Citation
Optimal Experimental Design Using A Consistent Bayesian Approach

Material Information

Title:
Optimal Experimental Design Using A Consistent Bayesian Approach
Series Title:
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 2017
Creator:
Walsh, Scott
Wildey, Tim
Jakeman, John
Publisher:
Journal of Risk and Uncertainty in Engineering Systems
Publication Date:
Physical Description:
Journal Article

Notes

Abstract:
We consider the utilization of a computational model to guide the optimal acquisition of experimental data to inform the stochastic description of model input parameters. Our formulation is based on the recently developed consistent Bayesian approach for solving stochastic inverse problems which seeks a posterior probability density that is consistent with the model and the data in the sense that the push-forward of the posterior (through the com- putational model) matches the observed density on the observations almost everywhere. Given a set a potential observations, our optimal experimental design (OED) seeks the observation, or set of observations, that maximizes the expected information gain from the prior probability density on the model parameters. We discuss the character- ization of the space of observed densities and a computationally efficient approach for rescaling observed densities to satisfy the fundamental assumptions of the consistent Bayesian approach. Numerical results are presented to compare our approach with existing OED methodologies using the classical/statistical Bayesian approach and to demonstrate our OED on a set of representative PDE-based models.
Acquisition:
Collected for Auraria Institutional Repository by the Self-Submittal tool. Submitted by Scott Walsh.
Publication Status:
In Press

Record Information

Source Institution:
Auraria Institutional Repository
Holding Location:
Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.

Downloads

This item is only available as the following downloads:


Full Text

PAGE 1

DRAFT OptimalExperimentalDesignUsingAConsistent BayesianApproach ScottN.Walsh UniversityofColoradoDenver, DepartmentofMathematicaland StatisticalSciences, Email:scott.walsh@ucdenver.edu TimM.Wildey SandiaNationalLaboratories, CenterforComputingResearch, Email:tmwilde@sandia.gov JohnD.Jakeman SandiaNationalLaboratories, CenterforComputingResearch, Email:jdjakem@sandia.gov ABSTRACT Weconsidertheutilizationofacomputationalmodeltoguidetheoptimalacquisitionofexperimentaldatato informthestochasticdescriptionofmodelinputparameters.Ourformulationisbasedontherecentlydeveloped consistent Bayesianapproachforsolvingstochasticinverseproblemswhichseeksaposteriorprobabilitydensity thatisconsistentwiththemodelandthedatainthesensethatthepush-forwardoftheposteriorthroughthecomputationalmodelmatchestheobserveddensityontheobservationsalmosteverywhere.Givenasetapotential observations,ouroptimalexperimentaldesignOEDseekstheobservation,orsetofobservations,thatmaximizes theexpectedinformationgainfromthepriorprobabilitydensityonthemodelparameters.Wediscussthecharacterizationofthespaceofobserveddensitiesandacomputationallyefcientapproachforrescalingobserveddensities tosatisfythefundamentalassumptionsoftheconsistentBayesianapproach.Numericalresultsarepresentedto compareourapproachwithexistingOEDmethodologiesusingtheclassical/statisticalBayesianapproachandto demonstrateourOEDonasetofrepresentativePDE-basedmodels. 1Introduction Experimentaldataisoftenusedtoinfervaluableinformationaboutparametersformodelsofphysicalsystems. However,thecollectionofexperimentaldatacanbecostlyandtimeconsuming.Forexample,exploratorydrilling canrevealvaluableinformationaboutsubsurfacehydrocarbonreservoirs,buteachwellcancostupwardsoftens ofmillionsofUSdollars.Insuchsituationswecanonlyaffordtogathersomelimitednumberofexperimental data,howevernotallexperimentsprovidethesameamountofinformationabouttheprocessestheyarehelping inform.Consequently,itisimportanttodesignexperimentsinanoptimalway,i.e.,tochoosesomelimitednumber ofexperimentaldatatomaximizethevalueofeachexperiment. Therstexperimentaldesignmethodsemployedmainlyheuristics,basedonconceptssuchasspace-lling andblocking,toselecteldexperiments[1].Whilethesemethodscanperformwellinsomesituations,these methodscanbeimproveduponbyincorporatinganyknowledgeoftheunderlyingphysicalprocessesbeinginferred ormeasured.Usingphysicalmodelstoguideexperimentselectionhasbeenshowntodrasticallyimprovethecost effectivenessofexperimentaldesignsforavarietyofmodelsbasedonordinarydifferentialequations[6],partial differentialequations[9]anddifferentialalgebraicequations[10].Whenmodelobservablesarelinearwithrespect tothemodelparametersthealphabeticoptimalitycriteriaareoftenused[11].Forexample A -optimalityto minimizetheaveragevarianceofparameterestimates, D -optimalitytomaximizethedifferentialShannonentropy, or G -optimalitytominimizethemaximumvarianceofmodelpredictions.Thesecriteriahavebeendevelopedin bothBayesianandnon-Bayesiansettings[11,12,14]. InthispaperwefocusattentiononBayesianmethodsforOEDthatcanbeappliedtobothlinearandnonlinearmodels[18].SpecicallywepursueOEDswhichareoptimalforinferringmodelparametersonnite-

PAGE 2

DRAFT dimensionalspacesfromexperimentaldataobservedatasetofsensorlocations.InthecontextofOEDforinference,analoguesofthealphabeticcriterion,forlinearmodelshavealsobeenappliedtononlinearmodels[9,14, 21,22].Incertainsituations,forexampleinnite-dimensionalproblemsrandomvariablesarerandomeldsor problemswithcomputationalexpensivemodels,OEDbaseduponlinearizationsofthemodelresponseandLaplace Gaussianapproximationsoftheposteriordistributionhavebeennecessary[14,21].Inothersettingsnon-Gaussian approximationsoftheposteriorhavealsobeenpursued[17]. ThismanuscriptpresentsanewapproachforOEDbasedupon consistent Bayesianinference,introducedin[23]. WeadoptanapproachforOEDsimilartotheapproachin[24]andseekanOEDthatmaximizestheexpected informationgainfromthepriortotheposterioroverthesetofpossibleobservationaldensities.AlthoughourOED frameworkisBayesianinnature,thisapproachisfundamentallydifferentfromthestatisticalBayesianmethods mentionedabove.TheaforementionedBayesianOEDmethodsusewhatwewillrefertoastheclassical/statistical Bayesianapproachforstochasticinferenceseee.g.,[25]tocharacterizeposteriordensitiesthatreectsanassumed errormodel.Incontrast,consistentBayesianinferenceassumesaprobabilitydensityontheobservationsisgiven andproducesaposteriordensitythatisconsistentwiththemodelandthedatainthesensethatthepush-forward oftheposteriorthroughthecomputationalmodelmatchestheobserveddensityalmosteverywhere.Wedirect theinterestedreaderto[23]foradiscussiononthedifferencesbetweentheconsistentandstatisticalBayesian approaches.ConsistentBayesianinferencehassomeconnectionswithmeasure-theoreticinference[26],whichwas usedforOEDin[27],butthetwoapproachesmakedifferentassumptionsandthereforetypicallygivedifferent solutionstothestochasticinverseproblem. TheconsistentBayesianapproachisappealingforOEDsinceitcanbeusedinanofine-onlinemode.ConsistentBayesianinferencerequiresanestimateofthepush-forwardoftheprior,whichalthoughexpensivecanbe computedofineorobtainedfromarchivalsimulationdata.Oncethepush-forwardofthepriorisconstructed, theposteriordensitycanbeapproximatedcheaply.Moreover,thispush-forwardofthepriordoesnotdependon thedensityontheobservationswhichenablesacomputationallyefcientapproachforsolvingmultiplestochastic inverseproblemsfordifferentdensitiesontheobservations.Thiscansignicantlyreducethecostofcomputingthe expectedinformationgainifthesetofcandidateobservationisknown apriori ThemainobjectivesinthispaperaretoderiveanOEDformulationusingtheconsistentBayesianframework andtopresentacomputationalstrategytoestimatetheexpectedinformationgainedforanexperimentaldesign. ThepursuitofacomputationallyefcientapproachforcouplingourOEDmethodwithcontinuousoptimization techniquesisanintriguingtopicthatweleaveforfuturework.Here,weconsiderbatchdesignoveradiscreteset ofpossibleexperiments.Batchdesign,alsoknownasopen-loopdesign,involvesselectingasetofexperiments concurrentlysuchthattheoutcomeofanyexperimentdoesnoteffecttheselectionoftheotherexperiments.Such anapproachisoftennecessarywhenonecannotwaitfortheresultsofoneexperimentbeforestartinganother,but islimitedintermsofthenumberofobservationswecanconsider. Theremainderofthispaperisoutlinedasfollows.InSection2wesummarizetheconsistentBayesianmethod forsolvingstochasticinverseproblems.InSection3wediscusstheinformationcontentofanexperiment,and presentourOEDformulationbaseduponexpectedinformationgain.Duringtheprocessofdeningtheexpected informationgainofagivenexperimentaldesign,caremustbetakentoensurethatthemodelcanpredictallofour potentialobserveddata.InSection4wediscusssituationsforwhichthisassumptionisviolatedandmeansfor avoidingthesesituations.NumericalexamplesarepresentedinSection5andconcludingremarksareprovidedin Section6. 2AConsistentBayesformulationforstochasticinverseproblems Weareinterestedinexperimentaldesignswhichareoptimalforinferringmodelparametersfromexperimental data.Inferringmodelparametersforasingledesignandrealizationofexperimentaldataisafundamentalcomponentofproducingsuchoptimaldesigns.InthissectionwesummarizetheconsistentBayesmethodforparametric inference,originallypresentedin[23].AlthoughBayesianinnature,theconsistentBayesianapproachdifferssignicantlyfromitsclassicalBayesiancounterpart[25,28,29]whichwasusedforOEDin[12,14,17,21,24,30]. Werefertheinterestedreaderto[23]forafulldiscussionofthesedifferences. 2.1Notation,Assumptions,andaStochasticInverseProblem Let M Y ; l denoteadeterministicmodelwithsolution Y l thatisanimplicitfunctionofmodelparameters l 2 L R n .Theset L representsthelargestphysicallymeaningfuldomainofparametervalues,and,forsimplicity, weassumethat L iscompact.Inpractice,modelersareoftenonlyconcernedwithcomputingarelativelysmallsetof quantitiesofinterestQoI, f Q i Y g m i = 1 ,whereeach Q i isareal-valuedfunctionaldependentonthemodelsolution Y .Since Y isafunctionofparameters l ,soaretheQoIandwewrite Q i l tomakethisdependenceexplicit.Given

PAGE 3

DRAFT asetofQoI,wedenetheQoImap Q l : = Q 1 l ; ; Q m l > : L D R m where D : = Q L denotesthe rangeoftheQoImap. Assume L ; B L ; L and D ; B D ; D aremeasurespaces.Weassume B L and B D aretheBorel s -algebras inheritedfromthemetrictopologieson R n and R m ,respectively.Themeasures L and D arevolumemeasures. WeassumethattheQoImap Q isatleastpiecewisesmoothimplyingthat Q isameasurablemapbetweenthe measurablespaces L ; B L and D ; B D .Forany A 2 B D ,wethenhave Q )]TJ/F52 7.3723 Tf 5.728 0 Td [(1 A = f l 2 L j Q l 2 A g 2 B L ; and Q Q )]TJ/F52 7.3723 Tf 5.729 0 Td [(1 A = A : Furthermore, B Q )]TJ/F52 7.3723 Tf 5.728 0 Td [(1 Q B forany B 2 B L ,althoughinmostcases B 6 = Q )]TJ/F52 7.3723 Tf 5.728 0 Td [(1 Q B evenwhen n = m Finally,weassumethatanobservedprobabilitymeasure, P obs D ,isgivenon D ; B D andisabsolutelycontinuous withrespectto D ,whichimpliesitcanbedescribedintermsofanobservedprobabilitydensity, p obs D .Thestochastic inverseproblemisthendenedasdeterminingaprobabilitymeasure, P L ,describedasaprobabilitydensity, p L suchthat,thepush-forwardmeasureagreeswith P obs D .Weuse P Q P L D todenotethepush-forwardof P L through Q l ,i.e., P Q P L D A = P L Q )]TJ/F52 7.3723 Tf 5.728 0 Td [(1 A : forall A 2 B D .Usingthisnotation,asolutiontothestochasticinverseproblemisdenedformallyasfollows: Denition1Consistency. GivenaprobabilitymeasureP obs D on D ; B D thatisabsolutelycontinuouswith respect D andadmitsadensity p obs D ,thestochasticinverseproblemseeksaprobabilitymeasureP L on L ; B L thatisabsolutelycontinuouswithrespectto L andadmitsaprobabilitydensity p L ,suchthatthesubsequent push-forwardmeasureinducedbythemap,Q l ,satises P L Q )]TJ/F52 7.3723 Tf 5.729 0 Td [(1 A = P Q P L D A = P obs D A ; foranyA 2 B D .WerefertoanyprobabilitymeasureP L thatsatises asa consistent solutiontothestochastic inverseproblem. Clearly,aconsistentsolutionmaynotbeunique,i.e.,theremaybemultipleprobabilitymeasuresthatare consistentinthesenseofDenition1.Thisisanalogoustoadeterministicinverseproblemwheremultiplesetsof parametersmayproducetheobserveddata.Auniquesolutionmaybeobtainedbyimposingadditionalconstraints orstructureonthestochasticinverseproblem.Inthispaper,suchstructureisobtainedbyincorporatingprior informationtoconstructauniqueBayesiansolutiontothestochasticinverseproblem. 2.2ABayesiansolutiontothestochasticinverseproblem FollowingtheBayesianphilosophy[33],weintroducea prior probabilitymeasure P prior L on L ; B L thatis absolutelycontinuouswithrespectto L andadmitsaprobabilitydensity p prior L .Thepriorprobabilitymeasure encapsulatestheexistingknowledgeabouttheuncertainparameters. Assumingthat Q isatleastmeasurable,thenthepriorprobabilitymeasureon L P prior L ,andthemap, Q ,induce apush-forwardmeasure P Q prior D on D ,whichisdenedforall A 2 B D P Q prior D A = P prior L Q )]TJ/F52 7.3723 Tf 5.728 0 Td [(1 A : Weutilizethefollowingexpressionfortheposterior, P post L B : = 8 < : P prior L B P obs D Q B P Q prior D Q B ; if P prior L B > 0 ; 0 ; otherwise ;

PAGE 4

DRAFT whichwedescribeintermsofaprobabilitydensitygivenby p post L l = p prior L l p obs D Q l p Q prior D Q l ; l 2 L : Wenotethatif p Q prior D = p obs D ,i.e.,ifthepriorsolvesthestochasticinverseproblem,thentheposteriordensitywill beequaltothepriordensity. Itwasrecentlyshownin[23]thattheposteriorgivenbydenesaconsistentprobabilitymeasureusinga contour s -algebra.Wheninterpretedasaparticulariteratedintegralof,theposteriordenesaprobabilitymeasure on L ; B L inthesenseofDenition1,i.e.,thepush-forwardoftheposteriormatchestheobservedprobabilitydensity.ApproximatingtheposteriordensityusingtheconsistentBayesianapproachonlyrequiresanapproximation ofthepush-forwardofthepriorprobabilityonthemodelparameters,whichisfundamentallyaforwardpropagationofuncertainty.Whilenumerousapproacheshavebeendevelopedinrecentyearstoimprovetheefciencyand accuracyoftheforwardpropagationofuncertaintyusingcomputationalmodels,inthispaperweonlyconsiderthe mostbasicofmethods,namelyMonteCarlosampling,tosamplefromtheprior.Weevaluatethecomputational modelforeachofthesamplesfromtheprioranduseastandardkerneldensityestimator[34]toapproximatethe push-forwardoftheprior. Giventheapproximationofthepush-forwardoftheprior,wecanevaluatetheposterioratanypoint l 2 L ifwe compute Q l .Thisprovidesseveralpossibilitiesforinterogatingtheposterior.InSection3.2,wecompute Q l onauniformgridofpointstovisualizetheposteriorafterwecomputethepush-forwardoftheprior.Thisdoes requireadditionalmodelevaluations,butvisualizingtheposteriorisrarelyrequiredandonlyusefulforillustrative purposesin1or2dimensions.Moreoften,weareinterestedinobtainingsamplesfromtheposterior.Thisis alsodemonstratedinSection3.2wherethesamplesfromthepriorareeitheracceptedorrejectedusingastandard rejectionsamplingprocedure.Foragiven l ,wecomputetheratio p post L l = M p prior L l ,where M isanestimate ofthemaximumoftheratioover L ,andcomparethisvaluewithasample, h ,drawnfromauniformdistribution on,1.Iftheratioislargerthan h ,thenweacceptthesample.Weapplytheaccept-rejectalgorithmtothe samplesfromthepriorandthereforethesamplesfromtheposteriorareasubsetofthesamplesusedtocompute thepush-forwardoftheprior.Sincewehavealreadycomputed Q l foreachofthesesamples,thecomputational costtoselectasubsetofthesamplesfortheposteriorisminimal.However,inthecontextofOEDweareprimarily interestedincomputingtheinformationgainedfromthepriortotheposteriorwhichonlyinvolvesintegratingwith respecttothepriorseeSection3.1anddoesnotrequireadditionalmodelevaluationsorrejectionsampling. Inpractice,weprefertousedatathatissensitivetotheparameterssinceotherwiseitisdifculttoinferuseful informationabouttheuncertainparameters.Specically,if m n andtheJacobianof Q isdeneda.e.in L and isfullranka.e.,thenthepush-forwardvolumemeasure D isabsolutelycontinuouswithrespecttotheLebesgue measure[23]. Fortherestofthisworkwemaintainthefollowingassumptionsneededtoproduceauniqueconsistentsolution tothestochasticinverseproblem: A1 Wehaveamathematicalmodelandadescriptionofourpriorknowledgeaboutthemodelinputparameters, A2 Thedataexhibitssensitivitytotheparametersa.e.in L ,hence,weusetheLebesguemeasure asthe volumemeasureonthedataspace, A3 Theobserveddensityisabsolutelycontinuouswithrespecttothepush-forwardoftheprior. Theassumptionconcerningtheabsolutecontinuityoftheobserveddensitywithrespecttothepriorisessential todeneasolutiontothestochasticinverseproblem[23].Whilethisassumptionmayappearratherabstract,it simplyassuresthatthepriorandthemodelcanpredict,withnon-zeroprobability,anyeventthatwehaveobserved. Sincetheobserveddensityandthemodelareassumedtobexed,thisisonlyanassumptionontheprior. Intheremainderofthiswork,wefocusonquantifyingthevalueoftheseposteriordensities.Weusethe Kullback-Leiblerdivergence[35,36],tomeasuretheinformationgainedabouttheparametersfromthepriortothe posterior.WecomputetheexpectedinformationgainofagivensetofQoIagivenexperimentaldesign,andthen determinetheOEDtodeployintheeld. 3Theinformationcontentofanexperiment WeareinterestedinndingtheOEDforinferringmodelinputparameters.Conceptually,adesignisinformativeiftheposteriordistributionofthemodelparametersissignicantlydifferentfromtheprior.Toquantify the informationgain ofadesignweusetheKullback-LeiblerKLdivergence[36]asameasureofthedifference betweenapriorandposteriordistribution.WhiletheKLdivergenceisbynomeanstheonlywaytocomparetwo

PAGE 5

DRAFT probabilitydensities,itdoesprovideareasonablemeasureoftheinformationgainedinthesenseofShannoninformation[37]andiscommonlyusedinBayesianOED[30].InthissectionwediscusshowtocomputetheKL divergenceanddeneourOEDformulationbaseduponexpectedinformationgainoveraspecicspaceofpossible observeddensities. 3.1Informationgain:Kullback-Leiblerdivergence Supposewearegivenadescriptionoftheuncertaintyontheobserveddataintermsofaprobabilitydensity p obs D Thisproducesauniquesolutiontothestochasticinverseproblem P post L thatisabsolutelycontinuouswithrespectto theLebesguemeasure L [23]andadmitsaprobabilitydensity, p post L .TheKLdivergenceoftheposteriorfromthe priorinformationgain,denoted I Q ,isgivenby I Q p prior L : p post L : = Z L p post L log p post L p prior L d L ; wherewechoosethevolumemeasure, L ,tobeascaledLebesguemeasuresuchthat,forthecompact L L L = 1.Thischoiceof L impactsthescalingof p prior L and p post L whichinturnproducesaninformationgainthatis independentofthemeasureoftheparameterspace,allowingustocompare I Q valuesobtainedusingdifferent models.Notethatbecause p prior L isxed, I Q issimplyafunctionoftheposterior I Q p prior L : p post L = I Q p post L ; andfromEq.theposteriorisafunctionoftheobserveddensity.Therefore,wewrite I Q asafunctionofthe observeddensity, I Q p post L = I Q p obs D : Theobservationthat I Q isafunctionofonly p obs D allowsustodenetheexpectedinformationgaininSection3.3 basedonaspecicspaceofobserveddensities. Givenahighdimensionalparameterspace,itmaybecomputationallyinfeasibletoaccuratelyapproximate theintegralinEq..Forexample,amulti-variatenormaldensitywithunitvariancein100-dimensionshasa maximumvalueof 1 = p 2 p 100 1 10 )]TJ/F52 7.3723 Tf 5.728 0 Td [(40 .However,wemaywritethisintegralintermsofdensitiesonthedata spaceevaluatedat Q l asfollows I Q p post L = Z L p post L l log p post L l p prior L l d L = Z L p prior L l p obs D Q l p Q prior D Q l log p obs D Q l p Q prior D Q l d L = Z L p obs D Q l p Q prior D Q l log p obs D Q l p Q prior D Q l dP prior L ; wherethesecondequalitycomesfromasimplesubstitutionusingEq.4.Givenasetofsamplesfromtheprior,we onlyneedtocomputethepush-forwardofthepriorinthedataspacetoapproximate I Q .Thisobservationprovides anefcientmethodforapproximating I Q givenahighdimensionalparameterspaceandalowdimensionaldata space.Infact,wefounditconvenienttousewheneverthepriorisnotuniform.IntheconsistentBayesian formulation,weevaluatethemodelatthesamplesgeneratedfromthepriortoestimatethepush-forwardofthe prior.Itisacomputationaladvantagetoalsousethesesamplestointegratewithrespecttothepriorratherthan integratingwithrespecttothevolumemeasurewhichwouldrequireadditionalmodelevaluations.

PAGE 6

DRAFT 3.2Amotivatingnonlinearsystem Considerthefollowing2-componentnonlinearsystemofequationswithtwoparametersintroducedin[26]: l 1 x 2 1 + x 2 2 = 1 x 2 1 )]TJ/F59 9.9626 Tf 9.125 0 Td [(l 2 x 2 2 = 1 TherstQoIisthesecondcomponent,i.e., Q 1 l = x 2 l .Theparameterrangesaregivenby l 1 2 [ 0 : 79 ; 0 : 99 ] and l 2 2 [ 1 )]TJ/F52 9.9626 Tf 9.238 0 Td [(4 : 5 p 0 : 1 ; 1 + 4 : 5 p 0 : 1 ] whicharechosenasin[26]toinduceaninterestingvariationintheQoI.We assumetheobserveddensityon Q 1 isatruncatednormaldistributionwithmean0.3andstandarddeviationof0.01, seeFigure1right. Wegenerate40,000samplesfromtheuniformprioranduseakerneldensityestimatorKDEtoconstruct anapproximationtotheresultingpush-forwarddensity,seeFigure1right.ThenweuseEq.toconstructan approximationtotheposteriordensityusingthesame40,000samples,seeFigure1left,andasimpleaccept/reject algorithmtogenerateasetofsamplesfromtheposterior,seeFigure1middle.Wepropagatethissetofsamples fromtheposteriorthroughthemodelandapproximatetheresultingpush-forwardoftheposteriordensityusing aKDE.InFigure1rightweseethepush-forwardoftheposterioragreesquitewellwiththeobserveddensity. Noticethesupportoftheposteriorliesinarelativelysmallregionoftheparameterspace.Theinformationgain fromthisposterioris I Q 1 p obs D 2 : 015. Fig.1.Approximationoftheposteriordensityobtainedusingthedata Q 1 leftwhichgives I Q 1 p obs D 2 : 015 ,asetofsamplesfromthe posteriormiddle,andacomparisonoftheobserveddensityon Q 1 withthepush-forwarddensitiesofthepriorandtheposteriorright. Next,weconsideradifferentQoItouseintheinverseproblem,andcomparethesupportofitsposteriortothe onewejustobserved.Specicallyconsider, Q 2 l = x 1 : Weassumetheobserveddensityon Q 2 isatruncatednormaldistributionwithmean1.015andstandarddeviationof 0.01.Weapproximatethepush-forwarddensityandtheposteriorusingthesame40,000samplesandagaingenerate asetofsamplesfromtheposteriorandpropagatethesesamplesthroughthemodeltoapproximatethepush-forward oftheposterior,seeFigure2. Althoughboth Q 1 and Q 2 havethesamestandarddeviationintheirobserveddensities,clearlythetwoQoI produceverydifferentposteriordensities.Theposteriorcorrespondingtodatafrom Q 2 hasamuchlargerregionof supportwithintheparameterspacecomparedtothatoftheposteriorcorrespondingto Q 1 .Thisisquantiedwith theinformationgainfromthisposterior I Q 2 p obs D 0 : 466.Giventhesetwomaps, Q 1 and Q 2 ,andthespecied observeddataoneachofthesedataspaces,thedata Q 1 ismore informative oftheparametersthanthedata Q 2 Next,weconsiderusingthedatafromboth Q 1 and Q 2 Q : L Q 1 ; Q 2 ,withthesamemeansandstandard deviationsasspeciedabove.Again,weapproximatethepush-forwarddensityandtheposteriorusingthesame 40,000samples,seeFigure3.Withtheinformationfromboth Q 1 and Q 2 weseeasubstantialdecreaseinthe supportoftheposteriordensity.Intuitively,thesupportoftheposteriorusingboth Q 1 and Q 2 isthesupportofthe posteriorusing Q 1 intersectedwiththesupportoftheposteriorusing Q 2 .Thisisquantiedintheinformationgain ofthisposterior I Q p obs D 2 : 98.

PAGE 7

DRAFT Fig.2.Approximationoftheposteriordensityobtainedusing Q 2 leftwhichgives I Q 2 p obs D 0 : 466 ,asetofsamplesfromtheposterior middle,andacomparisonoftheobserveddensityon Q 2 withthepush-forwarddensitiesofthepriorandtheposteriorright. Fig.3.Theapproximationofthepush-forwardofthepriorleft,theexactobserveddensityon Q 1 ; Q 2 middle,theapproximationofthe posteriordensityusingboth Q 1 and Q 2 rightwhichgives I Q p obs D 2 : 98 Inthescenarioinwhichwecanaffordtogatherdataonboth Q 1 and Q 2 ,webenetgreatlyintermsofreducing theuncertaintiesonthemodelinputparameters.However,supposewecouldonlyaffordtogatheroneoftheseQoI intheeld.Basedontheinformationgainfromeachposterior, Q 1 ismoreinformativeabouttheparametersthan Q 2 .However,considerascenarioinwhichtheobserveddatahasdifferentmeansinboth Q 1 and Q 2 .Duetothe nonlinearitiesofthemaps,itisnotnecessarilytruethat Q 1 isstillmoreinformativethan Q 2 .Ifwedonotknowthe meanofthedataforeither Q 1 or Q 2 ,thenwewanttodeterminewhichoftheseQoIwe expect toproducethemost informativeposterior. 3.3Expectedinformationgain Optimalexperimentaldesignmustselectadesignbeforeexperimentaldatabecomesavailable.Intheabsence ofdataweusethesimulationmodeltoquantifythe expected informationgainofagivenexperimentaldesign.Let O denotethespaceofdensitiesover D .Wewanttodenetheexpectedinformationgainassomekindofaverage overthisdensityspaceinameaningfulway.However,thisisfartoogeneralofaspacetousetodenetheexpected informationgain.Thisspaceincludesdensitiesthatareunlikelytobeobservedinreality.Therefore,werestrict O tobeaspacemorerepresentativeofdensitiesthatmaybeobservedinreality. WithnoexperimentaldataavailabletospecifyanobserveddensityonasingleQoI,weassumethedensity isatruncatedGaussianwithastandarddeviationdeterminedbysomeestimateofthemeasurementinstrument error.WithGaussiansofpossiblyvaryingstandarddeviationsspeciedforeachQoI,thisdenestheshapeofthe observeddensitiesweconsider.Welet O D denotethespaceofalldensitiesofthisshapecenteredin D = Q L O D = N q ; s 2 : q 2 D ; where N q ; s 2 isatruncatedGaussianfunctionwithmean q andstandarddeviation s .Moredetailsofthisdefinitionof O D areaddressedinSection4.Wecaneasilygeneralizeourdescriptionof O .Forexample,wecould

PAGE 8

DRAFT alsoconsiderthestandarddeviationoftheobserveddatatobeuncertain,inwhichcasewewouldalsoaverageover someintervalofpossiblevaluesfor s .However,inthisworkweonlyvarythecenteroftheGaussiandensities. Remark1. Wecanrestrict O inotherwaysaswell.Forexample,ifweexpecttheuncertaintyineachQoItobe describedbyauniformdensity,thenwedenetherestrictionon O accordingly.Thischoiceofcharacterization oftheobserveddensityspaceislargelydependentontheapplication.Theonlylimitationisthatwerequirethe measurespeciedontheobserveddensityspacetobedenedintermsofthepush-forwardmeasure,P Q prior D ,as describedbelow.InSection5.2wedescribeoneapproachfordeningarestrictedobserveddensityspacewhere theobserveddensityofeachQoIhasaGaussianproleandthestandarddeviationsarefunctionsofthemagnitudes ofeachQoI. Therestrictionofpossible p obs D tothisspecicspaceofdensitiesallowsustorepresenteachdensityuniquelywith asinglepoint q 2 D .Basedonourpriorknowledgeoftheparametersandthesensitivitiesofthemap Q ,the modelinformsusthatsomedataaremorelikelytobeobservedthanotherdata,thisisseenintheplotof p Q prior D inFigure3upperleft.Thisimplieswedonotwanttoaverageover D withrespectto or D ,butratherwith respecttothepush-forwardoftheprioron D P Q prior D .Thisrespectsthepriorknowledgeoftheparametersand thesensitivityinformationprovidedbythemodel.Wedenethe expectedinformationgain ,denoted E I Q ,asjust described, E I Q : = Z D I Q q p Q prior D q d = Z D I Q q dP Q prior D : FromEq., I Q itselfisdenedintermsofanintegral.Theexpandedformfor E I Q isthenaniteratedintegral, E I Q = Z D Z L p post L l ; q log p post L l ; q p prior L l d L dP Q prior D ; wherewemakeexplicitthat p post L isafunctionoftheobserveddensityand,byourrestrictionofthespaceof observeddensitiesinEq.,thereforeafunctionof q 2 D .WeutilizeMonteCarlosamplingtoapproximatethe integralinEq.asdescribedinAlgorithm1. Algorithm1 ApproximatingtheExpectedInformationGainofanExperiment 1.Givenasetofsamplesfromthepriordensity: l i i = 1 ;:::; N ; 2.Givenasetofsamplesfromthepush-forwarddensity: q j = Q l j j = 1 ;:::; N ; 3.Constructanobserveddensitycenteredateach q j 4.For j = 1 ;:::; M approximate I Q q j using: I Q q j 1 N N i = 1 p obs D Q l i p Q prior D Q l i log p obs D Q l i p Q prior D Q l i 5.Compute E I Q 1 M M j = 1 I Q q j ; Algorithm1appearstobeacomputationallyexpensiveproceduresinceitrequiressolving M stochasticinverseproblemsand,asnotedin[23],approximating p Q prior D canbeexpensive.In[23]andinthispaperweuse kerneldensityestimationtechniquestoapproximate p Q prior D whichdoesnotscalewellasthedimensionof D increases[34].Ontheotherhand,foragivenexperimentaldesign,weonlyneedtocomputethisapproximation once,aseach I Q inStep4ofAlgorithm1iscomputedusingthesamepriorandmap Q and,therefore,thesame p Q prior D .Inotherwords,thefactthattheconsistentBayesmethodonlyrequiresapproximatingthepush-forwardof thepriorimpliesthatthisinformationcanbeusedtoapproximateposteriorsfordifferentobserveddensitieswithout requiringadditionalmodelevaluations.Thissignicantlyimprovesthecomputationalefciencyoftheconsistent

PAGE 9

DRAFT BayesianapproachinthecontextofOED.Weleveragethiscomputationaladvantagethroughoutthispaperby consideringadiscretesetofdesignswhichallowsustocomputethepush-forwardforallofthecandidatedesigns simultaneously.Utilizingacontinuousdesignspacemightrequirecomputingthepush-forwardforeachiterationof theoptimizationalgorithm,sincethedesignslocationsoftheobservationsarenotknown apriori .Theadditional modelsimulationsrequiredtocomputethepush-forwardoftheprioratnewdesignpointsmightbeintractableif thenumberofiterationsislarge,howevertheneedfornewsimulationsmaybeavoided,ifthenewobservations canbeextractedfromarchivedstate-spacedata.Forexample,ifonestorestheniteelementsolutionsofaPDEat allsamplesofthepriorattherstiterationofthedesignoptimization,onecanevaluateobsevationsatnewdesign locations,whicharefunctionalsofthisPDEsolution,viainterpolationusingtheniteelementbasis. 3.4DeningtheOED WearenowinapositionfordeneourOEDformulation.Recallthatourexperimentaldesignisdenedasthe setofQoIcomputedfromthemodelandweseektheoptimalsetofQoItodeployintheeld.Givenaphysics basedmodel,priorinformationonthemodelparameters,aspaceofpotentialexperimentaldesigns,andageneric descriptionoftheuncertaintiesforeachQoI,wedeneourOEDasfollows. Denition2OED. Let Q representthedesignspace,i.e.,thespaceofallpossibleexperimentaldesigns,and Q z 2 Q beaspecicdesign.ThentheOEDistheQ z 2 Q thatmaximizestheexpectedinformationgain, Q opt : = argmax Q z 2 Q E I Q z : Aspreviouslymentioned,thefocusinthispaperisontheutilizationoftheconsistentBayesianmethodologywithin theOEDframework,sowedonotexploredifferentapproachesforsolvingtheoptimizationproblemgivenby Denition2andsimplyndtheoptimaldesignoveradiscretesetofcandidatedesigns. Remark2. ConsistentBayesianinferenceispotentiallywellsuitedtondingOEDincontinuousdesignspaces. TypicallyOEDbaseduponstatisticalBayesianmethodsusesMarkovChainMonteCarloMCMCmethodsto characterizetheposteriordistribution.MCMCmethodsdonotprovideafunctionalformfortheposteriorbutrather onlyprovidesamplesfromtheposterior.Consequently,gradient-freeorstochasticgradient-basedoptimization methodsmustbeusedtondtheoptimaldesign.IncontrastconsistentBayesianinferenceprovidesafunctional formfortheposteriorwhichallowstheuseofmoreefcientgradientbasedoptimizers.Exploringtheuseofmore efcientcontinuousoptimizationprocedureswillbethesubjectoffuturework. 4Infeasibledata TheOEDprocedureproposedinthismanuscriptisbaseduponconsistentBayesianinferencewhichrequiresthat theobservedmeasureisabsolutelycontinuouswithrespecttothepush-forwardmeasureinducedbythepriorand themodelassumptionA3.Inotherwords,anyeventthatweobservewithnon-zeroprobabilitywillbepredicted usingthemodelandpriorwithnon-zeroprobability.Duringtheprocessofcomputing E I Q ,itispossiblethat weviolatethisassumption.Specically,dependingonthemeanandvarianceoftheobservationaldensitywemay encounter p obs D 2 O D suchthat R D p obs D d < 1,i.e.,supportof p obs D extendsbeyondtherangeofthemap Q ,see Figure5upperright.Inthissectionwediscussthecausesofinfeasibledataandoptionsforavoidinginfeasible datawhenestimatinganoptimalexperimentaldesign. 4.1InfeasibledataandconsistentBayesianinference WheninferringmodelparametersusingconsistentBayesianinferencethemostcommoncauseforinfeasible dataisthatthemodelbeingusedtoestimatetheOEDisinadequate.Thatis,thedeviationbetweenthecomputationalmodelandrealityislargeenoughtoprohibitthemodelfrompredictingalloftheobservationaldata.The deviationbetweenthemodelpredictionandtheobservationaldataisoftenreferredtoasmodelstructureerror andcanoftenbeamajorsourceofuncertainty.Thisisanissueofmostifnotallinverseparameterestimation problems[29].Recentlytherehasbeenanumberofattemptstoquantifythiserrorseee.g.,[28]howeversuch approachesarebeyondthescopeofthispaper.Inthefollowingwewillassumethatthemodelstructureerrordoes notpreventthemodelfrompredictingalltheobservationaldata. 4.2InfeasibledataandOED ToestimateanapproximateOEDwemustquantifythe expected informationgainofagivenexperimentaldesign seeSection3.3.Theexpectationisoverallpossiblenormalobservationdensitieswithmean q 2 D andvariance

PAGE 10

DRAFT s ,denedbythespace.Whenthesupportof D isboundedthesedensitiesmayproduceinfeasibledata.The effectofthisviolationincreasesas q approachestheboundaryof D ToremedythisviolationofA3wemustmodifythesetofobservationaldensities.Inthispaperwechooseto normalize p obs D over D .Weredenetheobserveddensityspace O D sothatA3holdsforeachdensityinthespace, O D = N q ; s 2 C q : q 2 D ; where N q ; s 2 isatruncatedGaussianfunctionwithmean q andstandarddeviation s ,and C q istheintegralof N q ; s 2 over D withrespecttotheLebesguemeasureon D C q = Z D N q ; s 2 d : AsimilarapproachfornormalizingGaussiandensitiesovercompactdomainswastakenin[38]. 4.3Anonlinearmodelwithinfeasibledata Inthissection,weusethenonlinearmodelintroducedinSection3.2todemonstratethatinfeasibledatacanarise fromrelativelybenignassumptions.Supposetheobserveddensityon Q 1 isatruncatednormaldistributionwith mean0.3andstandarddeviationof0.04.Inthisonedimensionaldataspace,thisobserveddensityisabsolutely continuouswithrespecttothepush-forwardoftheprioron Q 1 ,seeFigure4left.Next,supposetheobserved densityon Q 2 isatruncatednormaldistributionwithmean0.982andstandarddeviationof0.01.Again,inthisnew onedimensionaldataspace,thisobserveddensityisabsolutelycontinuouswithrespecttothepush-forwardofthe prioron Q 2 ,seeFigure4right.Bothoftheseobservedensitiesaredominatedbytheircorrespondingpush-forward densities,i.e.,themodelcanreachalloftheobserveddataineachcase. Fig.4.Thepush-forwardandobserveddensitieson Q 1 leftandthepush-forwardandobserveddensitieson Q 2 right.Noticethesupport ofbothoftheobserveddensitiesiscontainedwithintherangeofthemodel,i.e.,theobserveddensitiesareabsolutelycontinuouswithrespect totheircorrespondingpush-forwarddensities. However,considerthedataspacedenedby bothQ 1 and Q 2 andthecorrespondingpush-forwardandobserved densitiesonthisspace,seeFigure5.Thenon-rectangularshapeofthecombineddataspaceisinducedbythe nonlinearityinthemodelandthecorrelationsbetween Q 1 and Q 2 .AsweseeinFigure5,theobserveddensity usingtheproductofthe1-dimensionalGaussiandensitiesis not absolutelycontinuouswithrespecttothepushforwarddensityon Q 1 ; Q 2 ,i.e.,thesupportof p obs D extendsbeyondthesupportof p Q prior D .ReferringtoEq., wenormalizethisobserveddensityover D ,seeFigure5right.Nowthatthenewobserveddensityobeysthe assumptionsneeded,wecouldsolvethestochasticinverseproblemasdescribedinSection2. 4.4Computationalconsiderations ThemaincomputationalchallengeintheconsistentBayesianapproachistheapproximationofthepush-forward oftheprior.Following[23],weuseMonteCarlosamplingfortheforwardpropagationofuncertainty.Whilethe

PAGE 11

DRAFT Fig.5.Thepush-forwardofthepriorforthemap Q : L Q 1 ; Q 2 introducedinSection3.2left,theobserveddensityusingtheproduct ofthe1-dimensionalGaussianmiddlewhichextendsbeyondtherangeofthemap,andthenormalizedobserveddensitythatdoesnot extendbeyondtherangeofthemapright. rateofconvergenceisindependentofthenumberofparametersdimensionof L ,theaccuracyinthestatisticsfor theQoImayberelativelypoorunlessalargenumberofsamplescanbetaken.Alternativeapproachesbasedon surrogatemodelscansignicantlyimprovetheaccuracy,butaregenerallylimitedtosmallnumberofparameters. Wealsoemploykerneldensityestimationtechniquestoconstructanon-parametricapproximationofthepushforwarddensity,butitiswell-knownthatthesetechniquesdonotscalewellwiththenumberofobservations dimensionof D [34]. Next,weaddressthecomputationalissueofnormalizing N q ; s 2 ,i.e., p obs D ,over D .Fromtheplotof p Q prior D inFigure5leftitisclearthedataspacemaybeacomplexregion.Normalizing p obs D ,asinFigure5right,over D wouldbecomputationallyexpensive.Fortunately,theconsistentBayesianapproachprovidesameanstoavoid thisexpense.NotethatfromEq.wehave, P post L L = P prior L L P obs D Q L P Q prior D Q L ; where P prior L L = P Q prior D Q L = 1whichimplies, P post L L = P obs D Q L : Therefore,normalizing p obs D over D isequivalenttosolvingtheinverseproblemandthennormalizing p post L where weusethetildeover p toindicatethisfunctiondoesnotintegrateto1becausewehaveviolatedA3over L Although L maynotalwaysbeageneralizedrectangle,A1implieswehaveacleardenitionof L andtherefore canefcientlyintegrate p post L over L andthennormalize p post L by p post L = p post L R L p post L d L : Infact,thisnormalizationfactorcanbeestimatedwithoutadditionalmodelevaluationsandwithoutusingthevalues ofthepriorortheposterior,whichmaynotbeusableinhigh-dimensionalspaces.Weobservethat P post L L = Z L p post L d L = Z L p obs D Q l p Q prior D Q l dP prior L : Thus,wecanusethevaluesof p obs D and p Q prior D computedforthesamplesgeneratedfromtheprior,whichwere usedtoestimatethepush-forwardofprior,tointegrate p obs D = p Q prior D withrespecttotheprior.

PAGE 12

DRAFT 5Numericalexamples Inthissectionweconsiderseveralmodelsofphysicalsystems.First,weconsiderastationaryconvectiondiffusionmodelwithasingleuncertainparametercontrollingthemagnitudeofthesourceterm.Next,weconsider atransienttransportmodelwithatwodimensionalparameterspacedeterminingthelocationofthesourceofa contaminant.Then,weconsiderainclusionproblemincomputationalmechanicswheretwouncertainparameters controltheshapeoftheinclusion.Finally,weconsiderahigh-dimensionalexampleofsingle-phaseincompressible owinporousmediawheretheuncertainpermeabilityeldisgivenbyaKarhunen-Loeveexpansion[39]. Ineachexample,wehaveaparameterspace L ,asetofpossibleQoI,andaspeciednumberofQoIwecan affordtogatherduringtheexperiment.Thisinturndenesadesignspace Q andwelet Q z 2 Q representa singleexperimentaldesignand D z = Q z L thecorrespondingdataspace.Foreachexperimentaldesign,welet s z representthestandarddeviationsdenedbytheuncertaintiesineachQoIthatcompose Q z and O D z representthe observeddensityspace. Alloftheseexampleshavecontinuousdesignspaces,soweapproximatetheOEDbyselectingtheOEDfroma largesetofcandidatedesigns.Thisapproachwaschosenbecauseitismuchmoreefcienttoperformtheforward propagationofuncertaintyusingrandomsamplingonlyonceandtocomputeallofthecandidatemeasurementsfor eachoftheserandomsamples.Alternatively,onecouldpursueacontinuousoptimizationformulationwhichwould requireafullforwardpropagationofuncertaintyforeachnewdesign.AsmentionedinSection3.4,onecouldlimit thenumberofdesignsusingagradient-basedorNewton-basedoptimizationapproach,butthisisbeyondthescope ofthispaper. 5.1Stationaryconvection-diffusion:uncertainsourceamplitude Inthissectionweconsideraconvection-diffusionproblemwithasingleuncertainparametercontrollingthe magnitudeofasourceterm.ThisexampleservestodemonstratethattheOEDformulationgivesintuitiveresults forsimpleproblems. 5.1.1Problemsetup Considerastationaryconvectiondiffusionmodelonasquaredomain: 8 > < > : )]TJ/F57 9.9626 Tf 7.741 0 Td [(D 2 u + vu = S ; x 2 W ; u n = 0 ; x 2 G N W ; u = 0 ; x 2 G D W ; with S x = A exp )]TJ 10.32 6.745 Td [(jj x src )]TJ/F57 9.9626 Tf 9.125 0 Td [(x jj 2 2 h 2 where W =[ 0 ; 1 ] 2 u istheconcentrationeld,thediffusioncoefcient D = 0 : 01,theconvectionvector v =[ 1 ; 1 ] and S isaGaussiansourcewiththefollowingparameters: x src isthelocation, A istheamplitude, h isthewidth.We imposehomogeneousNeumannboundaryconditionson G N rightandtopboundariesandhomogeneousDirichlet conditionson G D leftandbottomboundaries.Forthisproblem,wechoose x src =[ 0 : 5 ; 0 : 5 ] ,and h = 0 : 05.We let A beuncertainwithin [ 50 ; 150 ] ,thustheparameterspaceforthisproblemis L =[ 50 ; 150 ] .Hence,ourgoalis togathersomelimitedamountofdatathatprovidesthebestinformationabouttheamplitudeofthesource,i.e., reducesouruncertaintyin A .ToapproximatesolutionstothePDEinEq.18givenasourceamplitude A ,weuse aniteelementdiscretizationwithcontinuouspiecewisebilinearbasisfunctionsdenedonauniform 25 spatialgrid. 5.1.2Results Weassumethatwehavelimitedresourcesforgatheringexperimentaldata,specically,wecanonlyaffordto placeonesensorinthedomaintogatherasingleconcentrationmeasurement.Ourgoalistoplacethissinglesensor in W tomaximizetheexpectedinformationgainedabouttheamplitudeofthesource.Wediscretize W using2,000 uniformrandompointswhichproducesadesignspacewith2,000possibleexperimentaldesigns.Forthisproblem, welettheuncertaintyineachQoIbedescribedbyatruncatedGaussianprolewithaxedstandarddeviationof 0.1.Thisproducesobserveddensityspaces, O D z ,asdescribedinEq.13.

PAGE 13

DRAFT Fig.6.Theexpectedinformationgainoverthedesignspacewhichis W inthisexampleapproximatedusing50,200,1,000and5,000 samplesfromtheprior.Noticethehighervaluesinthecenterofthedomainandtowardsthetoprightinthedirectionoftheconvectionvector fromthelocationofthesource,thisisconsistentwithourintuition.Moreover,noticethesmallchangesinthedesignspaceasweincrease thenumberofsamplesfrom50to5,000.Thissuggestswecomputeaccurateapproximationstothedesignspaceusingasfewas50model evaluations. Wegenerate5,000uniformsamplesfromthepriorandsimulatemeasurementsofeachQoIforeachofthese 5,000samples.WeconsiderapproximatesolutionstotheOEDproblemusingsubsetsofthe5,000samplesofsize 50,200,1,000and5,000.Foreachexperimentaldesign,wecalculate E I Q z usingAlgorithm1andplot E I Q z asafunctionofthediscretizeddesignspaceinFigure6.Noticetheexpectedinformationgainisgreatestnearthe centerofthedomainnearthelocationofthesourceandinthedirectionoftheconvectionvectorawayfromthe source.Thisresultmatchesintuition,asweexpectdatagatheredinregionsofthedomainthatexhibitsensitivityto theparameterstoproducehighexpectedinformationgains. Wenotethat,forthisexample,asufcientlyaccurateapproximationtothedesignspaceandtheOEDisobtained usingonly50samplescorrespondingto50modelevaluations.InTable1weshowthetop5experimentaldesigns computedusingthefullsetof5,000samplesandcorresponding E I Q z foreachsetofsamples. 5.2Timedependentdiffusion:uncertainsourcelocation Inthissection,wecompareresultsfromastatisticalBayesianformulationofOEDtotheformulationdescribed inthispaper.Specically,weconsiderthemodelin[24]wheretheauthorusesaclassicalBayesianframework forOEDtodeterminetheoptimalplacementofasinglesensorthatmaximizestheexpectedinformationaboutthe locationofacontaminantsource.

PAGE 14

DRAFT DesignLocation502001,0005,000 0 : 558 ; 0 : 571 2.7582.7672.8152.826 0 : 561 ; 0 : 546 2.7522.7622.8092.820 0 : 582 ; 0 : 574 2.7292.7362.7822.793 0 : 549 ; 0 : 570 2.7282.7352.7812.792 0 : 593 ; 0 : 596 2.7262.7332.7792.790 Table1.Thetop5experimentaldesignschosenusingthefullsetof5,000samples.Foreachofthesedesigns,wecompute E I Q z for50, 200,1,000and5,000samples.Noticethechangein E I Q z foragivendesigndecreasesasweincreaseto5,000samples. 5.2.1Problemsetup Consideracontaminanttransportmodelonasquaredomain: 8 > < > : u t = 2 u + S ; x 2 W ; t > 0 ; u n = 0 ; x 2 W ; t > 0 ; u = 0 ; x 2 W ; t = 0 : with S x = s 2 p h 2 exp )]TJ/F14 7.3723 Tf 10.32 4.931 Td [(jj x src )]TJ/F57 7.3723 Tf 5.728 0 Td [(x jj 2 2 h 2 ; if0 t < t ; 0 ; if t t ; where W =[ 0 ; 1 ] 2 u isthespace-timeconcentrationeld,weimposehomogeneousNeumannboundaryconditions alongwithazeroinitialcondition,and S isaGaussiansourcewiththefollowingparameters: x src isthelocation, s istheintensity, h isthewidth,and t istheshutofftime. Ourgoalistogathersomelimitedamountofdatathatprovidesthebestinformationaboutthelocationofthe source,i.e.,reducesouruncertaintyin x src .Forthisproblem,wechoose s = 2 : 0 ; h = 0 : 05 ; and t = 0 : 3andlet x src beuncertainwithin [ 0 ; 1 ] 2 suchthat L =[ 0 ; 1 ] 2 .ToapproximatesolutionstothePDEinEq.19givenalocationof S i.e.,agiven x src ,weuseaniteelementdiscretizationwithcontinuouspiecewisebilinearbasisfunctionsdenedon auniform 25spatialgridandbackwardEulertimeintegrationwithastepsize D t = 0 : 004timesteps. 5.2.2Results Weassumethatwehavelimitedresourcesforgatheringexperimentaldata,specically,wecanonlyaffordto placeonesensorinthedomainandcanonlygatherasingleconcentrationmeasurementattime t = 0 : 24.Ourgoalis toplacethissinglesensorin W tomaximizetheexpectedinformationgainedaboutthelocationofthecontaminant source.Forsimplicity,wediscretize W usingan11 11regulargridofpointswhichproducesadesignspacewith 121possibleexperimentaldesigns.WelettheuncertaintyineachQoIbedescribedbyaGaussianprolewitha standarddeviationthatisafunctionofthemagnitudeoftheQoI,i.e., s i = 0 : 1 + 0 : 1 j q i j for i = 1 ::: M ; where M isthedimensionofthedataspace.Thisproducesobserveddensityspaces, O D z ,thatconsistoftruncated Gaussianfunctionswithvaryingstandarddeviations, O D z = N q ; s q 2 C q : q 2 D z : Wegenerate5,000uniformsamplesfromthepriorandsimulatemeasurementsofeachQoIforeachofthese 5,000samples.WeconsiderapproximatesolutionstotheOEDproblemusingsubsetsofthe5,000samplesofsize

PAGE 15

DRAFT Fig.7.Theexpectedinformationgainoverthedesignspacewhichis W inthisexampleapproximatedusing50,200,1,000and5,000 samplesfromtheprior.Noticethehighervaluesinthecornersandthegeneraltrendareconsistentwith[24]. DesignLocation502001,0005,000 0 ; 0 0.6870.8280.7380.741 1 ; 1 0.6530.7130.7470.740 0 ; 0 : 1 0.6870.8170.7330.736 0 : 1 ; 0 0.6870.8100.7280.735 1 ; 0 : 9 0.6480.7130.7420.735 Table2.Thetop5experimentaldesignschosenusingthefullsetof5,000samples.Foreachofthesedesigns,wecompute E I Q z for50, 200,1,000and5,000samples.Noticethechangein E I Q z foragivendesigndecreasesasweincreaseto5,000samples. 50,200,1,000and5,000.Foreachexperimentaldesign,weusethisdatatocalculate E I Q z usingAlgorithm1 andplot E I Q z asafunctionofthediscretizeddesignspaceinFigure7.Noticetheexpectedinformationgainis greatestnearthecornersofthedomainandsmallestnearthecenter,thisisconsistentwith[24].InTable2weshow thetop5experimentaldesigns,approximatedusingthefullsetof5,000samples,andcorresponding E I Q z foreach setofsamples. InFigure8weconsiderthreedifferentposteriorscomputedusingdatafromtheOEDapproximatedusing 5,000samples,i.e.,datagatheredbyasensorplacedinthebottomleftcornerofthedomain,whereeachposterior correspondstoadifferentpossiblelocationofthesource.Weseevaryinglevelsofinformationgaininthesethree scenarios,reiteratingthepointthatwechoosetheOEDbasedontheaverageoftheseinformationgains, E I Q Remark3. Althoughmanyoftheresultsinthissectionseemtomatchourintuitionaboutwhichmeasurement locationsshouldproducehighexpectedinformationgains,thismaynotalwaysbethecase.Inparticular,wehave

PAGE 16

DRAFT Fig.8.Posteriors,approximatedusing5,000samples,usingtheOEDforthreerealizationsofthelocationofthesource.Noticetheinformationgainchangessubstantiallyforeachposterior,however,thisexperimentaldesign,placementofthesensorinthebottomleftcorner, producesthemaximumaverageinformationgain, E I Q ,overallpossiblelocationsofthesource. foundthatourresultscandependonourchoiceofthevarianceinthetheobserveddensities s .If s ischosento belargerelativetotherangeofadataspace,thentheposteriorsproducedasweaverageover O D areallnearly thesameandpotentiallyproduceunusuallyhighinformationgainswhentheobserveddensitieshavesubstantial supportoverregionsofthedataspacewithverysmallprobabilityverysmallvaluesofthepush-forwardofthe prior.Anotherwaytothinkofthisisthepush-forwarddensitieshavehighentropyandbecause s islarge p obs D is veryclosetouniformandthisproducesposteriordensitieswithhighinformationgains.If s ischosentobesmall relativetotherangeofthedataspace,i.e.,ifweexpecttheexperimentstobeinformative,wedonotencounter thisissuebecauseweareintegratingover D withrespecttothepush-forwardmeasuresomostofourpotential observeddataliesinhighprobabilityregionsofthedataspace. 5.3AParameterizedInclusion Inthissection,weconsiderasimpleproblemincomputationalmechanicswherethepreciseboundaryofan inclusionisuncertain.Weparameterizetheinclusionandseektodeterminethelocationtoplaceasensorthat willmaximizetheinformationgainedregardingtheshapeoftheinclusion.Weusealinearelasticformulationto modeltheresponseofthemediatosurfaceforcesandmeasurehorizontalstressateachsensorlocation.Weassume thatthematerialpropertiesPoissonratioandYoung'smodulusaredifferentinsidetheinclusionandthatthese propertiesareknownaprior. 5.3.1Problemsetup Consideralinearelasticplanestrainmodel, 8 > < > : )]TJ/F59 9.9626 Tf 7.741 0 Td [( s u = 0 ; x 2 W =[ )]TJ/F52 9.9626 Tf 7.74 0 Td [(5 ; 5 ] [ 0 ; 2 ] ; u = g ; x 2 G D = f x ; y 2 W j x = 0 g ; s u n = t ; x 2 G N = W n G D ; where s u isgivenbythelinearelasticconstitutiverelation, s u = l u I + u + u T : WeexpressthisrelationintermsoftheLam eparameters, l and ,whicharerelatedtothePoissonratio, n ,and Young'smodulus, E ,viathefollowingexpressions, = E 2 1 + n ; l = E n 1 + n 1 )]TJ/F52 9.9626 Tf 9.125 0 Td [(2 n : Nowassumethatthereisaninclusionwithinthemediadenedbyanellipse I = x ; y 2 W j 1 a x )]TJ/F57 9.9626 Tf 9.125 0 Td [(x 0 2 + 1 b y )]TJ/F57 9.9626 Tf 9.125 0 Td [(y 0 2 1 ;

PAGE 17

DRAFT Fig.9.ThecomputationaldomainandPoissonratioshowingtheinclusionforaparticularrealizationoftheellipsoidparameters. where x 0 = y 0 = 0and a isuniformlydistributedon [ 0 : 5 ; 1 ] and b isuniformlydistributedon [ 0 : 25 ; 0 : 5 ] .The materialpropertiesareassumedtobeknownandaregivenby, n = 0 : 45 ; x ; y 2 I ; 0 : 3 ; otherwise ; ; E = 10 : 0 ; x ; y 2 I ; 40 : 0 ; otherwise ; : Thesematerialpropertieswerenotchosentoemulateanyparticularmaterials,justtodemonstratetheproposed OEDformulation. NextletusimposehomogeneousDirichletboundaryconditionsonthebottomboundaryandstressfreeboundaryconditionsonthesides,andimposeauniformtractioninthey-directionalongthetopboundary t top = 0 ; )]TJ/F52 9.9626 Tf 7.741 0 Td [(1 T .Andnallyassumethatwecanprobethemediaandmeasurethehorizontalstressatagivensensor location.Wedonotwanttopuncturetheinclusion,soweonlyconsidersensorlocationsoutsidetheboundsonthe inclusion.Equation22wassolvedusinganiteelementdiscretizationwithpiecewiselinearbasisfunctionsdened onauniform400 80meshresultinginasystemwith64,962degreesoffreedom.Thecomputationalmodelis implementedusingtheTrilinostoolkit[40]andeachrealizationofthemodelrequiresapproximately1secondusing 8processors. 5.3.2Results Asinpreviousexamples,weassumethatwehavelimitedresourcesforgatheringexperimentaldata,specically, wecanonlyaffordtoplaceonesensorinthedomaintogatherasinglestressmeasurement.Ourgoalistoplacethis singlesensortomaximizetheexpectedinformationgainedabouttheshapeoftheinclusion.Weselect2,000random sensorlocationsoutsidetheinclusionboundswhichproducesadesignspacewith2,000possibleexperimental designs.Forthisproblem,welettheprobabilitydensityfortheQoIbedescribedbyatruncatedGaussianprole withaxedstandarddeviationof0.001.Wegenerate1,000uniformsamplesfromthepriorandcomputethe horizontalstressateachsensorlocationforeachofthese1,000samples. First,wecomparetheposteriordensitiesfortwosensorlocations, 3 : 5294 ; 1 : 3049 and 1 : 3902 ; 1 : 2100 ,under theassumptionthatwehavealreadygathereddataatthesesensorlocations.Thepurposehereistodemonstratethat weobtaindifferentposteriordensitiesandthereforegaindifferentinformationfromeachsensor.Therstsensoris furtherfromtheinclusionsoweexpectthatthedatafromthesecondsensorwillconstraintheposteriormorethan thedatafromtherst.InFigures10and11,weplotthesamplesfromtheposteriorandthecorrespondingkernel densityestimateoftheposteriorfortherstandsecondsensorlocationsrespectively.Itisclearthatmeasuringthe horizontalstressclosertotheinclusionincreasestheinformationgainedfromthepriortotheposterior. WeconsiderapproximatesolutionstotheOEDproblemusingsubsetsofthe1,000samplesofsize10,50,100 and1,000.Foreachexperimentaldesign,weusethisdatatocalculate E I Q z usingAlgorithm1andplot E I Q z asafunctionofthediscretizeddesignspaceinFigure12.Noticetheexpectedinformationgainisgreatestnear thebottomofthedomainneartheinclusionandisreasonablysymmetricaroundtheinclusion.Alsonotethatthe expectedinformationgainisrelativelylargeinthebottomcornersofthedomain.Thisisduetothechoiceof boundaryconditionsforthemodelwhichinducesalargeamountofstressinthesecorners.InTable2weshowthe top5experimentaldesigns,approximatedusingthefullsetof1,000samples,andcorresponding E I Q z foreach setofsamples. 5.4AHigher-DimensionalPorousMediaExamplewithUncertainPermeability Inthissection,weconsideranexampleofsingle-phaseincompressibleowinporousmediawithaKarhunenLo eveexpansionoftheuncertainpermeabilityeld.ThepurposeofthisexampleistodemonstratetheOED formulationonaproblemwithahigh-dimensionalparameterspaceandmorethanonesensor.

PAGE 18

DRAFT Fig.10.Thesetofsamplesfromtheposteriorandthecorrespondingkerneldensityestimateoftheposteriorfortherstsensorlocation, 3 : 5294 ; 1 : 3049 Fig.11.Thesetofsamplesfromtheposteriorandthecorrespondingkerneldensityestimateoftheposteriorforthesecondsensorlocation, 1 : 3902 ; 1 : 2100 DesignLocation10501001,000 )]TJ/F52 9.9626 Tf 7.74 0 Td [(1 : 001 ; 0 : 961 2.3033.6624.0854.569 )]TJ/F52 9.9626 Tf 7.74 0 Td [(1 : 050 ; 1 : 035 2.3033.3913.8114.261 1 : 041 ; 0 : 959 2.3033.4373.8394.256 )]TJ/F52 9.9626 Tf 7.74 0 Td [(1 : 050 ; 1 : 130 2.3023.4183.6614.096 1 : 005 ; 0 : 870 2.2773.2463.5443.813 Table3.Thetop5experimentaldesignschosenusingthefullsetof1,000samples.Foreachofthesedesigns,wecompute E I Q z for10, 50,100and1,000samples. 5.4.1Problemsetup Considerasingle-phaseincompressibleowmodel: 8 > > > < > > > : )]TJ/F59 9.9626 Tf 7.741 0 Td [( K l p = 0 ; x 2 W = 0 ; 1 2 ; p = 1 ; x = 0 ; p = 0 ; x = 1 ; K p n = 0 ; y = 0and y = 1 :

PAGE 19

DRAFT Fig.12.Theexpectedinformationgainoverthedesignspacewhichis W inthisexampleapproximatedusing10,50,100and1,000 samplesfromtheprior.Noticethehighervaluesnearthelocationoftheinclusion. Here, p isthepressureeldand K isthepermeabilityeldwhichweassumeisascalareldgivenbyaKarhunenLo eveexpansionofthelogtransformation, Y = log K ,with Y l = Y + i = 1 x i l p h i f i x ; y ; where Y isthemeaneld.WeassumethemeanremovedrandommediaisgivenbyaGaussianprocesswhich impliesthatthe x i aremutuallyuncorrelatedrandomvariableswithzeromeanandunitvariance[41,42].The

PAGE 20

DRAFT DesignLocation501001,00010,000 0 : 48 ; 1 1.9662.0011.9922.008 0 : 5 ; 0 : 98 1.9521.9631.9932.007 0 : 46 ; 0 : 98 1.9301.9851.9862.006 0 : 52 ; 0 1.7771.9151.9992.006 0 : 56 ; 0 1.7511.8631.9962.006 Table4.Thetop5experimentaldesignschosenusingthefullsetof10,000samples.Foreachofthesedesigns,wecompute E I Q z for 50,100,1,000and10,000samples.Noticethechangein E I Q z foragivendesigndecreasesasweincreaseto10,000samples. eigenvalues, h i ,andeigenfunctions, f i ,arecomputednumericallyusingthefollowingcovariancefunction, C Y x ; x = s 2 Y exp )]TJ/F8 9.9626 Tf 8.936 6.744 Td [( x 1 )]TJETq1 0 0 1 316.521 555.267 cm[]0 d 0 J 0.398 w 0 0 m 4.453 0 l SQBT/F57 9.9626 Tf 316.521 549.544 Td [(x 1 2 2 h 1 )]TJ/F8 9.9626 Tf 10.32 6.744 Td [( x 2 )]TJETq1 0 0 1 369.089 555.267 cm[]0 d 0 J 0.398 w 0 0 m 4.453 0 l SQBT/F57 9.9626 Tf 369.089 549.544 Td [(x 2 2 2 h 2 ; where s Y and h i denotethevarianceandcorrelationlengthinthei th spatialdirectionrespectively.Weassumea correlationlengthof0 : 01ineachspatialdirectionandtruncatetheexpansionat100terms.Thischoiceoftruncation ispurelyforthesakeofdemonstration.Inpractice,theexpansionistruncatedonceasufcientfractionoftheenergy intheeigenvaluesisretained[39,41].Thistruncationgives100uncorrelatedrandomvariables, x 1 ;:::; x 1 00,with zeromeanandunitvariancewhichimplies L = R 1 00.ToapproximatesolutionstothePDEinEq.23weuse aniteelementdiscretizationwithcontinuouspiecewisebilinearbasisfunctionsdenedonauniform 50 spatialgrid. 5.4.2Results Inthissection,wepresentapproximatesolutionstoseveraldifferentdesignproblems.Webeginwiththe familiarproblemofchoosingasinglesensorlocationwithinthephysicaldomain.Then,weconsiderapproximating theoptimallocationofasecondsensor given thelocationoftherstsensor.Inthisway,wesolvethe greedy OED problemanddeterminethegreedyoptimallocationsof1-8sensorswithinthephysicaldomain.Wethenconsider solvingtheehaustiveOEDproblemwherewelimitthesensorsto25locationsandconsiderdeterminingtheoptimal locationof5availablesensors. First,assumethatwehavelimitedresourcesforgatheringexperimentaldata,specically,wecanonlyafford toplaceonesensorinthedomaintogatherasinglepressuremeasurement.Ourgoalistoplacethissinglesensor in W tomaximizetheexpectedinformationgainedabouttheamplitudeofthesource.Wediscretize W using1,301 piontsonagridwhichproducesadesignspacewith1,301possibleexperimentaldesigns.Forthisproblem,welet theuncertaintyineachQoIbedescribedbyatruncatedGaussianprolewithaxedstandarddeviationof0.01. Thisproducesobserveddensityspaces, O D z ,asdescribedinEq.13. Wegenerate10,000samplesfromthepriorandsimulatemeasurementsofeachQoI.Weconsiderapproximate solutionstotheOEDproblemusingsubsetsofthe10,000samplesofsize50,100,1,000and10,000.Foreach experimentaldesign,wecalculate E I Q z usingAlgorithm1andplot E I Q z asafunctionofthediscretizeddesign spaceinFigure13.Noticetheexpectedinformationgainisgreatestnearthetopandbottomofthedomainaway fromtheleftandrightedges.Thisresultmatchesintuition,asweexpectdatagatheredneartheleftandright edgestobelessinformativegiventheDirichletboundaryconditionimposedonthoseboundaries.Wenotethat,for thisexample,asufcientlyaccurateapproximationtothedesignspaceandtheOEDisobtainedusingonly1,000 samplescorrespondingto1,000modelevaluations.InTable1weshowthetop5experimentaldesignscomputed usingthefullsetof10,000samplesandcorresponding E I Q z foreachsetofsamples. Next,weconsiderthegreedyOEDproblemofplacing8sensorswithinthephysicaldomain.Wechoosetouse alloftheavailable10,000samplestosolvethisproblem.InFigure14,weseethedesignspaceasafunctionofthe previouslydeterminedlocationsofplacedsensors.Weobserveastrongsymmetrytothisproblem,asisexpected duetothesymmetryofthephysicalprocessdenedonthisdomainwiththegivenboundaryconditions.Inthe bottomrightofFigure14,noticetheverysmallrangeofthecolorbarindicatingthepossiblevaluesoftheexpected informationgain.Thissuggests,forthisexample,weexpectthereisalimitonthenumberofusefulsensorlocations forinforminglikelyparametervalues.

PAGE 21

DRAFT Fig.13.Theexpectedinformationgainoverthedesignspacewhichis W inthisexampleapproximatedusing50,100,1,000and10,000 samplesfromtheprior.Noticethesmallchangesinthedesignspaceasweincreasethenumberofsamplesfrom1,000to10,000.This suggestswecomputeaccurateapproximationstothedesignspaceusingasfewas1,000modelevaluations. Lastly,weconsidertheexhaustiveOEDproblemofplacing5sensorswithinthephysicaldomainand,for computationalfeasibility,restrictthepossiblelocationsofthese5sensorsto25pointsinthephysicaldomain,see Figure15.Wechoosetouse1,000samplestosolvethisproblem.InFigure15,weplotthedesignspaceforasingle sensorlocationusingthese1,000samplesandshowtheoptimallocationof1,2,3,4and5sensors.Theresultsare quitesimilartothegreedyresultspreviouslydescribed. 6Conclusion Inthismanuscript,wedevelopedanOEDformulationbasedontherecentlydeveloped consistent Bayesianapproachforsolvingstochasticinverseproblems.WeusedtheKullback-Leiblerdivergenceandtheposteriorobtained usingconsistentBayesianinferencetomeasuretheinformationgainofadesignandpresentadiscreteoptimization procedureforchoosingtheoptimalexperimentaldesignthatmaximizestheexpectedinformationgain.Theoptimizationprocedurepresentedinthispaperislimitedintermsofthenumberofobservationswecanconsider,but waschosentofocusattentiononthedenitionandapproximationoftheexpectedinformationgained.Moreefcientstrategies,utilizinggradient-basedmethodsoncontinuousdesignspaces,willbepursuedinafuturework.We discussedacharacterizationofthespaceofobserveddensitiesneededtocomputetheexpectedinformationgainand acomputationallyefcientapproachforrescalingobserveddensitiestosatisfytherequirementsoftheconsistent Bayesianapproach.Numericalexamplesweregiventohighlightthepropertiesandutilityofourapproach. 7Acknowledgments J.D.Jakeman'sworkwassupportedbyDARPAEQUIPS.

PAGE 22

DRAFT Fig.14.Theexpectedinformationgainoverthedesignspaceasafunctionofpreviouslychosensensorlocations.Notethattherangeof thecolorbarchangesintheprogressionofthegures.Inthebottomright,weseethegreedyoptimallocationofeightsensorswithinthe physicaldomain.

PAGE 23

DRAFT Fig.15.Inblack,weshowthepossiblelocationsofthesensorsandinwhiteweshowtheoptimallocationsfor1,2,3,4and5sensors. Inthebottom,weseetheoptimallocationofvesensorswithinthephysicaldomain.Notethateachcolorbarisindicativeoftheexpected informationgainfortherstsensorlocation,notfortheexpectedinformationgainformultiplesensors. References [1]Cox,D.R.,andReid,N.,2000. Thetheoryofthedesignofexperiments .CRCPress. [2]Fisher,R.A.,1966. TheDesignofExperiments ,8thed.Oliver&Boyd,Edinburgh,UnitedKingdom. [3]Pazman,A.,1986. FoundationsofOptimumExperimentalDesign .D.ReidelPublishingCo. [4]Pukelsheim,F.,1993. OptimalDesignofExperiments .JohnWiley&Sons,New-York. [5]Ucinski,D.,2005. Optimalmeasurementmethodsfordistributedparametersystemidentication .CRCPress, BocaRaton. [6]Korkel,S.,Kostina,E.,Bock,H.G.,andSchloder,J.P.,2004.Numericalmethodsforoptimalcontrol problemsindesignofrobustoptimalexperimentsfornonlineardynamicprocesses. OptimizationMethods andSoftware, 19 -4,pp.327.

PAGE 24

DRAFT [7]Chung,M.,andHaber,E.,2012.Experimentaldesignforbiologicalsystems. SIAMJournalonControland Optimization, 50 ,pp.471. [8]Bock,H.G.,K orkel,S.,andSchl oder,J.P.,2013. ParameterEstimationandOptimumExperimentalDesign forDifferentialEquationModels .SpringerBerlinHeidelberg,Berlin,Heidelberg,pp.1. [9]Horesh,L.,Haber,E.,andTenorio,L.,2010. OptimalExperimentalDesignfortheLarge-ScaleNonlinear Ill-PosedProblemofImpedanceImaging .JohnWiley&Sons,Ltd,pp.273. [10]Bauer,I.,Bock,H.G.,Krkel,S.,andSchlder,J.P.,2000.Numericalmethodsforoptimumexperimental designin f DAE g systems. JournalofComputationalandAppliedMathematics, 120 ,pp.125. [11]Haber,E.,Magnant,Z.,Lucero,C.,andTenorio,L.,2012.Numericalmethodsfora-optimaldesignswith asparsityconstraintforill-posedinverseproblems. ComputationalOptimizationandApplications, 52 pp.293. [12]Alexanderian,A.,Petra,N.,Stadler,G.,andGhattas,O.,2014.A-optimaldesignofexperimentsforinnitedimensionalbayesianlinearinverseproblemswithregularized l 0 -sparsication. SIAMJournalonScientic Computing, 36 ,pp.A2122A2148. [13]Haber,E.,Horesh,L.,andTenorio,L.,2008.Numericalmethodsforexperimentaldesignoflarge-scale linearill-posedinverseproblems. InverseProblems, 24 ,p.055012. [14]Alexanderian,A.,Petra,N.,Stadler,G.,andGhattas,O.,2016.Afastandscalablemethodfora-optimaldesignofexperimentsforinnite-dimensionalbayesiannonlinearinverseproblems. SIAMJournalonScientic Computing, 38 ,pp.A243A272. [15]Atkinson,A.,andDonev,A.,1992. OptimumExperimentalDesigns .OxfordUniversityPress. [16]Chaloner,K.,andVerdinelli,I.,1995.Bayesianexperimentaldesign:Areview. Statist.Sci., 10 ,08, pp.273. [17]Long,Q.,Scavino,M.,Tempone,R.,andWang,S.,2015.Alaplacemethodforunder-determinedbayesian optimalexperimentaldesigns. ComputerMethodsinAppliedMechanicsandEngineering, 285 ,pp.849. [18]Loredo,T.J.,andChernoff,D.F.,2003. BayesianAdaptiveExploration .SpringerNewYork,NewYork,NY, pp.57. [19]M uller,P.,Sans o,B.,andIorio,M.D.,2004.Optimalbayesiandesignbyinhomogeneousmarkovchain simulation. JournaloftheAmericanStatisticalAssociation, 99 ,pp.788. [20]Solonen,A.,Haario,H.,andLaine,M.,2012.Simulation-basedoptimaldesignusingaresponsevariance criterion. JournalofComputationalandGraphicalStatistics, 21 ,pp.234. [21]Long,Q.,Scavino,M.,Tempone,R.,andWang,S.,2013.Fastestimationofexpectedinformationgainsfor bayesianexperimentaldesignsbasedonlaplaceapproximations. ComputerMethodsinAppliedMechanics andEngineering, 259 ,pp.2439. [22]Haber,E.,Horesh,L.,andTenorio,L.,2010.Numericalmethodsforthedesignoflarge-scalenonlinear discreteill-posedinverseproblems. InverseProblems, 26 ,p.025002. [23]Butler,T.,Jakeman,J.,andWildey,T.,2016.Aconsistentbayesianformulationforstochasticinverseproblemsbasedonpush-forwardmeasures.SubmittedtoSIAMJ.Sci.Comput. [24]Huan,X.,2015.Numericalapproachesforsequentialbayesianoptimalexperimentaldesign. arXiv:1604.08320 [25]Stuart,A.M.,2010.Inverseproblems:Abayesianperspective. ActaNumerica, 19 ,5,pp.451. [26]Breidt,J.,Butler,T.,andEstep,D.,2012.Acomputationalmeasuretheoreticapproachtoinversesensitivity problemsI:Basicmethodandanalysis. SIAMJ.Numer.Analysis, 49 ,pp.1836. [27]Butler,T.,Pilosov,M.,andWalsh,S.,2016.Simulation-basedoptimalexperimentaldesign:Ameasuretheoreticperspective.InReview,http://digital.auraria.edu//IR00000054/00001. [28]Sargsyan,K.,Najm,H.N.,andGhanem,R.,2015.Onthestatisticalcalibrationofphysicalmodels. InternationalJournalofChemicalKinetics, 47 ,pp.246. [29]Kennedy,M.C.,andO'Hagan,A.,2001.Bayesiancalibrationofcomputermodels. JournaloftheRoyal StatisticalSociety:SeriesBStatisticalMethodology, 63 ,pp.425. [30]Huan,X.,andMarzouk,Y.M.,2013.Simulation-basedoptimalbayesianexperimentaldesignfornonlinear systems. JournalofComputationalPhysics, 232 ,pp.288317. [31]Chaloner,K.,andVerdinelli,I.,1995.Bayesianexperimentaldesign:Areview. InstituteofMathematical Statistics, 10 ,pp.273. [32]Huan,X.,andMarzouk,Y.,2014.Gradient-basedstochasticoptimizationmethodsinbayesianexperimental design. InternationalJournalforUncertaintyQuantication, 4 ,pp.479. [33]Tarantola,A.,2005. InverseProblemTheoryandMethodsforModelParameterEstimation .SIAM. [34]Wand,M.,andJones,M.,1994.Multivariateplug-inbandwidthselection. ComputationalStatistics, 9 pp.97. [35]Kullback,S.,andLeibler,R.A.,1951.Oninformationandsufciency. TheAnnalsofMathematical

PAGE 25

DRAFT Statistics, 22 ,pp.79. [36]vanErven,T.,andHarremoes,P.,2014.Renyidivergenceandkullback-leiblerdivergence. IEEETransactionsonInformationTheory, 60 ,pp.3797. [37]Cover,T.A.,andThomas,J.A.,2006. ElementsofInformationTheory .JohnWiley&Sons. [38]Bisetti,F.,Kim,D.,Knio,O.,Long,Q.,andTempone,R.,2016.Optimalbayesianexperimentaldesignfor priorsofcompactsupportwithapplicationtoshock-tubeexperimentsforcombustionkinetics. International JournalforNumericalMethodsinEngineering, 108 ,pp.136. [39]Zhang,D.,andLu,Z.,2004.Anefcient,high-orderperturbationapproachforowinrandomporousmedia viaKarhunen-Lo eveandpolynomialexpansions. JournalofComputationalPhysics, 194 ,pp.773. [40]Heroux,M.,Bartlett,R.,Hoekstra,V.H.R.,Hu,J.,Kolda,T.,Lehoucq,R.,Long,K.,Pawlowski,R.,Phipps, E.,Salinger,A.,Thornquist,H.,Tuminaro,R.,Willenbring,J.,andWilliams,A.,2003.AnOverviewof Trilinos.Tech.Rep.SAND2003-2927,SandiaNationalLaboratories. [41]Ganis,B.,Klie,H.,Wheeler,M.F.,Wildey,T.,Yotov,I.,andZhang,D.,2008.Stochasticcollocationand mixedniteelementsforowinporousmedia. Computermethodsinappliedmechanicsandengineering, 197 ,pp.3547. [42]Wheeler,M.F.,Wildey,T.,andYotov,I.,2011.Amultiscalepreconditionerforstochasticmortarmixed niteelements. ComputerMethodsinAppliedMechanicsandEngineering, 200 ,pp.1251. [43]Schwab,C.,andTodor,R.A.,2006.Karhunen-Lo eveapproximationofrandomeldsbygeneralizedfast multipolemethods. JournalofComputationalPhysics, 217 ,pp.100122.UncertaintyQuanticationin SimulationScience.