加入VIP
  • 专属下载特权
  • 现金文档折扣购买
  • VIP免费专区
  • 千万文档免费下载

上传资料

关闭

关闭

关闭

封号提示

内容

首页 随机偏微分方程数值解法

随机偏微分方程数值解法.pdf

随机偏微分方程数值解法

soul
2012-08-15 0人阅读 举报 0 0 暂无简介

简介:本文档为《随机偏微分方程数值解法pdf》,可适用于高等教育领域

ANUMERICALANALYST’SVIEWOFNUMERICALMETHODSFORSTOCHASTICPDESMaxGunzburgerDepartmentofScientificComputingFloridaStateUniversitygunzburgfsueduFallINTRODUCTORYREMARKSUncertaintyiseverywhere•Physical,biological,social,economic,financial,etcprocessesalwaysinvolveuncertainties•Asaresult,mathematicalmodelsoftheseprocessesshouldaccountforuncertainty•Accountingforuncertaintyinprocessesgovernedbypartialdifferentialequationscaninvolve–randomcoefficientsinthePDE,boundarycondition,andinitialconditionoperators–randomrighthandsidesinthePDE’s,boundaryconditions,andinitialconditions–randomgeometry,ie,randomboundaryshapes•Uncertaintyarisesbecause–availabledataareincompletetheyarepredictablebutaretoodifficult(perhapsimpossible)orcostlytoobtainbymeasurement→mediapropertiesinoilreservoirsoraquiferstheyareunpredictable→windshear,rainfallamounts–notallscalesinthedataandorsolutionscanorshouldberesolveditistoodifficult(perhapsimpossible)orcostlytodosoinacomputationalsimulation→turbulence,molecularvibrationssomescalesmaynotbeofinterest→surfaceroughness,hourlystockprices•Ofcourse,itiswellknownthattwoexperimentsrununderthe“same”conditionswillyielddifferentresultsModelingnoise•Whitenoise–inputdatavaryrandomlyandindependentlyfromonepointofthephysicaldomaintoanotherandfromonetimeinstanttoanother–uncertaintyisdescribedintermsofuncorrelatedrandomfields–examples:thermalfluctuationssurfaceroughnessLangevindynamics•Colorednoise–inputdatavaryrandomlyfromonepointofthephysicaldomaintoanotherandfromonetimeinstanttoanotheraccordingtoagiven(spatialtemporal)correlationstructure–uncertaintyisdescribedintermsofcorrelatedrandomfields–examples:rainfallamountsbonedensitiespermeabilitieswithinsubsurfacelayers•Randomparameters–inputdatadependonafinitenumberofrandomparameters–thinkofthiscaseas“knobs”inanexperiment–eachparametermayvaryindependentlyaccordingtoitsowngivenprobabilitydensity–alternately,theparametersmayvaryaccordingtoagivenjointprobabilitydensity–examples:homogeneousmaterialproperties,eg,Young’smodulus,Poisson’sratio,speedofsound,inflowmass•Ultimately,forallthreecases,onacomputeronesolvesproblemsinvolvingrandomparameters–inthewhitenoiseandcolorednoisecases,onediscretizesthenoisesothatthediscretizednoiseisapproximatedintermsofafinitenumberofparametersinthewhitenoisecase,thenumberofparametersdirectlydependsonthespatialandortemporalresolutionsofthenumericalschemeusedtosolvethePDEsinthecolorednoisecase,thenumberofparametersdependsmuchmoreweaklyonthespatialtemporalresolutionsUncertaintyquantification•Uncertaintyquantificationisthetaskofdeterminingstatisticalinformationaboutoutputsofasystem,givenstatisticalinformationabouttheinputsSYSTEMuncertaininputsuncertainoutputs–ofcourse,thesystemmayhavedeterministicinputsaswell•WeareinterestedinsystemsgovernedbypartialdifferentialequationsPDEuncertaininputsuncertainsolutionofthePDE–thesolutionofthepartialdifferentialequationdefinesthemappingfromtheinputvariablestotheoutputvariables•Often,solutionsofthePDEarenottheprimaryoutputquantityofinterest–quantitiesobtainedbypostprocessingsolutionsofthePDEaremoreoftenofinterestofcourse,onestillhastoobtainasolutionofthePDEtodeterminethequantityofinterestPDEuncertaininputsuncertainquantitiesofinterestPostprocessingofthesolutionofthePDEuncertainsolutionofthePDE•ArealizationoftherandomsystemisdeterminedbyspecifyingaspecificsetofinputvariablesandthenusingthePDEtodeterminethecorrespondingoutputvariables–thus,arealizationisasolutionofadeterministicproblem•OneisneverinterestedinindividualrealizationsofsolutionsofthePDEorofthequantitiesofinterest–oneisinterestedindeterminingstatisticalinformationaboutthequantitiesofinterest,givenstatisticalinformationabouttheinputs•Inputvariablescouldbedistributedindependently†orjointly,andinthelattercase,couldbecorrelatedoruncorrelated†Withoutproperjustificationandsometimesincorrectly,itisalmostalwaysassumedthattheparametersareindependentbasedonempiricalevidence,sometimesthisisajustifiableassumptionintheparametersare“knobs”case,butforcorrelatedrandomfields,itisjustifiableonlyfortheGaussiancaseingeneral,independenceisasimplifyingassumptionthatisinvokedforthesakeofconvenience,eg,becauseofalackofknowledgeIndependentvsuncorrelated•SupposewehaveNrandomvariables{yn}Nn=definedoveranNdimensionalsetΓ–ajointprobabilitydensityfunction(PDF)ρ(y,,yN)isamappingfromΓintotherealnumberssuchthatρ(y,,yN)≥forall{y,,yN}∈Γand∫Γρ(y,,yN)dy···dyN=•Forn=,,N,themeanorexpectedvalueofynisgivenbyµn=E(yn)=∫Γynρ(y,,yN)dy···dyN–ifµn=,thenyniscalledcentered•Thecovarianceof{yn}Nn=istheN×NmatrixCgivenbyCnn′=E((yn−µn)(yn′−µn′))=∫Γ(yn−µn)(yn′−µn′)ρ(y,,yN)dy···dyN=E(ynyn′)−E(yn)E(yn′)=E(ynyn′)−µnµn′–ifeitherynoryn′iscenteredCnn′=E(ynyn′)•Therandomvariables{yn}Nn=areindependentifthechoiceofvaluesofanysubsetofvariablesdoesnotdependonthechoicesmadeforthevaluesoftheremainingvariables–therandomvariables{yn}Nn=areindependentifandonlyifthejointPDFisaproductofthePDFsfortheindividualvariablesρ(y,,yN)=N∏n=ρn(yn)andΓ=Γ⊗Γ⊗···⊗ΓNwhere,forn=,,N,ρn(·)isamappingfromΓntotherealnumbersthatsatisfiesρn(yn)>∫Γnρn(yn)dyn=•Therandomvariables{yn}Nn=areuncorrelatedonlyifCnn′=σnδnn′∀n,n′=,,Nwhereσndenotesthevarianceofynσn=Cnn=E((yn−µn))=∫Γ(yn−µn)ρ(y,,yN)dy···dyN=E(yn)−µn–ifyniscenteredCnn=E(yn)•Independenceimpliesuncorrelated–ifn=n′,Cnn′=∫Γn(yn−µn)ρn(yn)dyn∫Γn′(yn′−µn′)ρn′(yn′)dyn′×N∏n′′=,n′′=n,n′′=n′∫Γn′ρn′′(yn′′)dyn′′=∫Γn(yn−µn)ρn(yn)dyn∫Γn′(yn′−µn′)ρn′(yn′)dyn′=E(yn−µn)E(yn′−µn′)=•Uncorrelateddoesnotnecessarilyimplyindependence–letybeuniformlydistributedon−,=⇒E(y)=E(y)=–lety=y–clearly,{y,y}isnotindependent–however,{y,y}isuncorrelatedC=E(yy)−E(y)E(y)=E(y)=•Whendoesuncorrelatedimplyindependence–ifandonlyifthevariablesfollowamultivariateGaussiandistribution•Realization=asolutionu(x,t~y)ofaPDEforaspecificchoice~y={yn}Nn=fortherandomparameters–again,thereisnointerestinindividualrealizations•Weusetheabbreviation~y={y,y,,yN}Quantitiesofinterest•OnemaybeinterestedinstatisticsofsolutionsofthePDE–averageorexpectedvalueu(x,t)=Eu(x,t·)=∫Γu(x,t~y)ρ(~y)d~y–covariancecov(x,tx′,t′)=E(u(x,t·)−u(x,t))(u(x′,t′·)−u(x′,t′))=∫Γ(u(x,t~y)−u(x,t))(u(x′,t′~y)−u(x′,t′))ρ(~y)d~y–variancevar(x,t)=cov(x,tx,t)–highermoments•OnemayinsteadbeinterestedinstatisticsofaquantityF(u(x,t,~y)x,t,~y)derivedfromthesolutionofthePDE–example:ifudenotesadisplacementfield,wemaywantstatisticsaboutthestress•OnemaybeinterestedinstatisticsofspatialtemporalintegralsofthesolutionofthePDE–foranyfixed~y,wehave,eg,J(t~y)=∫DF(u~y)dxorJ(x~y)=∫ttF(u~y)dtorJ(~y)=∫tt∫DF(u~y)dxdtwhereF(··)isgiven,Disaspatialdomain,and(t,t)isatimeinterval–quantitiesdefinedwithrespecttointegralsoverboundarysegmentsalsooccurinpractice–examplesthespacetimeaverageofuJ(~y)=∫tt∫Du(x,t~y)dxdtifudenotesavelocityfield,thenJ(t~y)=∫Du(x,t~y)·u(x,t~y)dxisproportionaltothetotalkineticenergy–again,oneisnotinterestedinthevaluesofthesequantitiesforspecificchoicesoftheparameters~yoneisinterestedintheirstatistics(means,variances,higherordermoments)–example:expectedvalueofthetotalkineticenergyE∫Du(x,t~y)·u(x,t~y)dx=∫Γ∫Du(x,t~y)·u(x,t~y)ρ(~y)dxd~y•Otherquantitiesofinterestdonotinvolveintegralsoverspatialtemporaldomains–importantexamplesarisein,eg,reliabilityorriskassessmenteg,whatistheexpectedvalueofthemaximumstresseg,whatistheprobabilitythatthemaximumstressexceedssomegivenvalueeg,whatistheprobabilitythatsealevelwillrisebymetersinthenextyears•Thus,aquantityofinterestcouldbethestatisticalaverageofthemaximumspatialtemporalvalueofaquantityderivedfromthesolutionofthePDEEmaxx,tF(u(x,t~y)x,t,~y)=∫Γmaxx,tF(u(x,t~y)x,t,~y)ρ(~y)d~y–higherstatisticalmoments(eg,variances)wouldbeofinterestaswell•AnotherquantityofinterestcouldbetheprobabilitythataquantityderivedfromthesolutionofthePDEexceedsagivenvalueprobG(u(x,t~y)x,t,~y)≥G–thisquantityofinterestcanalsobeexpressedintermsofanintegraloverparameterspacebecauseprobG(u(x,t~y)x,t,~y)≥G=∫Γχ≥ρ(~y)d~ywhereχ≥=ifG(u(x,t~y)x,t,~y)≥Gotherwise•Weseethatquantitiesofinterestrequireintegrationoverovertheparameterspace–eg,forsomeG(·),integralsofthetype∫ΓG(u(x,t~y)x,t,~y)ρ(~y)d~y•Ideally,onewantstodetermineanapproximationofthePDFforthequantityofinterest,ie,morethanjustafewstatisticalmomentsofsomeoutputquantity–thequantityofinterestisaPDF–oneway(butnottheonlyway)toconstructtheapproximatePDFistocomputemanystatisticalmomentsoftheoutputquantityso,again,wearefacedwithevaluatingstochasticintegralsQuadraturerulesforstochasticintegrals•Integralsofthetype∫ΓG(u(x,t~y))ρ(~y)d~ycannot,ingeneral,beevaluatedexactly•Thus,theseintegralsareapproximatedusingaquadraturerule∫ΓG(u(x,t~y))ρ(~y)d~y≈Q∑q=wqG(u(x,t~yq))ρ(~yq)forsomechoiceofquadratureweights{wq}Qq=(realnumbers)andquadraturepoints{~yq}Qq=(pointsintheparameterdomainΓ)–alternately,sometimestheprobabilitydensityfunctionisusedinthedeterminationofthequadraturepointsandweightssothatinsteadoneendsupwiththeapproximation∫ΓG(u(x,t~y))ρ(~y)d~y≈Q∑q=wqG(u(x,t~yq))•MonteCarlointegration–thesimplestrule=⇒–randomlyselectQpointsinΓaccordingtothePDFρ(~y)–evaluatetheintegrandateachofthesamplepoints–averagethevaluessoobtainedie,forallq,wq=Q–moreonMonteCarloandotherquadratureruleslaterBigdifficulty•Inpractice,oneusuallydoesnotknowmuchaboutthestatisticsoftheinputvariables–oneisluckyifoneknowsarangeofvalues,eg,maximumandminimumvalues,foraninputparameterinwhichcaseoneoftenassumesthattheparameterisuniformlydistributedoverthatrange–ifoneisluckier,oneknowsthemeanandvariancefortheinputparameterinwhichcaseoneoftenassumesthattheparameterisnormallydistributed–ofcourse,onemaybecompletelywronginassumingsuchsimpleprobabilitydistributionsforaparameterPDFswiththesamemeanandvariance•ThisleadstotheneedtosolvestochasticmodelcalibrationproblemsModelcalibration•Modelcalibrationisthetaskofdeterminingstatisticalinformationabouttheinputsofasystem,givenstatisticalinformationabouttheoutputs–eg,onecanuseexperimentalorfieldobservationstodeterminethestatisticalinformationabouttheoutputs–inparticular,onewantstoidentifytheprobabilitydensityfunctions(PDF)oftheinputvariables•Ofcourse,thesystemstillmapstheinputstotheoutputs–thus,determiningtheinputPDFisaninverseproblem–usuallyinvolvesaniterationinwhichguessesfortheinputPDFareupdated–severalwaystodotheupdate,eg,Baysean,maximumlikelyhood,SYSTEMuncertaininputsuncertainoutputsPDFknownPDFtobedeterminedUncertaintyquantification–directproblemuncertaininputsPDFtobedetermineduncertainoutputsPDFknowninitialguessfortheinputPDFsystemoutputupdatedinputPDFSYSTEMcomparerandupdaterModelcalibration–inverseproblem•Modelcalibrationproblemsareaparticularcaseofmoregeneralstochasticinverse,orparameteridentification,orcontrol,oroptimizationproblemsinitialuncertaininputssystemoutputupdatedinputsSYSTEMfeedbacklawFeedbackcontroloptimalinputs(controls)andsystemstatesOPTIMIZERsystemobjectiveOptimalcontrolOBSERVATIONSABOUTTHESELECTURES•Ofgreatestinterest(tous)arenonlinearproblemshowever–sowefocusonmethodsthatareusefulinthenonlinearsetting–however,wedosometimescommentonspecialfeaturesofsomemethodsthatonlyholdforlinearproblems•Bothtimedependentandsteadystateproblemsareofinterest–forthesakeofsimplifyingtheexposition,weconsidermostlysteadystateproblems–however,almosteverthingwehavetosayappliesequallywelltotimedependentproblemsWHITENOISEUNCORRELATEDRANDOMFIELDS•Whitenoisereferstothecaseofuncorrelatedrandomfieldsη(x,tω)forwhichwehave†E(η(x,tω))=andE(η(x,tω)η(x′,t′ω))=δ(t−t′)δ(x−x′)–ateverypointinspaceandateveryinstantintime,η(x,tω)isindependentandidenticallydistributedonedeterminesη(x,tω)atanypointinspaceandanyinstantintimebysamplingaccordingtoagivenprobabilitydistribution–theGaussiancaseistheonethatoftenarisesinpractice(sometimesbecauseofalackofinformation)†ThezeromeanandunitvarianceassumptionsarenotrestrictiveDiscretizingwhitenoise•Incomputersimulations,onecannotsampletheGaussiandistributionateverypointofthespatialdomainandateveryinstantoftime–whitenoisetermsarereplacedbydiscretizedwhitenoisetermsdiscretizedwhitenoiseismoreregularthatwhitenoise•Amongthemeansavailablefordiscretizingwhitenoise,gridbasedmethodsarethemostpopular•Todefineasinglerealizationofthediscretizedwhitenoise,we–subdividethespatialdomainDintoNspacesubdomains–subdividethetemporalinterval,TintoNtimetimesubintervals–then,inthensthspatialsubdomainhavingvolumeVnsandinthentthtemporalsubintervalhavingduration∆tnt,setηapproximate(x,t{yns,nt})=√∆tnt√Vnsyns,ntwhereyns,ntareindependentGaussiansampleshavingzeromeanandunitvariance•AdditionalrealizationsaredefinedbyresamplingoverthespacetimegridRealizationsofdiscretizedwhitenoiseatasametimeintervalinasquaresubdividedinto,,,,,,,andtrianglesRealizationsofdiscretizedwhitenoiseattwodifferenttimeintervalsinasquaresubdividedintothesamenumberoftriangles•Thus,thediscretizedwhitenoiseispiecewiseconstantinspaceandtime•Notethatthepiecewiseconstantfunctionismuchsmootherthantherandomfielditapproximates•ItcanbeshownthatlimNspace→∞,Ntime→∞E(ηapproximate(x,t{yns,nt})ηapproximate(x′,t′{yns,nt}))=E(η(x,t)η(x′,t′))=δ(x−x′)δ(t−t′)•Thewhitenoisecasehasbeenreducedtoacaseofalargebutfinitenumberofparameters–wehavetheN=NspaceNtimeparametersyns,ntwherens=,,Nspaceandnt=,,Ntime–ifwerefinethespatialgridandorreducethetimestep,thenumberofparametersincreasesPDE’SFORCEDBYWHITENOISE•Formally,wecanwriteanevolutionequationwithwhitenoiseforcingas∂u∂t=A(ux,t)f(x,t)B(ux,t)η(x,tω)inD×(,TwhereAisapossiblynonlineardeterministicoperatorfisadeterministicforcingfunctionBisapossiblynonlineardeterministicoperatorηisthewhitenoiseforcingfunction–amongmanyothercases,A,f,andBcantakecareofcaseswithmeans=andvariances=•IfBisindependentofu,wehaveadditivewhitenoise∂u∂t=A(ux,t)f(x,t)b(x,t)η(x,t)–inpractice,oftenbisaconstant•IfBdependsonu,wehavemultiplicativewhitenoise–ofparticularinterestisthecaseofBlinearinu∂u∂t=A(ux,t)f(x,t)b(x,t)uη(x,t)•Someobservations–solutionsarenotsufficientlyregularfortheequationsjustwrittentomakesensetherenownedItocalculusisintroducedtomakesenseofdifferentialequationswithwhitenoiseforcing–whitenoiseneednotberestrictedtoforcingtermsinthePDEinpractice,itcanalsoappearinthecoefficientsofthePDEsandboundaryandinitialconditionsinthedatainboundaryandinitialconditionsinthedefinitionofthedomain•SpatialdiscretizationofthePDEcanbeeffectedviaafiniteelementmethodbasedonatriangulationofthespatialdomainDtemporaldiscretizationiseffectedviaafinitedifferencemethod,eg,abackwardEulermethod–itisnaturaltousethesamegridsinspaceandtimeasareusedtodiscretizethewhitenoise–thus,ifonerefinesthefiniteelementgridandthetimestep,onealsorefinesthegridandtimestepforthewhitenoisediscretization•Oncearealizationofthediscretizednoiseischosen,ie,onceonechoosestheNspaceNtimeGaussiansamplesηns,nt,arealizationofthesolutionofthePDEisdeterminedbysolvingadeterministicproblem•Forexample,considertheproblem∂u∂t=∆uf(x,t)b(x,t)uη(x,tω)inD×(,Tu=in∂D×(,Tu(x,)=u(x)inD–subdivide,TintoNtimesubintervalsofduration∆tnt,nt=,,Ntime–subdivideDintoNspacefiniteelements{Dns}Nspacens=–defineafiniteelementspaceSh⊂H(D)withrespecttothegrid{Dns}Nspacens=–chooseanapproximationu(,h)(x)totheinitialdatau(x)–sample,fromastandardGaussiandistribution,theNspaceNtimevaluesyns,nt,ns=,,Nspaceandnt=,,Ntime–setu()h(x)=u(,h)(x)–then,fornt=,,Ntime,determineu(nt)h(x)∈Shfrom∫Du(nt)h−u(nt−)h∆tntvhdx∫D∇u(n)h·∇vhdx=∫Dfvhdx√∆tnt√AnsNs∑ns=∫Dnsyns,ntvhdxforallvh∈ShnotethatwehaveusedabackwardEulertimesteppingscheme•Thisisastandarddiscretefiniteelementsystemfortheheatequation,albeitwithanunusualrighthandside•DuetothelackofregularityofsolutionsofPDE’swithwhitenoise,theusualnotionsofconvergenceoftheapproximatesolutiontotheexactsolutiondonothold,eveninexpectation–onehastobesatisfiedwithveryweaknotionsofconvergenceCOLOREDNOISECORRELATEDRANDOMFIELDS•Wenowconsidercorrelatedrandomfieldsη(x,tω)–ateachpointxinaspatialdomainDand

用户评价(0)

关闭

新课改视野下建构高中语文教学实验成果报告(32KB)

抱歉,积分不足下载失败,请稍后再试!

提示

试读已结束,如需要继续阅读或者下载,敬请购买!

文档小程序码

使用微信“扫一扫”扫码寻找文档

1

打开微信

2

扫描小程序码

3

发布寻找信息

4

等待寻找结果

我知道了
评分:

/45

随机偏微分方程数值解法

仅供在线阅读

VIP

在线
客服

免费
邮箱

爱问共享资料服务号

扫描关注领取更多福利