissampleset.Denoteby I thesetofallindicesfromthediscretization partition f D i g 1 i M inAlgorithm 1 suchthat Q ( )= Q s; 0 ( ) for 2 Q 1 ( D i ) ,andforeach i 2I suppose V j = ( V j ) forall j 2C i .Then, P ;M;N;h ( A )= P ;M ( A ) ; 8 A 2B \ [ i 2I Q 1 ( D i ) : Proof Theconditionofequalityonthespeciedcontoureventsimp liesthatthesecontour eventscannecessarilybewrittenastheniteunionoftheVo ronoicellsusedinconstructing P ;M;N;h .Theconclusionthenfollowsimmediatelybythediscretefo rmoftheansatzusedto dene P ;M Giventheabovetworesults,weidentifytheapproximatesol utionobtainedfromAlgorithm 1 asbeingthe exact solution(neglectinganyerrorinapproximating ( V j ))toa separatestochasticinverseproblemwherethemap Q ( )hasbeenreplacedby Q s; 0 ( ).We denotethisexactsolutiontothesurrogatemap Q s; 0 obtainedbyAlgorithm 1 as P ;M;s 0 whichimmediatelyprovesthefollowing Corollary5.3. IftheconditionsofCorollary 5.1 aremet,then P ;M;s 0 ( A )= P ;M;N;h ( A ) ; 8 A 2B :

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 11 Remark5.1 Wecanchangeallequalityof Q ( )and Q s; 0 ( )intheabovecorollariesto a.e.equalitiessubjecttotheconstraintthatthesampleru sedtoconstructthesamplesets ( j ) 1 j N neversamplesfromthesetofmeasurezeroonwhichthesemaps maydier. Ingeneral,thesizeofthesampleset ( k ) 1 k N s 0 forwhichwenumericallysolvethe modeltodetermine Q h ( )maybeconstrainedbyacomputationalbudget.If N s 0 istoosmall, thenthenitesamplingerrorissignicant.However,evalu ationofthesurrogateischeapand wecanchoosetouse Q s; 0 constructedfromthe N s 0 samplesasinEq.( 5.1 )withinAlgorithm 1 usingadierentsetof N N s 0 samples.Then,for1 j N ,evaluationof Q h ( ( j ) )is numericallydonevianearest-neighborsearchesof ( j ) to ( k ) 1 k N s 0 insteadofadditional numericalsolutionofthemodel.Whilethiscanbedonetomak ethenitesamplingerror negligible,thedeterministicerrorislikelysignicantl yworsesinceeachsampleisnowpolluted byboththeerror s ( )and h ( )presentinthesurrogate. 6.AdjointBasedAPosterioriErrorEstimationandSurrogateEnhancement. Wesummarizethegeneralvariationalanalysisanduseofadjointp roblemstoderiveaposteriorierror estimatesforQoI.Theinterestedreadershouldreferto[ 4 23 1 25 19 52 ]formoreinformationonthetheoryandimplementationofadjointbased aposteriorierrorestimatesin general,andto[ 11 10 12 45 9 ]formoreinformationontheapplicationtocertainclasses of surrogatemodels.Foramorethoroughintroductiontotheth eoryandapplicationofadjoints ingeneral,werecommend[ 40 41 36 18 ]. 6.1.Adjointbasedaposteriorierrorestimatesandderivatives. Forthesakeofsimplicity,weinitiallyassumethesolutiontothemodel M ( u ; )=0isdenedbythesolutionto theparameterizedlinearsystem A ( ) u = b ( ) ; (6.1) whereforeach 2 R m b ( ) 2 R n and A ( ) 2 R n n isinvertible.Then,foreach 2 thereexistsasolution u ( ) 2 R n .WealsoinitiallyassumetheQoImapisgivenbyasingle scalarfunctionaldenedby Q ( )= h u ( ) ; i where 2 R n and h ; i denotesthestandard Euclideaninnerproduct.TheadjointproblemtoEq.( 6.1 )is A ( ) > = ; (6.2) where ( )isthesolutiontotheadjointproblem(oftencalledagener alizedGreen'svector), and isdeterminedbytheQoIandindependentof .Supposeforaxed 2 we numericallysolveEq.( 6.1 )toobtain u h ( ) u ( )andsubsequentlycompute Q h ( ) Q ( ). Recallthat Q;h ( ):= Q h ( ) Q ( ) :

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DRAFT 12 T.ButlerandS.Mattis Withouttheexactvalueof Q ( ), Q;h ( )isuncomputable.Usingastandardvariational analysisandpropertiesofinnerproductsandlinearoperat ors,wehave Q;h ( )= h u h ( ) u ( ) ; i = D u h ( ) u ( ) ;A ( ) > ( ) E = h A ( ) u h ( ) A u ( ) ; ( ) i = h A ( ) u h ( ) b ( ) ; ( ) i : (6.3) When ( )isgiven,thenEq.( 6.3 )iscomputableandgivestheexacterror Q;h ( ).Generally, ( )isapproximatedby h ( )andreplacementof ( )with h ( )inEq.( 6.3 )givesa computableaposteriorierrorestimate,whichwedenoteby e Q;h ( ).Typically,wecompute h ( )usingahigherordermethodthanusedtocompute u h ( ). Let i denotethe i thcomponentofthevector for1 i m .Then,dierentiating Eq.( 6.1 )withrespectto i andfollowingasimilarsetofsteps,wearriveat @ i Q h ( )= h @ i b ( ) [ @ i A ( )] u ( ) ; ( ) i : (6.4) Thepartialderivativesof b ( )and A ( )canoftenbedeterminedbyalgorithmic/automatic dierentiation,e.g.,see[ 6 ].Subsequently,thisimpliesthatthegradientoftheQoIwi th respecttotheparameter ,denotedby r Q ( ),canbeapproximatedbysolvingboththe modelandadjointexactlyonceandthencomputinganitenum berofinnerproductsgiven byEq.( 6.4 ). Remark6.1 Theaboveapproachcanbeappliedtomostmodelsdenedbyali nearoperator whereonlyafewspecicdetailschange.Forexample,whenth emodelisgivenbyapartial dierentialequation,andaniteelementmethodisusedtoco mpute u h ( ),thenwegenerally solve h ( )eitheronarenedmeshorusinghigherorderelementstoavo idGalerkinorthogonality.Twoexcellentandcomprehensivereferencesonthis subjectare[ 5 ]and[ 7 ].Theresult isanaposteriorierrorestimatewithasimilarformtoEq.( 6.3 )wheretheEuclideaninner productisreplacedbyadualitypairing.Remark6.2 Whentheoperatordeningthemodelisnonlinear,wemustlin earizethemodel operatorpriortodeningtheadjointproblem.Oneapproach usesthesamelinearoperator thatisusedincomputingastepofNewton'smethod,inwhichc asetheremaindertermis typicallyahigherorderperturbationtermthatisoftenneg lected[ 7 ]. 6.2.Enhancingsurrogateswithsolutionstoadjointproblems. Traditionally,aposteriorierrorestimatesofQoIfromdierentialequationmodelsd erivedbyavariationalanalysis andadjointswereusedtoguidelocal h -or p -adaptivity,i.e.,meshororderrenement,respectively,inthenumericalsolutiontothemodel(e.g.,se e[ 7 ]andthereferencestherein).We considerusingtheadjointsolutionstoenhancethesurroga temodel Q s; 0 givenbyEq.( 5.1 ) usingatwo-stageapproach.FromCorollaries 5.1 { 5.3 ,wesubsequentlyinterpreteachstageof enhancementaseitherimplementingaformof p -adaptivityor h -adaptivityofAlgorithm 1

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 13 Intherststage,let N s 1 denotethenumberofnumericalsolutionstothemodel M ( u ; )= 0suchthatwecanalsosolve N s 1 adjointproblemsandcomputethenecessaryinnerproducts toconstruct r Q h ( )foreachofthe N s 1 samplesin .Wethenconstructthepiecewisedened,rst-order,surrogate Q s; 1 ( )= X 1 k N s 1 h Q h ( ( k ) )+ r Q h ( ( k ) )( ( k ) ) i V ( k ) ( ) : (6.5) ApplyingAlgorithm 1 where Q h ( ( i ) )isreplacedby Q s; 1 ( ( i ) )for1 i N isinterpretedas atypeof p -adaptivityforthealgorithm. Ifwecansolvethe N s 1 adjointproblemswithahigher-ordermethodtoproduceaset of reliableaposteriorierrorestimates, n e Q;h ( ( k ) ) o 1 k N s 1 ; thenweproceedtostagetwo.Instagetwo,wecorrectforthep ersistentlocalbiasduetothe error Q;h ( ( k ) )pollutingtheevaluationofany 2V k byusingtheenhancedsurrogate ^ Q s; 1 ( )= X 1 k N s 1 h Q h ( ( k ) )+ e Q;h ( ( k ) )+ r Q h ( ( k ) )( ( k ) ) i V ( k ) ( ) : (6.6) Insomecases,calculatingthederivativesinstage1maybec omputationallyexpensiveor diculttoimplement,whiletheerrorestimatesforstagetw omaybemoreeasilycomputable. Insuchcases,anenhancedpiecewiseconstantsurrogatecan beconstructedbyonlyperforming stage2by ^ Q s; 0 ( )= X 1 k N s 1 h Q h ( ( k ) )+ e Q;h ( ( k ) ) i V ( k ) ( ) : (6.7) Wedemonstratetheeectivenessoftheseapproachesonreduc ingerrorsinthecomputationofprobabilitiesofeventsin B inSection 7 ,andendthissectionwithafewnotes. Theuseofevenhigher-order(local)surrogateapproximati onsordierentclassesofsurrogate responsesurfacesbasedonpiecewise-denedtechniquesre centlystudiedintheliterature(e.g., piecewise-denedpolynomialchaosapproximations)isbri erydiscussedintheconcludingremarksandisthesubjectoffuturework.Here,wesimplynotet hattheaboveapproachis motivatedbytheinterpretationsoftheAlgorithm 1 providedbyCorollaries 5.1 { 5.3 ,anditis currentlyunclearhowtheothersurrogatemethodsshouldbe interpretedwithinthecontext oftheerrorsarisingfromAlgorithm 1 .Wecouldalsolookatcorrectingfornumericalerrors intheapproximationof r Q h ( )aswasdonein[ 13 ].However,thisgenerallyrequiressolving adjointstotheapproximateadjointinordertoestimatesuc herrors.Whilemanycomputationalmodelsarenowbeingdevelopedwithadjointcapabili ties,theyaregenerallydeveloped withoutthistypeofsensitivityerrorestimationinmind,s oweneglecterrorsinthesensitivity computationshere. 6.3.Measure-TheoreticInversionwithEnhancedSurrogateModels. Thepurposefor generatingenhancedsurrogatesistoreduceerrorinthesol utionofstochasticinverseproblems. Wecanreplacetheunenhancedsurrogate Q s; 0 ( )withenhancedsurrogatesforallofthe

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DRAFT 14 T.ButlerandS.Mattis theorypresentedinSection 5.2 .Stageoneenhancementdecreasestheerrorinthesolution associatedwith s ( )becauseofthehigher-orderlocalpolynomialapproximati on.Stagetwo enhancementdecreasestheerrorinthesolutionassociated with h ( )bydecreasingtheeect ofnumericalerroronthesolutionbyaddingerrorestimates RecallfromSection 5.2 that,wedenote P ;s 0 ;M for A 2B asthesolutionobtainedby Algorithm1usingthesurrogate Q s; 0 ( ).Likewise,wedene ^ P ;M;s 0 asthesolutionusing thepiecewiseconstantenhancedsurrogate ^ Q s; 0 ( ).Wedene P ;M;s 1 asthesolutionusing thepiecewiselinearsurrogate Q s; 1 ( ).Wedene ^ P ;M;s 1 asthesolutionusingtheenhanced piecewiselinearsurrogate ^ Q s; 1 ( ).Inpractice,viaAlgorithm1,thesevaluesarecalculated approximatelyby P ;M;N;h ( A )with N N s 0 asdescribedinSection 5.2 .Inthefollowing section,wegenerateenhancedsurrogatesandusethemforst ochasticinversion.Theresulting solutionsandtheirerrorsareanalyzed.ThePythonpackage BET[ 34 ]hastoolsforgenerating surrogatesandsolvingstochasticinverseproblemsandwas usedforgeneratingtheseresults. 7.NumericalResults.7.1.Example1:. Thepurposeofthisexampleistoillustratehowtheerrorfro mchoice ofsurrogate s ( )andtheerrorinthesurrogatefromthenumericalsolutiono fthemodel h ( )bothcanpollutethesolutionofthestochasticinversepro blem.Whenthenumber ofgeneratingsamples N s 0 issmall, s ( )tendstobelarge,sonomatterhowaccuratethe numericalmodelis,therewillbeerrorinthesolutiontothe stochasticinverseproblem. Also, s ( )decreasesas N s 0 increases( h -renement)andasthethepolynomialorderofthe approximationincreases( p -renement),andweseethisintheresults.Allmethodseven tually leveloas N s 0 increasesandthepiecewiselinearsurrogatesdosomuchmor equickly.The remainingerrorisdueto h ( ).Theimpactof h ( )isreducedbyenhancingwitherror estimates.Thesolutionsfromenhancedsurrogateslevelo tovaluesmuchclosertotheexact solution.Weseethatusingtheenhancedpiecewiselinearsu rrogate,bothtypesoferrorare reducedwithrelativelyfewsamples.Wenowprovidethedeta ilsforthisexample. Considerthemodeldenedbythelinearsystem A ( ) u = b ( )(7.1) where, A ( )= e cos( ) sin( )2 e (7.2) and b ( )= sin(10 )2 e T ; (7.3) for 2 =[ 0 : 5 ; 1].TheQoImap, Q isdenedastheaverageofthesolution u by, Q ( )= > u ; (7.4) where =[0 : 5 ; 0 : 5] > ,deninganonlinear1D-to-1Dmapfrom to D = Q ( ).Theapproximatemap Q h ( )isevaluatedbyapproximatelysolvingthesystemdenedby Eqs.( 7.2 )and ( 7.3 )for u h using2Gauss-SeideliterationsandevaluatingtheQoIfunc tionusingEq.( 7.4 ).

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 15 Theadjointproblemisdenedbysolvingthelinearsystem A ( ) > = (7.5) for .Wesolvetheadjointsystemusingdirectnumericalinversi on.Usingstandardaposteriorierrorestimationtechniques,weestimatethenumeric alerrorin Q h ( )by Q;h ( )= > ( b ( ) A ( ) u h )(7.6) where isthesolutiontotheadjointproblemcorrespondingwith .Wecanalsoestimate thederivative @ Q h ( )= > ( @ b ( ) @ A ( ) u h )(7.7) where @ A ( )= e sin( ) cos( )2 e (7.8) and @ b ( )= 10 cos(10 )2 e > : (7.9) Figure7.1:TheQoImapforExample1solvedexactly( Q ( )),approximately( Q h ( )),and enhancedwiththeerrorestimate( Q h ( )+ Q;h ( )). Figure 7.1 showstheQoImap.Noticethatthemapishighlynonlinearwit hrespectto .The\exact" Q ( )wascalculatedbysolvingEq.( 7.1 )usingdirectnumericalinversion. TheapproximateQoImap Q h ( )wascalculatedforseveralvaluesof .Weseethatin someregionsof Q h ( )isaverygoodapproximationandinsomeplacesitisavery badapproximation.Withoutperformingerrorestimationsu chbehavioroftheerroroften cannotbepredicted.Figure 7.1 alsoshowstheenhancedQoImapforseveralvaluesof

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DRAFT 16 T.ButlerandS.Mattis where Q;h ( )wascalculatedbysolvingEq.( 7.6 )andaddedbackto Q h ( ).Weseethatthe enhancedapproximateQoImapmatchesupextremelywellwith the\exact"solution.This illustratesthatenhancingwitherrorestimatesgivesmuch moreaccuratepointwiseevaluations oftheQoImap.Wenowlookathowitimprovessurrogatemodels n r r r r n Figure7.2:ExactQoImap( Q ( )),surrogateQoImap( Q s;h ( )),andenhancedsurrogateQoI map( ^ Q s ( ))using N =21generatingsamplesfora(a)piecewiseconstantsurroga teand(b) piecewiselinearsurrogate. n r r r n Figure7.3:ExactQoImap( Q ( )),surrogateQoImap( Q s;h ( )),andenhancedsurrogate QoImap( ^ Q s ( ))using N =101generatingsamplesfora(a)piecewiseconstantsurrog ate and(b)piecewiselinearsurrogate. AsdescribedinSection 5.2 ,wegeneratepiecewiseconstantandpiecewiselinearsurro gate modelsusingarelativelysmallsetofsamples ( i ) 1 i N s 0 .Thederivativescalculated inEq.( 7.7 )areusedtogeneratethepiecewiselinearsurrogate.Figur es 7.2 and 7.3 show thesurrogatemodelsgeneratedusing N =21and N =101uniformsamplesrespectively. Noticethattheenhancedsurrogates ^ Q s; 0 ( )and ^ Q s; 1 ( )aremuchbetterapproximations

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 17 N s 0 P ;M;s 0 ( A ) j P ( A ) P ;M;s 0 ( A ) j ^ P ;M;s 0 ( A ) j P ( A ) ^ P ;M;s 0 ( A ) j 101.00000e+002.37255e-011.00000e+002.37255e-01307.01989e-016.07566e-028.40285e-017.75396e-02 1007.15283e-014.74620e-027.98403e-013.56580e-023006.80119e-018.26257e-027.72898e-011.01527e-02 10006.84054e-017.86906e-027.72541e-019.79637e-0330006.78654e-018.40909e-027.60572e-012.17289e-03 100006.79327e-018.34182e-027.61442e-011.30337e-03 Table7.1:Probabilitiesoftheset A calculatedwithapiecewiseconstantsurrogatewithand withoutenhancementforExample1.Theerrorsarecalculate dwithrespecttothereference value P ( A )=7 : 627450e 01. of Q ( )thantheregularsurrogates Q s; 0 ( )and Q s; 1 ( ).Theunderlyingnumericalerror, h ( ),persistsintheregularsurrogates,evenasthethethenum berofgeneratingsamplesis increased.Asthenumberofgeneratingsamplesisincreased ,theenhancedsurrogateshave thepropertythat s ( )decreasesandarecloserto Q ( )point-wise.Noticethattheratethat s ( )isreducedwithrespectto N s 0 dependsgreatlyonthedelityofthesurrogatemodel (i.e.piecewiseconstantsvs.piecewiselinears).Withonl y21generatingsamples(seeFigure 7.2 (b)),theenhancedlinearsurrogate ^ Q s; 1 ( )isacloseapproximationto Q ( ),andwith101 generatingsamples(seeFigure 7.3 (b))appearstobeapoint-wiseaccurateapproximationto Q ( ).Evenwith101generatingsamples, s ( )remainssignicantfor ^ Q s; 0 ( )Thisillustrates thevalueofcalculatinggradientsusingadjointsolutions .Theadjointsolutionsmustalready becalculatedtoestimateerrors,anditoftenrequireslitt leadditionalcomputationaleortto thencalculatederivativesusingtheseadjointsolutions. Wenowposeastochasticinverseproblemforthismodel.Supp osetheprobabilitymeasure P D ontheoutputspace( D ; B D )isuniformontheinterval[ 0 : 25 ; 0 : 25].Sincethedistributionisuniform,wechoosearathersimplediscretizationof P D by P D ;M ,wheretheinterval [ 0 : 25 ; 0 : 25]denesonecell.WeuseAlgorithm 1 tonumericallysolvethestochasticinverse problemusingpiecewiseconstantandpiecewiselinearappr oximationsof Q ( )withandwithoutenhancementforarangeofnumbers N s 0 ofuniformi.i.d.generatingsamplesin D .We usetheresultingprobabilitymeasurestogenerateapproxi mationsoftheprobabilityofset A =[ 0 : 5 ; 0 : 25] 2B .AsdescribedinSection 5.2 ,Algorithm 1 issolvedusing N N s 0 of so-calledemulateduniformi.i.d.samplesin ,andforthisproblem, N =1 E 6. Thereferencevalueof P ( A )iscalculatedbyevaluatingthe\exact" Q ( )(usingdirect numericalinversion)at1 E 6i.i.d.samples,andwerefertothisreferencevalueasthe\ truth". Itshouldbenotedthatthisvaluestillhasasmallamountofn umericalerrorduetoerror inMonteCarlointegration.Table 7.1 showsthecalculatedprobabilitiesandabsoluteerrors usingapiecewiseconstantsurrogatewithandwithoutenhan cement.Noticethatbothvalues leveloasthenumberofgeneratingsamplesisincreased;ho wever,thecalculatedprobability withoutenhancementlevelsotoavaluethatisdierentfrom truthbecauseoftheunderlying eectof h ( ).Thecalculationwithenhancementhasverylittleerror.T able 7.2 showsthe samevaluescalculatedusingapiecewiselinearsurrogate. Noticethatbothprobabilityvalues

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DRAFT 18 T.ButlerandS.Mattis N s 0 P ;M;s 1 ( A ) j P ( A ) P ;M;s 1 ( A ) j ^ P ;M;s 1 ( A ) j P ( A ) ^ P ;M;s 1 ( A ) j 108.99760e-011.37015e-011.00000e+002.37255e-01307.60447e-012.29854e-038.61388e-019.86429e-02 1007.07169e-015.55758e-027.70030e-017.28480e-033006.72109e-019.06357e-027.55968e-016.77739e-03 10006.83757e-017.89878e-027.67382e-014.63713e-0330006.78233e-018.45122e-027.62434e-013.11191e-04 100006.79486e-018.32596e-027.61476e-011.26897e-03 Table7.2:Probabilitiesoftheset A calculatedwithapiecewiselinearsurrogatewithand withoutenhancementforExample1.Theerrorsarecalculate dwithrespecttothereference value P ( A )=7 : 627450e 01. nr rr r rrr Figure7.4:Absoluteerrorsincalculatedprobabilitiesfo rExample1. leveloaswell,andatafasterratethantheresultsusingth epiecewiseconstants.The resultsnotusingenhancement,however,donotlevelototh etruevalue.Theerrorinthe resultusingtheenhancedsurrogatereducesveryquicklyto errorvaluesthatareessentially negligible.Figure 7.4 showstheabsoluteerrorswithrespecttonumberofgenerati ngsamples. 7.2.Example2:. Forthenextexample,weconsiderahigher-dimensionalprob lembased onadiscretizedpartialdierentialequationthatwasrstc onsideredin[ 10 ].Considerthe ellipticpartialdierentialequation: 8><>: r ( K ( x;y; ) r u )= f ( x;y; ) ; ( x;y ) 2 D u =0 ; ( x;y ) 2 D K ( x;y; ) r u n =0 ; ( x;y ) 2 @D n D (7.10) where x and y arethespacialcoordinates, D =[0 ; 1] [0 ; 1]and D isthelinesegmentdened by f x =0 ; 0 y 1 g .Let f i g 7i =1 beuncertainparameterswith i 2 [ 1 ; 1]for i =1 ; 2 ;:::; 7

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 19 andthus =[ 1 ; 1] 7 K ( x;y; )isparameterizedby K ( x;y; )=15+ 6 X k =1 k (sin( kx )+cos( ky )) ; and f ( x;y; )= e 7 .Weusetheopen-sourcesoftwareFEniCS[ 39 3 ]todiscretizeEq.( 7.10 ) bycontinuouspiecewiselinearniteelementsonauniformt riangulationof D consistingof 800trianglesand441degreesoffreedom.Foreach ,thisniteelementdiscretiztionproduces alinearsystemoftheform A ( ) u = b ( ) ; (7.11) wherethematrix A ( )iscommonlycalledthe\stinessmatrix"andissparse,symm etric, andpositive-denite.TheQoIisthesolutionat( x;y )=(0 : 8 ; 0 : 6),sowedene ( x;y )= 400 exp( 400( x 0 : 8) 2 +400( y 0 : 6) 2 )(7.12) andinterpolateitontotheniteelementspacetoobtainthe vector .TheapproximateQoI mapisdenedby Q h ( )= T u h ; (7.13) where u h isthenumericalsolutiontoEq.( 7.11 ).Acorrespondingadjointproblemis A ( ) T = : (7.14) Usingstandardaposteriorierrorestimationtechniques,w eestimatethenumericalerrorin Q h ( )by Q;h ( )= T ( b ( ) A ( ) u h )(7.15) where isthesolutiontotheadjointproblemcorrespondingwith .SimilarlytoEq.( 7.7 ), wecanalsocalculatederivativeinformation. TheforwardproblemgivenbyEq.( 7.11 )issolvedapproximatelyby10iterationsofthe conjugategradientmethodandtheadjointproblemgivenbyE q.( 7.14 )issolvedbydirect numericalinversion.Forthestochasticinverseproblemwe supposetheprobabilitymeasure P D on( D ; B D )isuniformontheinterval[0 : 05 ; 0 : 1].Similartothepreviousexample,wediscretize P D by P D ;M ,wheretheinterval[0 : 05 ; 0 : 1]denesonecell.Wenumericallysolvethestochastic inverseproblemusingpiecewiseconstantandpiecewiselin earapproximationsof Q ( )with andwithoutenhancementforarangeofnumbers N s 0 ofuniformi.i.d.generatingsamples in .Wethenapproximatetheprobabilitymeasureofaset A where A =[ 0 : 9 ; 0 : 9] 7 usingthecomputedapproximationof P byAlgorithm 1 .Forreference,the\exact" Q ( ) wascalculatedbysolvingtheforwardproblemusingamuchn ermeshwith2601degrees offreedomandsolvingthelinearsystemwithdirectnumeric alinversionfor50000uniform i.i.dsamples.Anapproximatevalueof P ( A )=4 : 40598e 01wascalculatedbyusingan enhancedpiecewiselinearsurrogategeneratedfromtheser esults. N =1E7uniformi.i.d. emulatedpointswereusedincalculations. Tables 7.3 and 7.4 showthecomputedprobabilitiesanderrorscalculatedusin gpiecewise constantandlinearsurrogates,respectively.Figure 7.5 showstheabsoluteerrorversusthe

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DRAFT 20 T.ButlerandS.Mattis N s 0 P ;M;s 0 ( A ) j P ( A ) P ;M;s 0 ( A ) j ^ P ;M;s 0 ( A ) j P ( A ) ^ P ;M;s 0 ( A ) j 1000.00000e+004.40598e-014.46305e-015.70706e-032000.00000e+004.40598e-014.54361e-011.37626e-024500.00000e+004.40598e-014.58927e-011.83291e-02 10002.12878e-012.27720e-014.50069e-019.47116e-0321002.08349e-012.32249e-014.41398e-017.99553e-0445002.04305e-012.36293e-014.39352e-011.24588e-03 100002.00959e-012.39640e-014.35974e-014.62462e-03210001.90763e-012.49835e-014.35520e-015.07852e-03500001.86423e-012.54176e-014.34838e-015.76068e-03 Table7.3:Probabilitiesoftheset A calculatedwithapiecewiseconstantsurrogatewithand withoutenhancementforExample2.Theerrorsarecalculate dwithrespecttothe\true" value P ( A )=4 : 40598e 01. N s 0 P ;M;s 1 ( A ) j P ( A ) P ;M;s 1 ( A ) j ^ P ;M;s 1 ( A ) j P ( A ) ^ P ;M;s 1 ( A ) j 1004.85626e-014.50275e-024.91527e-015.09285e-022004.99731e-015.91324e-024.85011e-014.44128e-024505.02789e-016.21906e-024.77081e-013.64825e-02 10004.76159e-013.55608e-024.64649e-012.40512e-0221004.80641e-014.00429e-024.56990e-011.63915e-0245004.74045e-013.34470e-024.52858e-011.22598e-02 100004.72203e-013.16045e-024.48637e-018.03873e-03210004.57799e-011.72003e-024.45198e-014.59946e-03500004.49942e-019.34375e-034.41226e-016.27748e-04 Table7.4:Probabilitiesoftheset A calculatedwithapiecewiselinearsurrogatewithand withoutenhancementforExample2.Theerrorsarecalculate dwithrespecttothe\true" value P ( A )=4 : 40598e 01. numberofgeneratingsamplesforthesurrogates.Thisexamp leillustratessomeoftheeect ofdimensiononsurrogates.Theprobabilitycalculatedusi ngtheunenhancedpiecewiseconstantssurrogate P ;M;s 0 ( A )hasagreatdealoferror,evenas N s 0 increases.Thisisbecause largevaluesof s ( )and h ( )arepollutingthesolutionofthestochasticinverseprobl em. Piecewiseconstantsurrogatescanbeespeciallybadinhigh erdimensionalproblems,especially ifgradientsarelarge.Theenhancedpiecewiseconstantsur rogateapproximatestheprobabilitymuchbetter.Thesolutionusingtheunenhancedpiecewis elinearsurrogateismuchbetter thanthepiecewiseconstant;however,theunderlying h ( )stillpollutesthesolution.Aswith thepreviousexample,weseethatthesolutionusingtheenha ncedpiecewiselinearsurrogate levelsotothecorrectvalue,andtheerrorreducesrelativ elyquickly.

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 21 n nrn n nrn Figure7.5:Absoluteerrorsofcalculatedprobabilitiesfo rExample2. 7.3.Example3:. Weconsideratime-dependentpartialdierentialequationm odelling groundwatercontaminanttransport: 8>><>>: @u @t + v r u r ( D r u )=0 ;t 2 (0 ;T ] ; ( x;y ) 2 n r u n =0 ;t 2 (0 ;T ] ; ( x;y ) 2 @ n u = u 0 ;t =0 ; ( x;y ) 2 n (7.16) wheretheunknown u istheconcentrationofacontaminant, isporosity, v =[ v x ;v y ] T isthe Darcyvelocity, D isthedispersiontensor. D isasymmetrictensordenedby D xx = L v 2 x j v j + T v 2 y j v j + D ; (7.17) D yy = L v 2 y j v j + T v 2 x j v j + D ; (7.18) and D xy = D yx =0where L isthelateraldispersion, T isthetransversedispersion, and D istheunderlyingmoleculardiusion.Thedomainisn=[49700 0 : 0 ; 502000 : 0] [537000 : 0 ; 541000 : 0].Theinitialcondition u 0 isdenedasanapproximatepointsourceat ( x s ;y s )=(498250 : 0 ; 538000 : 0)by u 0 = M exp( ( x x s ) 2 ( y y s ) 2 ) ; (7.19) where Q isthesourcemagnitude,and isascalingfactor.Supposethatthecomponentsof theDarcyvelocity v x and v y areuncertainandalloftheothermodelparametersareknown Letthespaceofunknownvaluesof[ v x ;v y ]be =[10 ; 50] 2 .Supposewehavegeophysically reasonableparameters =0 : 25, L =70, T =7, D =0 : 01,and M =1000.TheQoIisan approximatesolutionatsomepoint( x r ;y r )attime T =10denedby = exp( ( x x r ) 2 ( y y r ) 2 ) :

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DRAFT 22 T.ButlerandS.Mattis Letthemeasurementlocationbe( x r ;y r )=(499563 : 69 ; 538995 : 82). Weusetheopen-sourcesoftwareFEniCS[ 39 3 ]tosolvethesystemusingthespace-time continuousGalerkinmethodusinglinearelementsinspacea ndtime,usinga50x50spatial meshandatimestepof0.1.Theinnerproductforthisproblem isthespace-time L 2 inner product h U;V i = Z T 0 Z n U ( x;y;t ) V ( x;y;t ) d n dt: Thecorrespondingadjointproblemis 8>><>>: @ @t v r r ( D r )=0 ;t 2 [0 ;T ) ; ( x;y ) 2 n r n =0 ;t 2 [0 ;T ) ; ( x;y ) 2 @ n = ;t = T; ( x;y ) 2 n : (7.20) Theadjointproblemissolvedbackwardintime.Wesolvethea djointproblemonthesame 50x50meshastheforwardproblem,butwithquadraticelemen ts.Solvinginthisenriched spaceisrequiredforcalculatingerrorestimates.Lineare lementswithatimestepof0.1are usedintime.ErrorestimatesarecomputedasdescribedinEq .( 6.3 )andderivativesare computedasdescribedinEq.( 6.4 ).Fortheforwardandadjointproblems,linearsytemsare solveddirectly. Forthestochasticinverseproblemwesupposetheprobabili tymeasure P D on( D ; B D )is uniformontheinterval[0 : 005 ; 0 : 021].Here,wediscretize P D by P D ;M ,wheretheinterval [0 : 05 ; 0 : 1]isdividedinto3uniformintervals.The\exact"solution wascalculatedbysolving themodelona100x100meshwithtimestepsof0.05.Correspon dingadjointproblemswere solvedandusedtocalculateerrorestimatesandderivative stogenerateahighlyaccurate piecewiselinearsurrogatewith10000generatingsamples. Theset A isdenedby[25 ; 35] 2 Anapproximatevalueof P ( A )=2 : 21992e 01wascalculatedbyusinganenhancedpiecewise linearsurrogategeneratedfromtheseresults. N =1E7uniformi.i.d.emulatedpointswere usedincalculations.Tables 7.5 and 7.6 showthecomputedprobabilitiesanderrorscalculated usingpiecewiseconstantandlinearsurrogates,respectiv ely.Figure 7.6 showstheabsolute errorversusthenumberofgeneratingsamplesforthesurrog ates.Aswiththeothercases,the resultsusingsurrogateswithoutenhancementlevelotoav aluepollutedwithnumericalerror. Boththeenhancedsurrogatesproduceestimateswhoseerror isreducedrelativelyquickly. 8.ConclusionsandFutureWork. Weprovidedare-interpretationofanon-intrusive samplebasedmeasure-theoreticalgorithmforcomputingno n-parametricprobabilitymeasures solvingstochasticinverseproblems.Thisallowedustoide ntifytheapproximatesolutionofthe stochasticinverseproblemastheexactsolutionofanappro ximateproblemformulatedwith respecttosurrogateresponsesurfaces.Thismotivatedthe useofadjointbasedtechniques toenhancesurrogateresponsesurfacesintwoways.First,b yincreasingthelocalorderof approximations,andthenbycorrectingfornumericalerror .Numericalresultsdemonstrates theimprovedaccuracyinenhancingsurrogatestoapproxima teprobabilitiesofparameter events. Therearemanyinterestingfuturedirectionstotakethiswo rk.Perhapsthemostobvious directionwouldbetoexaminepiecewisesurrogateapproxim ationsbasedonglobalsurrogate

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 23 N s 0 P ;M;s 0 ( A ) j P ( A ) P ;M;s 0 ( A ) j ^ P ;M;s 0 ( A ) j P ( A ) ^ P ;M;s 0 ( A ) j 1001.87975e-013.40169e-021.89844e-013.21485e-022002.24155e-012.16290e-032.19715e-012.27707e-034502.23614e-011.62220e-032.26974e-014.98183e-03 10002.36665e-011.46730e-022.15488e-016.50368e-0321002.36307e-011.43144e-022.22693e-017.00983e-0445002.33974e-011.19820e-022.24348e-012.35608e-03 100002.35103e-011.31110e-022.22965e-019.73110e-04 Table7.5:Probabilitiesoftheset A calculatedwithapiecewiseconstantsurrogatewithand withoutenhancementforExample3.Theerrorsarecalculate dwithrespecttothe\true" value P ( A )=2 : 21992e 01. N s 0 P ;M;s 1 ( A ) j P ( A ) P ;M;s 1 ( A ) j ^ P ;M;s 1 ( A ) j P ( A ) ^ P ;M;s 1 ( A ) j 1001.87730e-013.42617e-021.97466e-012.45266e-022002.21392e-016.00038e-042.20162e-011.82992e-034502.22753e-017.61095e-042.28662e-016.67031e-03 10002.38815e-011.68224e-022.16272e-015.72033e-0321002.37003e-011.50107e-022.23634e-011.64136e-0345002.34705e-011.27130e-022.24910e-012.91785e-03 100002.35274e-011.32817e-022.23033e-011.04071e-03 Table7.6:Probabilitiesoftheset A calculatedwithapiecewiselinearsurrogatewithand withoutenhancementforExample3.Theerrorsarecalculate dwithrespecttothe\true" value P ( A )=2 : 21992e 01. nr rr r rrr Figure7.6:Absoluteerrorsofcalculatedprobabilitiesfo rExample3.

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DRAFT 24 T.ButlerandS.Mattis approachesasdiscussedbelow.Whilethereismuchworkinth euseofpolynomialchaosexpansionstoconstructglobalsurrogates[ 53 50 30 ],inordertoensuretheerror s ( )isbounded bysomethresholdsothatresultsareuseful,ahigh-orderpo lynomialapproximationmaybe requiredwhichsubsequentlyimpliesthatthevalueof N s 0 maybeprohibitivelylargeand beyondthelimitsprovidedbythecomputationalbudget.For example,inpolynomialchaos approaches,thetotalnumberofcoecientsweneedtocomput eageneral p -ordersurrogate overan m -dimensionalparameterspaceisgivenby( m + p )! = ( m p !)[ 53 11 ].Computation ofthepolynomialcoecientsinturnrequirescomputationo fintegralsovertheparameter space,whichisoftendonewithnumericalquadrature.IfaQo Iisparticularlysensitiveto thenalargeorder p oftheglobalpolynomialchaossurrogateapproximationmay berequired tocontrol s ( ).Thisinturnrequireshigherordernumericalquadrature, whichincreases thetotalnumber N ofnumericalsolutionstothemodel.In[ 42 ],foraBayesianinference problemofparameterstoacontaminanttransportmodelwith relativelyfewparameters,it wassuggestedthatapiecewisepolynomialchaossurrogateb econstructedonaregularpartitioningof toimproveaccuracy.Thegeneralideawasthat s ( )canbemadesmallif thesurrogateisdenedbylocalizedlow-ordersurrogateap proximationstotheQoIresponse surface,andthatthiswascomputationallyfeasiblesincei tisofteneasiertosolveseverallower orderproblemsthanasignicantlyhigherorderproblem.Ac curacyintheglobalposterior oftheBayesiansolutionfromsuchasurrogatewassubsequen tlyobserved.Thisprovidesa motivationforfutureworkinthedesignoflow-orderpolyno mialsurrogatesconstructedby othermeansthanthoseshowninthisworkforgoal-orientedm easure-theoreticinversion. 9.Acknowledgments. ThismaterialisbaseduponworksupportedbytheU.S.Depart mentofEnergyOceofScience,OceofAdvancedScienticCo mputingResearch,Applied MathematicsprogramunderAwardNumbersDE-SC00009279and DE-SC0009286aspartof theDiaMonDMultifacetedMathematicsIntegratedCapabili tyCenter. T.ButlerworkisalsosupportedinpartbytheNationalScien ceFoundation(DMS1228206). REFERENCES [1] M.AinsworthandJ.Oden Aposteriorierrorestimationinniteelementanalysis ,ComputerMethods inAppliedMechanicsandEngineering,142(1997),pp.1{88. [2] R.C.AlmeidaandJ.T.Oden Solutionverication,goal-orientedadaptivemethodsfor stochastic advection-diusionproblems ,Comput.MethodsAppl.Mech.Engrg.,199(2010),pp.2472{2 486. [3] M.Alns,J.Blechta,J.Hake,A.Johansson,B.Kehlet,A.Logg ,C.Richardson,J.Ring, M.Rognes,andG.Wells TheFEniCSProjectVersion1.5 ,ArchiveofNumericalSoftware,3 (2015). [4] I.BabuskaandW.C.Rheinboldt A-posteriorierrorestimatesfortheniteelementmethod ,InternationalJournalforNumericalMethodsinEngineering,12( 1978),pp.1597{1615. [5] W.BangerthandR.Rannacher AdaptiveFiniteElementMethodsforDierentialEquations BirkhauserVerlag,2003. [6] R.A.Bartlett,D.M.Gay,andE.T.Phipps AutomaticDierentiationofC++CodesforLargeScaleScienticComputing ,SpringerBerlinHeidelberg,Berlin,Heidelberg,2006,pp .525{532. [7] R.BeckerandR.Rannacher Anoptimalcontrolapproachtoaposteriorierrorestimatio ninnite elementmethods ,ActaNumer.,10(2001),pp.1{102.

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DRAFT SurrogateModeling,Error,andMeasure-TheoreticInversion 25 [8] J.Breidt,T.Butler,andD.Estep AMeasure-TheoreticComputationalMethodforInverseSens itivityProblemsI:MethodandAnalysis ,SIAMJournalonNumr.Anal.,49(2011),pp.1836{1859. [9] C.M.Bryant,S.Prudhomme,andT.Wildey Errordecompositionandadaptivityforresponse surfaceapproximationsfrompdeswithparametricuncertai nty ,SIAM/ASAJournalonUncertainty Quantication,3(2015),pp.1020{1045. [10] T.Butler,P.Constantine,andT.Wildey Aposteriorierroranalysisofparameterizedlinearsystemsusingspectralmethods ,SIAMJournalonMatrixAnalysisandApplications,33(2012 ),pp.195{ 209. [11] T.Butler,C.Dawson,andT.Wildey Aposteriorierroranalysisofstochasticdierentialequat ions usingpolynomialchaosexpansions ,SIAMJournalonScienticComputing,33(2011),pp.1267{1 291. [12] T.Butler,C.Dawson,andT.Wildey Propagationofuncertaintiesusingimprovedsurrogatemod els SIAM/ASAJournalonUncertaintyQuantication,1(2013),p p.164{191. [13] T.Butler,D.Estep,andJ.Sandelin AComputationalMeasureTheoreticApproachtoInverse SensitivityProblemsII:APosterioriErrorAnalysis ,SIAMJournalonNumr.Anal.,50(2012), pp.22{45. [14] T.Butler,D.Estep,S.Tavener,C.Dawson,andJ.Westerink AMeasure-TheoreticComputationalMethodforInverseSensitivityProblemsIII:Mu ltipleQuantitiesofInterest ,SIAM/ASA JournalonUncertaintyQuantication,2(2014),pp.174{20 2.doi:10.1137/130930406. [15] T.Butler,D.Estep,S.Tavener,T.Wildey,C.Dawson,andL.G raham SolvingStochastic InverseProblemsusingSigma-AlgebrasonContourMaps .Inpreparation,2014. [16] T.Butler,L.Graham,D.Estep,C.Dawson,andJ.Westerink Denitionandsolutionofa stochasticinverseproblemforthemanning's n parametereldinhydrodynamicmodels ,Adv.in WaterResour.,78(2015),pp.60{79. [17] T.Butler,A.Huhtala,andM.Juntunen Quantifyinguncertaintyinmaterialdamagefromvibrationaldata ,JournalofComputationalPhysics,283(2015),pp.414{435 [18] D.Cacuci SensitivityandUncertaintyAnalysis:Theory ,vol.I,Chapman&Hall/CRC,1997. [19] V.Carey,D.Estep,andS.Tavener Aposteriorianalysisandadaptiveerrorcontrolformultis cale operatordecompositionsolutionofellipticsystemsi:Tri angularsystems ,SIAMJournalonNumerical Analysis,47(2009),pp.740{761. [20] P.G.Constantine,E.Dow,andQ.Wang Activesubspacemethodsintheoryandpractice:Applicationstokrigingsurfaces ,SIAMJournalonScienticComputing,36(2014),pp.A1500{ A1524. [21] S.Cotter,M.Dashti,andA.Stuart Approximationofbayesianinverseproblems ,SIAMJournal ofNumericalAnalysis,48(2010),pp.322{345. [22] C.DellacherieandP.Meyer ProbabilitiesandPotential ,North-HollandPublishingCo.,Amsterdam, 1978. [23] D.Estep Aposteriorierrorboundsandglobalerrorcontrolforappro ximationofordinarydierential equations ,SIAMJournalonNumericalAnalysis,32(1995),pp.1{48. [24] D.EstepandD.Neckels Fastandreliablemethodsfordeterminingtheevolutionofu ncertainparametersindierentialequations ,JournalofComputationalPhysics,213(2006),pp.530{556 [25] D.Estep,S.Tavener,andT.Wildey Aposteriorianalysisandimprovedaccuracyforanoperator decompositionsolutionofaconjugateheattransferproble m ,SIAMJournalonNumericalAnalysis, 46(2008),pp.2068{2089. [26] G.Evensen Sequentialdataassimilationwithanonlinearquasi-geost rophicmodelusingmontecarlo methodstoforecasterrorstatistics ,JournalofGeophysicalResearch:Oceans(1978{2012),99( 1994), pp.10143{10162. [27] G.Evensen TheensembleKalmanlter:theoreticalformulationandpra cticalimplementation ,Ocean Dynamics,53(2003),pp.343{367. [28] J.Gentle RandomNumberGenerationandMonteCarloMethods ,Springer,2003. [29] C.J.Geyer Practicalmarkovchainmontecarlo ,StatisticalScience,7(1992),pp.473{483. [30] R.GhanemandJ.Red-Horse Propagationofprobabilisticuncertaintyincomplexphysi calsystems usingastochasticniteelementapproach ,Phys.D,133(1999),pp.137{144. [31] R.GhanemandP.Spanos StochastifFiniteElements:ASpectralApproach ,SpringerNewYork,1991. [32] W.Gilks,S.Richardson,andD.Spiegelhalter MarkovChainMonteCarloinPractice ,Chapman %Hall,1996.

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DRAFT 26 T.ButlerandS.Mattis [33] L.Graham,T.Butler,S.Walsh,C.Dawson,andJ.Westerink Ameasure-theoreticalgorithm forestimatingbottomfrictioninacoastalinlet:Casestud yofbayst.luoisduringhurricanegustav (2008) ,MonthlyWea.Rev.,(2016).InRevision. [34] L.Graham,S.Mattis,S.Walsh,T.Butler,M.Pilosov,andD.M cDougall BET:Butler, Estep,TavenerMethodv2.0.0 ,Aug.2016. [35] R.E.Kalmanetal. Anewapproachtolinearlteringandpredictionproblems ,Journalofbasic Engineering,82(1960),pp.35{45. [36] C.Lanczos LinearDierentialOperators ,DoverPublications,1997. [37] O.P.LeMa ^ tre,O.M.Knio,H.N.Najm,andR.G.Ghanem Uncertaintypropagationusing wiener-haarexpansions ,J.Comput.Phys.,197(2004),pp.28{57. [38] O.P.LeMa ^ tre,H.N.Najm,R.G.Ghanem,andO.M.Knio Multi-resolutionanalysisofwienertypeuncertaintypropagationschemes ,J.Comput.Phys.,197(2004),pp.502{531. [39] A.Logg,K.-A.Mardal,andG.Wells ,eds., AutomatedSolutionofDierentialEquationsbythe FiniteElementMethod ,SpringerBerlinHeidelberg,2012. [40] G.I.Marchuk Adjointequationsandanalysisofcomplexsystems ,Kluwer,1995. [41] G.I.Marchuk,V.I.Agoshkov,andV.P.Shutyaev AdjointEquationsandPerturbationAlgorithms inNonlinearProblems ,CRCPress,BocaRaton,FL,1996. [42] Y.M.Marzouk,H.N.Najm,andL.A.Rahn Stochasticspectralmethodsforecientbayesian solutionofinverseproblems ,J.Comput.Phys.,224(2007),pp.560{586. [43] S.A.Mattis,T.D.Butler,C.N.Dawson,D.Estep,andV.V.Ves silinov Parameterestimation andpredictionforgroundwatercontaminationbasedonmeas uretheory ,WaterResourcesResearch, 51(2015),pp.7608{7629. [44] F.Nobile,R.Tempone,andC.G.Webster Asparsegridstochasticcollocationmethodforpartialdierentialequationswithrandominputdata ,SIAMJournalonNumericalAnalysis,46(2008), pp.2309{2345. [45] S.PrudhommeandC.M.Bryant Adaptivesurrogatemodelingforresponsesurfaceapproxim ations withapplicationtobayesianinference ,AdvancedModelingandSimulationinEngineeringSciences 2(2015),pp.1{21. [46] W.T.M.R.H.Cameron Theorthogonaldevelopmentofnon-linearfunctionalsinse riesoffourierhermitefunctionals ,AnnalsofMathematics,48(1947),pp.385{392. [47] C.RobertandG.Casella MonteCarloStatisticalMethods ,Springer,2004. [48] A.M.Stuart Inverseproblems:ABayesianperspective ,ActaNumerica,19(2010),pp.451{559. [49] A.Tarantola InverseProblemTheoryandMethodsforModelParameterEsti mation ,siam,2005. [50] X.WanandG.E.Karniadakis Beyondwiener|askeyexpansions:Handlingarbitrarypdfs ,J.Sci. Comput.,27(2006),pp.455{464. [51] N.Wiener Thehomogeneouschaos ,AmericanJournalofMathematics,60(1938),pp.897{936. [52] T.Wildey,S.Tavener,andD.Estep Aposteriorierrorestimationofapproximateboundaryruxe s CommunicationsinNumericalMethodsinEngineering,24(20 08),pp.421{434. [53] D.XiuandG.E.Karniadakis Thewiener{askeypolynomialchaosforstochasticdierenti alequations SIAMJournalonScienticComputing,24(2002),pp.619{644