JournalofComputationalPhysics229(2010)48–4663ContentslistsavailableatScienceDirect
JournalofComputationalPhysicsjournalhomepage:www.elsevier.com/locate/jcpNumericalapproachforquantificationofepistemicuncertainty
JohnJakemana,MichaelEldredb,1,DongbinXiuc,*,2aDepartmentofMathematics,AustralianNationalUniversity,AustraliaSandiaNationalLaboratories,Albuquerque,NM87185,UnitedStatescDepartmentofMathematics,PurdueUniversity,WestLafayette,IN47907,UnitedStatesbarticleinfoabstract
Inthefieldofuncertaintyquantification,uncertaintyinthegoverningequationsmayassumetwoforms:aleatoryuncertaintyandepistemicuncertainty.Aleatoryuncertaintycanbecharacterisedbyknownprobabilitydistributionswhilstepistemicuncertaintyarisesfromalackofknowledgeofprobabilisticinformation.Whileextensiveresearcheffortshavebeendevotedtothenumericaltreatmentofaleatoryuncertainty,littleatten-tionhasbeengiventothequantificationofepistemicuncertainty.Inthispaper,wepro-poseanumericalframeworkforquantificationofepistemicuncertainty.Theproposedmethodologydoesnotrequireanyprobabilisticinformationonuncertaininputparame-ters.Themethodonlynecessitatesanestimateoftherangeoftheuncertainvariablesthatencapsulatesthetruerangeoftheinputvariableswithoverwhelmingprobability.Toquan-tifytheepistemicuncertainty,wesolveanencapsulationproblem,whichisasolutiontotheoriginalgoverningequationsdefinedontheestimatedrangeoftheinputvariables.Wedis-cusssolutionstrategiesforsolvingtheencapsulationproblemandthesufficientconditionsunderwhichthenumericalsolutioncanserveasagoodestimatorforcapturingtheeffectsoftheepistemicuncertainty.Inthecasewhereprobabilitydistributionsoftheepistemicvariablesbecomeknownaposteriori,wecanusetheinformationtopost-processthesolu-tionandevaluatesolutionstatistics.Convergenceresultsarealsoestablishedforsuchcases,alongwithstrategiesfordealingwithmixedaleatoryandepistemicuncertainty.Severalnumericalexamplesarepresentedtodemonstratetheprocedureandpropertiesoftheproposedmethodology.Ó2010ElsevierInc.Allrightsreserved.Articlehistory:Received7October2009Receivedinrevisedform8February2010Accepted3March2010Availableonline15March2010Keywords:UncertaintyquantificationEpistemicuncertaintyGeneralizedpolynomialchaosStochasticcollocationEncapsulationproblem1.IntroductionMathematicalmodelsareusedtosimulateawiderangeofsystemsandprocessesinengineering,physics,biology,chem-istryandenvironmentalsciences.Thesesystemsaresubjecttoawiderangeofuncertainties.Theeffectsofsuchuncertaintyshouldbetracedthroughthesystemthoroughlyenoughtoallowonetoevaluatetheireffectsontheintendeduseofthemodelusually,butnotalways,relatedtopredictionofmodeloutputs.Therearetwoformsofmodeluncertainty:aleatoryandepistemic.Aleatoryuncertaintyarisesfromtheinherentvariationassociatedwiththesystemunderconsiderationandisirreducible.Epistemicuncertaintyrepresentsanylackofknowledge*Correspondingauthor.Tel.:+176962846.E-mailaddresses:john.jakeman@anu.edu.au(J.Jakeman),mseldre@sandia.gov(M.Eldred),dxiu@purdue.edu(D.Xiu).1SandiaisamultiprogramlaboratoryoperatedbySandiaCorporation,aLockheedMartinCompany,fortheUnitedStatesDepartmentofEnergy’sNationalNuclearSecurityAdministrationunderContractDE-AC04-94AL85000.2TheworkofD.XiuwassupportedbyAFOSR,DOE/NNSA,andNSF.0021-9991/$-seefrontmatterÓ2010ElsevierInc.Allrightsreserved.doi:10.1016/j.jcp.2010.03.003J.Jakemanetal./JournalofComputationalPhysics229(2010)48–466349orinformationinanyphaseoractivityofthemodelingprocess[13]andisreduciblethroughtheintroductionofadditionalinformation.Thesourcesofaleatoryuncertaintyaretypicallyrepresentedusingaprobabilisticframeworkunderwhichthealeatoryuncertaintycanberepresentedbyafinitenumberofrandomvariableswithsomeknowndistribution.Thesourcesofale-atoryuncertaintyincludebothuncertaintyinmodelcoefficients(parametricuncertainty)anduncertaintyinthesequenceofpossibleevents(stochasticuncertainty).Stochasticuncertaintyisentirelyaleatorybynature.Parametricuncertaintycanalsobecompletelyaleatoryifthecompletedistributionofallthemodelparametersareknownapriori.Frequently,strongstatisticalinformationsuchasprobabilitydistributionfunctionsorhigh-orderstatisticalmomentsisnotavailable.Experimentaldataneededtoconstructthisinformationisoftenexpensiveandconsequentlynodata,oronlyasmallcollectionofdatapoints,maybeobtainable.Inthesecases‘‘expertopinion”isusedinconjunctionwiththeavailabledatatoproduceweakinferentialestimatesofparametriccharacteristics,oftenintheformoflowerandupperbounds.Othersourcesofepistemicuncertaintyincludelimitedunderstandingormisrepresentationofthemodeledprocess,knowncom-monlyas‘‘modelform”uncertainty.Inclusionof‘‘enough”additionalinformationabouteitherthemodelparametersorstructurecanleadtoareductioninthepredicteduncertaintyofamodeloutput.Consequently,wecanconsiderepistemicuncertaintyasproviding(conservative)boundsonanunderlyingaleatoryuncertainty,wherereductionandconvergencetothetruealeatoryuncertainty(or,insomecases,aconstantvalue)canbeobtainedgivensufficientadditionalinformation.Untilrecently,mostuncertaintyanalysishasfocusedonaleatoryuncertainty.Numerousmethodshavebeendevelopedthatprovideaccurateandefficientestimatesofthisformofuncertainty.Inparticular,stochasticGalerkin(SG)[2,9,28]andstochasticcollocation(SC)[1,5,8,16,21,25,27]methodsprovideaccuraterepresentationsofaleatoryuncertaintyandhavetheabilitytodealwithsteepnon-lineardependenceofthesolutiononrandommodeldata.Foradetailedreviewonthemeth-ods,see[26].Incomparisontothequantificationofaleatoryuncertainty,theanalysisofepistemicuncertaintyhasprovedmorechal-lenging.Probabilisticrepresentationsofepistemicuncertaintyareinappropriate,sincethecharacterizationofinputepiste-micuncertaintythroughwell-definedprobabilitydistributionsimposesalargeamountofunjustifiedstructureontheinfluenceoftheinputsonthemodelpredictions.Thiscanresultinstrongerinferencesthanarejustifiedbytheavailabledata.Evidencetheory[12],possibilitytheory[4]andintervalanalysis[11,19]havebeenproposedasmoreappropriatealter-natives,wheretheyarelistedindescendingorderbasedontheamountofimposedinputstructure.Oftheaforementionedmethods,evidencetheoryisthemostcloselyrelatedtoprobabilitytheory.Evidencetheorystartsfrombasicprobabilityassignmentsontheinputs,propagatesthesedescriptionsthroughamodelusingstandardsamplingtechniques,andproducesestimatesofthelowestandhighestprobabilitiesofthemodelobservables.Theseestimatesdefinecumulativebeliefandcumulativeplausibilityfunctionsthatrepresenttheuncertaintyintheoutputmetrics,wherebeliefprovidesameasureoftheamountofinformationthatsupportsaneventbeingtrueandplausibilitymeasurestheabsenceofinformationthatsupportstheeventbeingfalse.Theevidencetheoryrepresentationofuncertaintyapproachestheprob-abilisticrepresentationastheamountofinformationabouttheinputdataincreases[12].Possibilitytheoryiscloselyrelatedtofuzzysettheoryand,similartoevidencetheory,utilizestwodescriptionsoflike-lihood,necessityandpossibility.Thesetwomeasuresarebaseduponthepropertiesofindividualelementsoftheuniversalsetofevents,unlikeplausibilityandbeliefwhicharederivedfromthepropertiesofsubsetsoftheuniversalset.Formoredetails,see[10].Evidenceandpossibilitytheoryrequireaggregationofdatafrommultiplesourcesintoaformatconsistentwiththecho-sentechnique.Inpractice,thiscanbedifficultandtimeconsuming.Intervalanalysis[17],ontheotherhand,onlyrequiresupperandlowerboundsontheuncertaininputdata.Samplingand/oroptimization[5,19]isthenusedtogenerateupperandlowerbounds(intervals)onthemodeloutputsfrompredefinedintervalsontheinputdata.Theapplicationofevidencetheory,possibilitytheoryandintervalanalysistonon-linearandcomplexproblemsoftenre-quiresaprohibitivelylargenumberofsamplesandtypicallyunderestimatestheoutputextrema.Globalsurrogatemodelshavebeenusedinanattempttoalleviatethisproblem[10];however,theperformanceoftheseapproachesishighlydepen-dentontheaccuracyofthesurrogatemodelandconstructioncostscanbehighwhenglobalaccuracyisrequiredandcon-vergenceratesarenotexponentiallyfast.Inmorerecentwork,surrogateswithadaptiverefinementstrategieshavebeencombinedwithstochasticcollocationmethods[5,6,19,20]inordertosegregatealeatoryquantificationwithstochasticexpansionsfromepistemicquantificationusingoptimization-basedintervalestimation.Thechoiceoftheaforementionedmethodsdependsontheamountofavailableinformationwhichcanbeutilizedtochar-acterizetheinputuncertainty.Consequentlythischoiceishighlyproblemdependent.Hereweproposeanewandmoregen-eralframeworktonumericallyquantifyepistemicuncertainty.Thisproposedmethodcandealwithvaryingamountsofinformationontheinputdatafromsimpleboundstofullprobabilisticdescriptions,andthuscanseamlesslyhandletheproblemswithbothepistemicandaleatoryuncertainties.Furthermoretheproposedapproachutilizestheclassicalapprox-imationtheoryinmulti-dimensionalspaceandachieveshighefficiencythanthemethodscurrentlyavailable.Unlikemanyexistingnumericalmethodsforquantifyingepistemicuncertainty,theproposedmethodrequiresonlyanapproximationoftherangesoftheinputdatathatencapsulatesthe‘‘true”boundsoftheinputdata.Wethenproposesolvingan‘‘encapsulationproblem”whichgeneratesasolutiontothegoverningequationsinadomainthatenclosesthetrue(andunknown)probabilityspace.Hereamulti-dimensionalpolynomialexpansioncanbeemployedtoapproximatethesolutiononthelargerencapsulationspace.Weshowthatifsucharepresentationconvergesintheencapsulationspacethenthismethodwillalsoconvergeinthetrueprobabilityspace.Furthermore,convergenceismaintainedeveninthepresenceof4650J.Jakemanetal./JournalofComputationalPhysics229(2010)48–4663dependenciesbetweeninputdata.Wealsodemonstratenumericallythatifthedistributionsoftheinputdataarefoundaposteriori,thepolynomialapproximationofthesolutionstatisticswillconverge.InSection2,wepresentthenecessarymathematicalframeworkforquantifyingepistemicuncertaintyandinSection3wediscusstheconstructionandsolutionoftheencapsulationproblem.InparticularwefocusonpolynomialbasedGalerkinandcollocationmethodsandillustratehowthesemethodscanbeusedtoconstructefficientandaccurateapproximationsofthesolutiontotheencapsulationproblem.Wealsoextendtheencapsulationapproachtomodelswithmixedepistemicandale-atoryuncertaintyanddiscusshowtoextractandinterpretimportantstatisticalinformation.Numericalexamplesarepre-sentedinSection4andweconcludethepaperinSection5.2.ProblemsetupLetD&R‘;‘¼1;2;3,beaphysicaldomainwithcoordinatesx¼ðx1;...;x‘ÞandletT>0bearealnumber.Weconsiderthefollowinggeneralstochasticpartialdifferentialequation8>:v¼v0;DÂft¼0gÂIZ;ð2:1ÞwhereLisa(non-linear)differentialoperator,Bistheboundaryconditionoperator,v0istheinitialcondition,andZ¼ðZ1;...;ZdÞ2IZ&Rd;dP1,areasetofrandomvariablescharacterizingtherandominputstothegoverningequation.Thesolutionisthereforeastochasticquantity,½0;TÂIZ!Rnv:vðx;t;ZÞ:Dð2:2ÞWithoutlossofgenerality,hereafterweassume(2.1)isascalarsystemwithnv¼1.Wealsomakeafundamentalassump-tionthattheproblem(2.1)iswell-posedinIZ.Mostoftheexistingstudiesadoptaprobabilisticformulationtoquantifyaleatoryuncertainty.Thatis,itistypicallyas-sumedthatthedistributionoftherandomvariablesZisknown,withthemostwidelyadoptedapproachassumingthemar-ginaldistributionsofZiareknownandallZiareindependentfromeachother.Inthispaper,however,weconsiderthecasewheretheuncertaintyisepistemic.Thatis,thedistributionfunctionsofZiarenotknown,primarilyduetoourlackofunder-standingandcharacterizationofthephysicalsystemgovernedbythesystemofequations(2.1).ThefocusisonthedependenceofthesolutionontheuncertaininputsZ;therefore,wepresentsolutionsforfixedlocationxandtimetandomitthesevariableswheneverpossible.3.MethodologyWenowpresentamethodforsolvingsystem(2.1)subjecttoepistemicuncertaininputs.Theproposedmethodologyisathree-stepprocedurewhichinvolvesidentifyingtherangesoftheuncertaininputs,generatinganaccuratepolynomialapproximationofthesolutionto(2.1)withinestimatedrangesandpost-processingtheresults.Notethatnoprobabilitydis-tributioninformationwillbeutilizedinthesolutionprocedure.3.1.RangeestimationUnliketraditionalaleatoryuncertaintyquantification,theproposedmethodonlyrequiresanestimateoftherangesoftherandominput.Thegoalistoidentifyarange,orbound,thatissufficientlylargesuchthatthe‘‘true”,andyetunknown,rangeoftheinputuncertaintyismostlyincorporated.Wenowillustratetheideamoreprecisely.ForeachrandomvariableZi;i¼1;...;d,letIZi¼½ai;bi;ai:DÂft¼0gÂIX;u¼u0;ð3:10ÞwhereIXistheboundedhypercubedefinedin(3.6).Thisiseffectivelythesameproblem(2.1)definednowontheencapsu-lationsetIXthatcoverstheoriginalrandomparametersetIZwithprobabilityatleast1Àd.ThenewproblemiswelldefinedinIXbecausewehaveassumedtheestimatedrangeofeachIXistaysintherangeofwell-posednessallowedbythegoverningequation.Sinceproblem(2.1)and(3.10)areexactlythesameinthecommondomainIo,wehavethefollowingtrivialresult,uðÁ;nÞ¼vðÁ;nÞ;8n2Io:ð3:11ÞWeremarkthatfortheencapsulationproblem(3.10)wedonotassignanyprobabilityinformationtovariablesX.3.3.SolutionfortheencapsulationproblemForsolutionoftheencapsulationproblem(3.10),weagainfocusonlyonthedependenceonthevariablesX,whichnowresidesinthehypercubeIX#Rd.4652J.Jakemanetal./JournalofComputationalPhysics229(2010)48–4663Foranyfixedlocationxandtimet,uðXÞ:IX!R:ð3:12ÞAcriticalrequirementfortheproposedmethodologyistheneedforthenumericalapproximationof(3.10)toconvergepoint-wise.LetunðXÞbeanumericalsolution,wheretheindexnisassociatedwithdiscretizationparametersusedintheapproximation.Wethenrequiren,kuÀunkL1ðIXÞ!0;n!1;ð3:13ÞwherekÁk1isthestandardL1normdefinedaskukL1ðIXÞ¼supjuðXÞj:X2IXWhenuissufficientlysmooth,suchkindofpoint-wiseconvergentapproximationcanbeobtained,atleastinprinciple.Whiletherearechoicesforsolvingtheencapsulationproblem(3.10),herewewillfocusonpolynomialapproximationun,wheretheindexnistypicallyassociatedwiththehighestdegreeofpolynomialsusedintheapproximation.Thiskindofmethodsaredirectextensionofpolynomialapproximationtheoryandthegeneralizedpolynomialchaos(gPC)methodsusedprimarilyforaleatoryuncertaintyanalysis.Againweemphasizethatthekeyofchoosingaparticularmethodisthat,despiteitscomputationalefficiency,itshouldprovideanaccurateapproximationintheL1normof(3.13).Withoutlossofgeneralityandmerelyfornotationalconvenience,hereafterweassumetheencapsulationsetIXisahypercubeIX¼½À1;1d;dP1:ð3:14ÞNotethiscanalwaysbeaccomplishedbyproperlytranslatingandscalingofthevariablesXin(3.10).3.3.1.CollocationapproachThesolutionuðXÞtotheencapsulationproblem(3.10)isdecoupledintheparameterspaceIX.Subsequentlywecansolve(3.10)atasetofdiscretenodesandthenconstructapolynomialapproximationofuthatinterpolatesthesolutionateachnode.Thissocalledcollocationapproachhasbeenusedextensivelytoquantifyaleatoryuncertainty[1,25,27]LetHn¼fXjgmj¼1&IXbeasetof(prescribed)nodes,wheremP1isthenumberofnodes.Byadoptingthecollocationmethodology,weenforce(3.10)atthenodeXj;j¼1;...;m,andsolve8j>:DÂft¼0g:u¼u0;ð3:15ÞItiseasytoseethatforeachj,(3.15)isadeterministicproblemwithfixedvaluesofX.Therefore,solvingthesystemposesnodifficultyprovidedonehasawell-establisheddeterministicalgorithm.Letuj¼uðÁ;XjÞ;j¼1;...;m,bethesolutionoftheaboveproblemandfujgmj¼1betheensembleofsolutionsobtainedbysolving(3.15).Throughuseofthesolutionensemble,wethenseektoconstructunðXÞ2PðXÞ,wherePðXÞisaproperpoly-nomialspace,sothattheconvergenceproperty(3.13)canbeachieved.Whilethegeneralstrategyisstraightforward,theoptionsforpracticalimplementationarelimited.Multivariateapprox-imationisachallengingareawithmanyopenissues.Here,wedescribeamoreestablishedmethodbasedonsparsegridinterpolation[3],whichhasbeenusedextensivelyinquantifyingaleatoryuncertaintyfollowingtheworkof[27].miSparsegridinterpolation,isbaseduponacombinationofone-dimensionalinterpolationformula.LetHi1¼fX1i;...;XigmbeasetofdistinctnodesinthedirectionXiandfuðXjigj¼i1thenumericalsolutionatthesenodes.Wecanapproximatetheone-dimensionalcomponentofthesolutionuovertherangeofXiusingthefollowinginterpolationformulaUi½u¼miXj¼1uðXjiÞÁWjiðXiÞ;ð3:16ÞwheremiisthenumberofcollocationnodesandWjiistheinterpolatingbasiswhichsatisfiesthediscreteorthogonalityprop-ertyWjiðXkiÞ¼djk,TheLagrangepolynomialsandthepiecewiselinearbasisaretwocommonlyusedbases.Inthemultivariatecased>1wecanapproximateubytheNth-levelSmolyakformula([18])UN¼XNÀdþ16jij6NðÀ1ÞNÀjijÁdÀ1NÀjijÁðUi1ÁÁÁUidÞ;ð3:17Þwherei¼ði1;...;idÞandjij¼i1þÁÁÁþid.See[24]fordetailedderivationoftheformula.TocomputetheinterpolatingsolutionuNðXÞ¼UN½u;oneonlyneedstoevaluatethefunctionuonthesparsegrid,J.Jakemanetal./JournalofComputationalPhysics229(2010)48–46634653HN;d¼[NÀdþ16jij6NðHi11ÂÁÁÁÂH1dÞ:ið3:18ÞToachievebetterefficiency,thenodalsetsshouldbenestedsothatHi&Hiþ1andHN;d&HNþ1;d.ThismeansthattoincreasethelevelofinterpolationfromNtoNþ1weneedonlysolve(3.10)onthenewsetofpointsHNþ1;dnHN;d.Unlikeafulltensorproductconstruction,whichsuffersfromthecurseofdimensionalityinthatthenumberofnodesgrowsexponentiallywiththedimensiond,thenumberofnodesrequiredbytheSmolyakformulaonlygrowslogarithmicallywithd.Differentsparsegridinterpolationscanbeconstructedbasedonthechoiceofone-dimensionalinterpolation(3.16).OnepopularchoiceisClenshaw–Curtisinterpolation,whichutilizestheLagrangepolynomialbasisdefinedontheextremaoftheChebyshevpolynomials.Foranychoiceofmi;16i6d,thesenodesaregivenbyXji¼ÀcospðjÀ1ÞmiÀ1;j¼1;...;mi;ð3:19ÞToensurethenodalsetsarenested,wechoosem1¼1andmi¼2iÀ1þ1;fori>1Therehavebeenextensivestudiesontheapproximationpropertiesofsparsegrids,particularlythosebasedonClenshaw–Curtisabscissas.Here,weciteoneoftheearlyresultsfrom[3].ForfunctionsinspacedjijF‘d¼ff:½À1;1!Rj@fcontinuous;ij6‘;8jg;withnormkfk¼maxfkDafk1;a2Nd0;ai6‘g;theinterpolationerrorfollowskIÀUNk6Cd;‘mÀ‘ðlogmÞð‘þ2ÞðdÀ1Þþ1;ð3:20ÞwheremisthetotalnumberofnodesrequiredbythesparsegridinterpolationHN;d.(Notethereisingeneralnoexplicitfor-mulaform.)WhenquantifyingaleatoryuncertaintytheClenshaw–Curtissparsegridsareonlyappropriatewhentheunderlyingran-domvariablespossessuniformdistributions.Howeverwhenquantifyingepistemicuncertaintythisrequirementcanbere-moved.TheClenshaw–Curtisgridmaynotbeoptimalforthe‘‘true”unknowndistribution,however,theresultingapproximationwillstillexhibittherequiredpoint-wiseconvergence,albeitataslowerrate.SubsequentlyClenshaw–Curtissparsegridinterpolation,orsparsegridinterpolationforthatmatter,iscertainlynottheonlychoiceforthecollocationap-proach.Inpractice,anyvalidinterpolationapproachcanbeemployed,solongasonecanestablishitsconvergenceinthepoint-wisesenseof(3.13).3.3.2.GalerkinapproachWebrieflyremarkthat(3.10)canalsobesolvedbytheGalerkinapproach.IntheGalerkinapproach,weseekanumericalsolutionunðXÞinapolynomialspacesuchthattheresidualof(3.10)isorthogonaltothepolynomialspace.WhilemostoftheconvergenceoftheGalerkinsolutionisintheweightedLpnormonIX,itispossibletohavethesolutionconvergepoint-wiseuniformly,whichiswhatwerequire.Thisusuallyimposesstrongersmoothnessconditionsonthetruesolutionu.Forexam-ple,forstochasticdiffusionequationwithlinearformrandomdiffusivity,itwasshownthatthesolutionisanalyticintermoftherandominputs[1],andnumericalsolutionconverginginpoint-wisesensecanbeusedforsamplingnon-Gaussianpro-cess[23].SinceitisnotpossibletodiscusstheconvergenceoftheGalerkinapproachwithoutspecifyingtheformof(3.10),wewillnotengageinmorediscussionsonthis.Wealsoremarkthatthereexistsapseudo-spectralcollocationmethod[25],alsoknownasnon-intrusivegPCmethod.Thoughthismethodisofcollocationtype,itsconvergenceisusuallyinLpnorm,similartoGalerkin.Thereforeitisnotpos-sibletodiscussitsL1errorwithoutspecifyingtheunderlyinggoverningequationsandwewillnotfocusonthismethodaswell.3.3.3.PiecewisesmoothcaseInthediscussionsabovewehaverequiredthesolutiontoconvergein1-normintheentiredomainIX.Thisrequiressuf-ficientglobalsmoothnessofu,whichisratherstronginmanypracticalproblems.Infact,thediscussionscanbegeneralizedtopiecewisesmoothfunctionofu.Thatis,thereexistsafinitedecompositionofIkX;k¼1;...;m,suchthatm[k¼1IkX¼IX;IiX\\IjX¼;;i–j;andineachsubdomainIkX;k¼1;...;m;uðXÞissmooth.46J.Jakemanetal./JournalofComputationalPhysics229(2010)48–4663Inthiscase,asuitablenumericalapproachintheglobalsense(e.g.thesparsegridinterpolation)canbeappliedtoeachksubdomainseparatelyandobtainaconvergentsolution(inthe1-norm)uknðXÞineachsubdomainIX.Agloballyconvergentsolutioncanthenbeconstructedby‘‘patching”thesubdomainsolutionstogether.unðXÞ¼mXk¼1uknðXÞIIkðXÞ;Xð3:21ÞwhereIAðsÞ¼1ifs2A;IAðsÞ¼0otherwise,istheindicatorfunction.ItiseasytoseethatthissolutionwillconvergetouintheentiredomainofIXinthe1-norm.Notethatduetothenatureoftheproblem(3.10),thereisnocontinuityrequirementofthesolutionacrossthesubdomaininterfaces.Therefore,atleastontheconceptuallevelsolvingthesubdomainproblemisstraightforward.Fromthepracticalpointofview,themulti-element,orpiecewise,approximationtechniquesdevelopedforaleatoryuncertaintycanbeborrowed.Theseincludetheworkof[2,7,14,15,22].Hereafterwewillrestrictourselvestoglob-allysmoothproblemstoemphasizethenewconceptualideasrelatedtoepistemicuncertaintyquantification.3.4.EpistemicuncertaintyanalysisWhenunðXÞ,thepolynomialapproximationofthetruesolutionuðXÞ,isobtainedfor(3.10)andconvergesinthe1-norm(3.13),itcanserveasanaccurateandpoint-wisemodel.Wecanthenapplyvariousoperationsonuninsteadofu.Notetheoperationsonundonotrequireustosolvethegoverningequationsanymore—theycanbetreatedaspost-processingsteps.AssuminginformationaboutthedistributionoftherandominputsZisknownaposteriori,thenwecanevaluatethesolu-tionstatisticsbyusingtheun.ThiscanbeachievedbyevaluatingthestatisticsofunusingtheprobabilityofZinthedomainIodefinedin(3.7).LetqZðsÞ¼dFZðsÞ;s2IZ,betheprobabilitydensityfunctionoftheepistemicuncertaininputZ,whichwasnotknownpriortothecomputationsbutisnowknownafterthecomputations.Then,forexample,themeanofthetruesolu-tionvðZÞ,l,E½ðvÞðZÞ¼ZZIZvðsÞqZðsÞds;ð3:22Þcanbeapproximatedbyln,IounðsÞqZðsÞds:ð3:23ÞThefollowingresultcanbeestablishedTheorem3.1.Assumethesolutionof(2.1),vðZÞ,isboundedandletCv¼kvkL1ðIZÞ.LetunðXÞbeanapproximationtothesolutionuðXÞof(3.10)andconvergeintheformof(3.13)anddenoten¼kunðXÞÀuðXÞkLjlÀlnj6nþCvd:1ðIXÞ:ð3:24ÞThenthemeanofvin(3.22)andthemeanofunin(3.23)satisfyð3:25ÞProof.Wefirstextendthedomainofdefinitionofv,q,anduntoIþ,followingthedefinitionsofthedomainsin(3.7),anddefineqþðsÞ¼IIZðsÞqZðsÞ;vþðsÞ¼IIZðsÞvðsÞ;s2Iþ;anduþnðsÞ¼IIXðsÞunðsÞ;Zs2Iþ:ZNaturally,qþisaprobabilitydensityfunctiononIþ.Then(3.22)canbeexpressedasl¼IZvðsÞqZðsÞds¼IþvþðsÞqþðsÞds;ZIowhichcanbesplitintotwopartsl¼ZIovþðsÞqðsÞdsþþZIÀvþðsÞqðsÞds¼þvðsÞqZðsÞdsþZIoZIÀvþðsÞqþðsÞds:ZIÀ\\IZð3:26ÞByusing(3.23),wehavelÀln¼ZIoðvðsÞÀunðsÞÞqZðsÞdsþZIÀvðsÞqþðsÞds¼ðuðsÞÀunðsÞÞqZðsÞdsþvðsÞqZðsÞds;ð3:27Þwheretheproperty(3.11)hasbeenused.Utilizingthecondition(3.9),themainresult(3.25)isestablished.hJ.Jakemanetal./JournalofComputationalPhysics229(2010)48–46634655Thesecondtermin(3.27),Cvd,isaresultoftruncatingthe‘‘tails”ofthetrueprobabilitydistributionofZbyusingaboundedhypercubeIXtoencapsulateapossiblyunboundeddomainIZ.Forunboundeddomains,forwhichIÀ\\IZ–;,thetermCvdcanbemadearbitrarilysmallbychoosingasufficientlybighypercubeforIX.IfZisinaboundeddomainandthedomainIXcancompletelyencapsulateIZthenthetermCvddisappears.Inthiscase,theerrorinthemeanwouldconvergetozeroaslongasunconvergesintheformof(3.13).Weremarkthattheencapsulationproblemcanalsobeusedtoobtainupperandlowerboundsonthemodelobservables.Consequentlysolvingtheencapsulationproblemcanbeusedasanalternativetotechniquessuchasintervalanalysis[17].Theaccuracyoftheseestimatesisbeyondthescopeofthispaperandisleftforfuturework.3.5.MixedaleatoryandepistemicuncertaintyanalysisInpractice,situationsmayariseforwhichthedistributionsofsomeoftherandomvariablescharacterizingtheinputareknownandsomearenot.Theencapsulationmethodologyproposedherecaneasilybeextendedtosuchcasespossessingmixedaleatoryandepistemicuncertainty.Letusconsiderstochasticdifferentialequationswiththefollowingform8>:v¼v0;DÂft¼0gÂIYÂIZ;ð3:28ÞwhereYisasetofrandomvariableswithknownprobabilitydistributionFYðyÞ¼PðY6yÞ;y2IY#Rr;rP1,andZ2Rsareasetofrandomvariableswithunknowndistribution.Asinthepurelyepistemiccasewefirstbeginbydefiningandsolvinganencapsulationproblem.WiththisaimweagaindefinetheencapsulationsetIXaccordingto(3.6)whichencapsulatesIZ,the‘‘true”andunknownsupportofZ,withproba-bilityatleast1Àd.Theencapsulationproblemisthen8>:DÂft¼0gÂIYÂIX:u¼u0;ð3:29ÞUnlikeinthepurelyepistemiccase,theencapsulationproblemisnowdefinedintermsoftheepistemicandaleatoryvari-ables.Thisencapsulationproblemcanbesolvedintwowaysdependingonwhetheronewantstosolvetheepistemicandaleatoryproblemsseparatelyorsimultaneously:Separateconstruction:DifferentmethodscanbeemployedtoquantifytheepistemicuðÁ;XÞandaleatoryuncertaintyekðYÞbeanapproximationto^mðXÞbeanapproximationtouðÁ;XÞafterfixingallvariablesotherthanXanduuðÁ;YÞ.LetuuðÁ;YÞ,wheretheindicesmandkdenotingthelevelofapproximations.ThenuðX;YÞcanbeapproximatedbyatensorekðYÞ.Thatis,^mðXÞanduproductofu^mðXÞuekðYÞ;unðX;YÞ¼uwheretheindexndependsonmandk.(Inthecaseofpolynomialapproximation,ncanbeeitherthehighestpolynomialekorthetotalorderofthemixedpolynomialsofu^manduek.)Theconstructionallowsustousedifferent^manduorderinu^mðXÞconverginginL1normintheepi-methodsforXandY.Forexample,wecanmixanaccuratecollocationsolutionuekðYÞconverginginL2stemicvariableXwithanaccuratestochasticGalerkinsolutionuqYnorminthealeatoryvariableY.Simultaneousconstruction:Insteadoftreatingtheepistemicandaleatoryvariablesseparately,wecanconsiderthealea-toryvariablesasepistemicandsolvetheepistemicencapsulationproblem(3.10),withIXdefinedinsuchawaythatitencapsulatesbothIZandIY.ForraleatoryvariablesYandsepistemicvariablesZ,wedefineÀÁÀsÁÂÂIX¼ÂrIIYXi¼1ii¼1i;ð3:30ÞwhereIXiareboundedintervalsthatencapsulateIZiwithoverwhelmingprobability.ThesamemethodsfortheepistemicencapsulationproblemcanbeusedtogenerateanapproximationinIX.TheprobabilisticinformationassociatedwiththealeatoryvariablesYcanbeintroducedinpost-processing.AdditionalprobabilisticinformationforZcanbeprocessedaposterioriwhenknown.Weremarkthatthisapproachrequirespoint-wiseaccuracyintheentirespace(3.30).ThismaybetoostrongbecauseingeneralaccuracyinmeansquaresenseinthealeatoryvariablesYissufficient.Point-wiseaccu-racyisparticularlyhardtoachievewhenthealeatoryvariablesareunbounded.Inthiscasewemayneedtotruncatethedomainofthealeatoryvariablesandthisleadstoadditional‘‘truncation”error.Therefore,thesimultaneousapproachismoreappropriatewhenallthevariablesarebounded.AftertakingintoaccounttheprobabilitydistributionofthealeatoryrandomvariablesY,thesolutionbecomesafunctionoftheepistemicvariableZ.Forexample,themeansolutionislðsÞ¼EY½vðY;ZÞ¼Zvðy;sÞdFYðyÞ;s2IZ:4656J.Jakemanetal./JournalofComputationalPhysics229(2010)48–4663ThiscanbeapproximatedbylnðsÞ¼EY½unðY;XÞ¼Zunðy;sÞdFYðyÞ;s2IX:NotethatnoprobabilityinformationisassignedtothevariablesXandZpriortoanycomputations.4.NumericalexamplesInthissection,weprovideseveralnumericalteststoillustratetheimplementationandconvergenceoftheproposedmethodology.Inallexamples,wefirstseekpolynomialapproximationtothesolutionsintermsoftheepistemicvariables.Wethen,inpost-processingsteps,assumecertainprobabilitydistributioninformationisknownaposterioriandtheneval-uatethesolutionstatisticsofthetruesolutionandthenumericalapproximationsandexaminetheaccuracyofthemethods.Inallexamples,weutilizeglobalpolynomialapproximations.4.1.OrdinarydifferentialequationConsiderdvðtÞ¼ÀZ1v;dtvð0Þ¼Z2;ð4:1ÞwheretheparametersZ1andZ2arerandomvariablesrepresentingtheinputuncertainty.Theexactsolutionisvðt;ZÞ¼Z2expðÀZ1tÞ:ð4:2ÞLetusassumethatthedistributions(anddependence)ofZ1andZ2areunknown,exceptthattheboundsoftheparameterscanbeestimatedwitharangethatissufficientlywide.Theencapsulationproblemisutðt;XÞ¼ÀX1u;uð0Þ¼X2;ð4:3ÞwhereX¼ðX1;X2Þ2½À1;12afterscaling.Here,weusetheGalerkinmethodbasedonLegendrepolynomialstosolve(4.3).ThisimpliesthenumericalsolutionwillconvergeintheL2norm.However,sincethesolutionisanalytic,point-wiseconvergencecanalsobeachieved.Forcompar-ison,wealsopresentresultsusingasparsegridapproximationofubaseduponthetensorproductofLagrangepolynomialsdefinedattheClenshaw–Curtisabscissas.Bothmethodsprovidefastconvergingpolynomialapproximationsofthesolution.Weillustratetheconvergenceofthemeanandvarianceoftheseapproximationswhenthemarginalandjointprobabilitydistributionsarefoundaposterioribelow.Fig.4.1.ConvergenceoftherelativeerrorinthemeanandvarianceforthelinearODEwithtwodimensional(d¼2)independentinputwithvaryingaposterioridistributions,Z1;Z22betað0;1;1;1Þ(solidlines),Z12betað0;1;2;5ÞandZ22betað0;1;1;1Þ(dashedlines).(a)ConvergencewithGalerkinpolynomialorder.(b)Convergencewithcollocationsparsegridlevel.J.Jakemanetal./JournalofComputationalPhysics229(2010)48–466346574.1.1.IndependentcaseLetusassumethe‘‘true”(andyetunknown)distributionofZisZ12betað0;1;a1;b1Þ;Z22betað0;1;a2;b2Þ,whereZ1andZ2areindependentandbetaða;b;a;bÞisthebetadistributionontheinterval[a,b]withdistributionparametersaandb.Ana-lyticalexpressionsforthemomentsofvexistandcanbeusedtotestconvergenceofournumericalsolutionsoftheencap-sulationprobleminX.NoteinthiscaseIZ¼½0;1½0;1andiscompletelyencapsulatedbyIX¼½À1;12.Themomentsofthenumericalapproximationswereobtainedbyusingmulti-dimensionaltensorproductGauss–Jacobiquadrature.Specificallyanappropriatehigh-orderone-dimensionalquadraturerule,determinedbythenowknowndistri-butionofZ,waschosenforeachindependentvariableandthenatensorproductoftheseruleswasusedtoconstructasetofmulti-dimensionalquadraturenodesandassociatedweights.Theorderofthequadraturerulewaschosentomatchtheor-deroftheapproximatingpolynomial.Samplingthepolynomialexpansionatthequadraturenodesisapost-processingpro-cessandonlyrequirestheevaluationofalgebraicexpressionsandisthusinexpensivecomparedtothecostofevaluatingthetruemodel.InFig.4.1therelativeerrorinthefirsttwomomentsareshownforvaryingvaluesofaiandbi.Hereandthroughouttheremainderofthispaperrelativeerrorisdefinedtobetheabsolutedifferencebetweentheapproximateandtruevaluenor-malizedbythetruevalue.AstheorderoftheLegendre–Galerkinpolynomialexpansionandtheapproximationlevelofthecollocationsparsegridincreases,theerrorsconvergeexponentiallyfastbeforereachingsaturationlevels.4.1.2.DependentcaseLetusassumethe‘‘true”(andyetunknown)distributionofZisZ12betað0;1;a1;b1Þ;Z2isdependentonZ1.SpecificallyletusassumethatZ2¼Z1.ThisimpliesthatIZ¼½0;1andcanbeentirelyencapsulatedbyIX¼½À1;12.Againanalyticalexpressionsforthemomentsofvexistandcanbeusedtotestconvergenceofournumericalapproximations,whosemo-mentswereobtainedbyselectinganappropriatehigh-orderone-dimensionalquadraturerule,determinedbythenowknowndistributionofZ,forthevariablesZ1.Thisone-dimensionalquadraturerulewasthenusetoevaluatethemomentsoftheapproximationsalongthelineZ1¼Z2.Fig.4.2plotstheerrorinthefirsttwomomentsforvaryingvaluesofaiandbi.Astheapproximationlevelofthecollo-cationsparsegridincreasestheerrorsconvergeexponentiallyfastbeforereachingsaturationlevels.4.1.3.ChoiceofpolynomialbasisInSections4.1.1and4.1.2,multi-dimensionalLagrangeandLegendrepolynomialswereusedtoproduceanapproxima-tiontothesolutionoftheencapsulationproblem.Howeveranytypeofpolynomialthatsatisfies(3.13)canbeused.Itwasshownin[28]that,inthecontextofaleatoryuncertaintyquantification,ifthepolynomialbasisusedtoapprox-imateastochasticsolutionischosenaccordingtothedistributionoftheunderlyingrandomvariables,betterapproximationaccuracycanbeachieved.Iftheoptimalbasisisnotchosen,therateofconvergencewilldeteriorate.Here,weinvestigatetheeffectofthechoiceoftheapproximatingpolynomialontheconvergenceofthemeanandvarianceofsolutionssubjecttoepistemicuncertainty.Letusassumethe‘‘true”(andyetunknown)distributionsofZiareindependent.Fig.4.3showstherateofconvergenceintheestimatesofvarianceforvarioustypesofpolynomialapproximationsoftheencapsulationproblem(4.3).Whentheopti-malpolynomialbasisisused,estimatesofthevarianceareobtaineddirectlyfromthebasiscoefficients.ThevarianceoftheFig.4.2.ConvergenceoftherelativeerrorinthemeanandvarianceforthelinearODEwithtwodimensionalðd¼2Þindependentinputwithvaryingaposterioridistributions,Z1;2betað0;1;1;1Þ(solidlines),Z12betað0;1;2;5Þ(dashedlines).InallcasesZ2¼Z1.(a)ConvergencewithGalerkinpolynomialorder.(b)Convergencewithcollocationsparsegridlevel.4658J.Jakemanetal./JournalofComputationalPhysics229(2010)48–4663100LegendreOptimal10−210Relative Error−410−610−810−1010−121234567Approximation OrderFig.4.3.ConvergenceoftherelativeerrorinthevarianceforthelinearODEwithtwoindependentinputvariables.ConvergenceisshownwithrespecttotheorderofthegPCexpansionortwochoicesofstochasticpolynomialexpansionsandfortwodifferentinputdistributions;Z1;Z22betaðÀ1;1;1;1Þ(solidlines)andZ1;Z22betaðÀ1;1;2;2Þ(dashedlines).(non-optimal)Legendreapproximationwascalculatedusingahigh-ordertwo-dimensionaltensorproductGauss–Jacobiquadraturerule.Whenthetypeofpolynomialexpansionischosentomatchthedistributionoftheinputvariables,afasterofconvergenceisobtainedthanifanothertypewaschosen.Thenatureofepistemicuncertaintymeansthatanoptimalbasiscannotbecho-senaprioriandsomeaccuracypenaltymayhavetobepaidduetothelackoffullprobabilisticinformationatthetimeofexpansioncomputation.If,bycoincidence,thebasischosentoapproximatetheencapsulationproblemmatchestheweight-ingfunctionsoftheunderlyingrandomvariables,thentheoptimalconvergenceratewillbeachieved.Inmostcases,how-ever,wemustselectabasisthatprovidesareasonablecompromisegiventheinformationavailable;e.g.ifonlyboundsareprovidedandthereisnojustificationtoweighterrorsunequallywithinthesebounds,thenaLegendrebasisisthenaturalchoice.4.2.RandomoscillatorThissectioninvestigatestheperformanceofstochasticcollocationtoquantifyepistemicuncertaintyinadampedlinearoscillatorsubjecttoexternalforcingwithsixunknownparameters.Thatis,dxdt22ðt;ZÞþcdxþkx¼fcosðxtÞ;dtð4:4Þsubjecttotheinitialconditionsxð0Þ¼x0;_ð0Þ¼x1;xð4:5Þwhereweassumethedampingcoefficientc,springconstantk,forcingamplitudefandfrequencyx,andtheinitialcondi-tionsx0andx1arealluncertain,andletZ¼ðc;k;f;x;x0;x1Þ2R6betheepistemicvariables.Theencapsulationproblemisthendx2dtxð0Þ¼X5;ðt;XÞþX12dxþX2x¼X3cosðX4tÞ;dt_ð0Þ¼X6;xð4:6Þð4:7ÞwhereX¼ðX1;...;X6Þ2½À1;16(uponscaling)aretheencapsulationvariables.WeemploysparsegridLagrangeinterpola-tionattheClenshaw–Curtisabscissastosolvetheencapsulationproblem.J.Jakemanetal./JournalofComputationalPhysics229(2010)48–466346594.2.1.EpistemicuncertaintywithdependentinputsToagainillustratetheconvergenceofmoments,letusassumethe‘‘true”(andyetunknown)distributionofZisdependentonZ3andZ6isdependentonZ5.Forexample,Zi2betaðai;bi;ai;biÞ;i¼1;3;5andZ2isdependentonZ1;Z4isZ21andZ2¼4þ0:01;Z32betað0:08;0:12;1;1ÞandZ4¼10Z3,andletusassumeZ12betað0:08;0:12;3;2ÞZ52uniformð0:45;0:55ÞandZ6¼ðZ5À0:5Þ.MomentscanbeevaluatedbycollapsingtheexpansioninXbysubstitutionbasedupontheknownfunctionaldependenceandthenapplyingalower-dimensionalquadraturerule.Here,weemployedathree-dimensionalquadraturerulebaseduponthetensorproductofone-dimensionalrulesforZ1;Z3andZ5.Valuesfortheremainingvariableswereselectedbaseduponthefunctionaldependencespecifiedabove.Fig.4.4plotstheerrorinthefirsttwomomentsatt¼20.Astheorderoftheapproximationlevelofthesparsegridin-creasestheerrorsconvergeexponentiallyfastbeforereachingsaturationlevels.4.2.2.EpistemicuncertaintywithknowncovarianceInpracticeonemayoftenencounteruncertaintyarisingfromasetofrandomvariableswithnormallydistributedmar-ginaldistributionsandknowncovariance.ConsiderX¼ðX1;...;X6Þ$Nð0;CÞwherethecovariancematrixisatri-diagonalmatrixwithnon-zeroentriesr11¼0:03;r22¼0:0009;r33¼0:0003;r44¼0:01;r55¼0:001;r66¼0:0025;r12¼0:05r11;r21¼r22;r34¼0:02r44;r43¼0:2r44,andr56¼r65¼0:1r55.Unlikethepreviousexamplestheepistemicvariablesarenowunbounded.Consequentlywemustconstructanapprox-imationtotheencapsulationproblemwhichcapturesthetrueinputrangewithoverwhelmingprobability.Here,weinves-tigatethechoiceofsizeoftheboundinghyper-regionontheaccuracyoftheofsolutionmoments.Fig.4.5plotstheerrorinthefirsttwomomentsatt¼20.Astheapproximationlevelofthecollocationsparsegridincreasestheerrorsconvergeexpo-nentiallyfastbeforereachingsaturationlevels.However,theaccuracyatwhichsaturationoccursisdependentonhow‘‘well”theinputspaceisencapsulated.Astheencapsulationprobabilityincreases,thatisddecreases,thebestpossibleaccu-racythatcanbeobtainedbysolvingtheencapsulationproblemincreases.Itmustbenotedthattheconvergencerateslowswithdecreasingd,becauseitincreasesthesizeoftheencapsulationdomain.Ingeneralinterpolationofalargerdomainre-quiresmoreevaluationsofthegoverningequationstoachieveacomparableaccuracy.Theexactmomentsofthesolutionwereobtainedbyapplyinghighordersix-dimensionalGauss–Hermitesparsegridquadraturetothegoverningequations.MomentsoftheSCapproximationwereobtainedbyapplyingthesamequadratureruletothenumericalsolutionoftheencapsulationproblem.TheGauss–Hermitesparsegridquadratureassumesindepen-dentGaussianvariables.ACholeskydecompositionofthecovariancematrixwasusedtogenerateasetofdependentreal-izationsofZ.4.2.3.Mixedaleatory-epistemicuncertaintyNowletusconsidertheuncertaintyofthesolutionto(4.4)wherethedistributionsofsomeofthevariablesareknownandthedistributionsofothervariablesareunknown.Asimpletwostepiterativeprocedurecanbeusedtogeneratesuchanensembleofstatistics.Inthiscasewewishtogen-erateanensembleofCDFsofthesolutiontothegoverningequationsattimet¼20.Inthefirststepofeachiterationapar-ticularvalueofeachepistemicvariableischosenfromwithintheirassumedranges.FixingthesevalueswethenrandomlyFig.4.4.Convergenceoftherelativeerrorinthemeanandvarianceforthedampedharmonicoscillatorwithsixdependentinputs.ConvergenceisshownwithrespecttotheapproximationleveloftheSCsparsegrid.466010−1J.Jakemanetal./JournalofComputationalPhysics229(2010)48–466310δ∼ 10110−21−δ∼ 10−110101010101010100δ∼ 10−5δ∼ 109−δ∼ 10−5δ∼ 109−−110Relative Error−3−210−4Relative Error1234Approximation Order56−310−5−410−6−510−7−610−8−71234Approximation Order56(a)Mean.(b)Variance.Fig.4.5.Convergenceoftherelativeerrorinthemeanandvarianceforthedampedharmonicoscillatorwithsixdependentinputswithknowncovariance.Convergenceiswithrespecttotheorderofthecollocationsparsegridandthetruncationprobabilityd.Fig.4.6.Anensembleof25CDFsofthesolutiontothedampedharmonicoscillatorsubjecttomixedaleatoryandepistemicuncertainty.EachCDFcorrespondstooneparticularsetofepistemicvariablesvalues.samplefromthealeatoryvariablesinastandardprobabilisticmanner.Thesesamplesarethenusedtoevaluatethepolyno-mialapproximationoftheencapsulationproblem.Followingthisheuristic,eachsetofepistemicvariablesgeneratesafulldistributionaldescriptionandcorrespondingstatisticalmetrics,suchasmoments,fortheoutputquantities.Z2LetthevariablesZ1;Z2;Z3,andZ5haveknowndistributionswithZ1$betað0;1;0;0Þ;Z2¼41þ0:01;Z3$betað0;1;1;1ÞandZ5$betað0;1;2;1ÞandletthetworemaininginputvariablesZ4andZ6beepistemicvariablesthatliewithinthefollow-ingranges:Z42½0:8;1:2andZ62½À0:05;0:05.Nowletusconstructasparsegridcollocationapproximationtotheencap-sulationproblemusingthesimultaneousconstructionoutlinedinSection3.5.Fig.4.6plotsanensembleofCDFsofthenumericalsolutionto(4.4)for25realizationsofthetwoepistemicvariables.EachepistemicvariablewasassumedtotakefivediscretevaluesZkiequallyspacedthroughouttheircorrespondingrangesk½ai;bi;i¼4;6.SpecificallywechooseZi¼aiþkðbiÀaiÞ=4;k¼0;...;4.Foreachofthe25combinationsofZki,Monte-CarlosamplingofthealeatoryvariablesisusedtogenerateaCDFfromevaluationofthepolynomialapproximation.Itisevidentthatthetwoepistemicvariableshavealargeinfluenceonthedistributionoftheoutput.Thiscouldindicatethateffortshouldbefocusedonmoreaccuratelyreducingtherangeofthesevariables.J.Jakemanetal./JournalofComputationalPhysics229(2010)48–46631010101010Relative Error101010-34661−4−5−6−7−8−9−1010−111010101010−12−13−14−15−16123Approximation Order45Fig.4.7.Convergenceoftherelativeerrorinthemeanandvarianceforthehomogeneousdiffusionequationwithsixdependentinputs.ConvergenceiswithrespecttotheorderofthegPCexpansion,atthespatiallocationx¼0:8425247397.4.3.HomogeneousdiffusionequationInthissection,weconsiderthehomogeneousdiffusionequationinone-spatialdimensionsubjecttoepistemicuncer-taintyinthediffusivitycoefficient.Attentionisrestrictedtotheone-dimensionalphysicalspacetoavoidunnecessarycom-plexity.Theproceduredescribedherecaneasilybeextendedtohigherphysicaldimensions.ConsiderthefollowingproblemwithdP1randomdimensions: !dduaðx;ZÞðx;ZÞ¼0;dxdxðx;ZÞ2ð0;1ÞÂIZð4:8Þsubjecttothephysicalboundaryconditionsuð0Þ¼0;uð1Þ¼0:ð4:9ÞFurthermoreassumethattherandomdiffusivitysatisfiesaðx;ZÞ¼1þrdXk¼11k2p2cosð2pkxÞZk;ð4:10ÞwhereZk2½À1;1;k¼1;...;dareindependentanduniformlydistributedrandomvariables.Theformof(4.10)issimilartothatobtainedfromaKarhunen–LoèveexpansionandsatisfiestheauxiliarypropertiesE½aðx;ZÞ¼1and1Àr6
