Numerical Approximation of Flows in Random Porous Media

153 Pages·2016·8.69 MB·English

by  Tesei, Francesco

Numerical Approximation of Flows in Random Porous Media
THÈSE NO 6860 (2016)
PRÉSENTÉE LE 12 FÉVRIER 2016
À LA FACULTÉ DES SCIENCES DE BASE
MATHICSE - GESTION
PROGRAMME DOCTORAL EN MATHÉMATIQUES
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES
PAR
Francesco TESEI
acceptée sur proposition du jury:
Prof. V. Panaretos, président du jury
Prof. F. Nobile, directeur de thèse
Prof. O. Ernst, rapporteur
Prof. R. Scheichl, rapporteur
Prof. A. Abdulle, rapporteur
Suisse
2015

Abstract
Theobjectiveofthisthesisistodevelopefficientnumericalschemestosuccessfullytackleproblems
arisingfromthestudyofgroundwaterflowsinaporoussaturatedmedium;wedealthereforewith
partialdifferentialequations(PDE)havingrandomcoefficientsandweareinterestedincomputing
statisticsrelatedtospecificquantitiesofinterest(QoI),e.g.alinearfunctionalofthesolutionofthe
PDEorthesolutionitself.Wemainlyconsidertheapproximationofthepressureinthemedium
throughastochasticDarcyproblemwithrandomlognormallydistributedpermeabilityrelyingon
Matérn-typecovariancefunctionstotakeintoaccountawiderangeofpossiblesmoothnessof
thepermeabilityfield.Oncetheproblemhasbeenreformulatedintermsofacountablenumber
ofrandomvariables,weanalyzesparsegridpolynomialapproximationsoftheQoI.Wepropose
differentstrategiestoexploittheanisotropicityoftheQoIwithrespecttothedifferentrandom
entries;tothisendweconsider“apriori”and“aposteriori”strategiestodrivetheexplorationof
themulti-indexsetthatdefinesthesparsegrid,associatingaprofittoeachmulti-indexeitherby
usingexplicittheoreticalestimatesorbyactuallysolvingthePDEandcomputingontheflythe
correspondingsparsegridinterpolant.Weshowonseveralnumericalexamplestheeffectiveness
ofthisstrategyintreatingthecaseofsmoothpermeabilityfields.Inordertocoveralsothecaseof
roughinputpermeabilitiesweconsider,instead,MultiLevelMonteCarlotechniquesbasedonthe
useofasuitablecontrolvariate.Suchacontrolvariateisobtainedfromthesolutionofanauxiliary
Darcyproblemwitharegularizedinputpermeabilitywhichleadstopressuredistributionsthatare
smootherandlessoscillatorythantheoriginalones,butstillhighlycorrelatedwiththem.Weuse
thenasparsegridapproximationtocomputeeffectivelythemeanofthecontrolvariateandprovide
explicitboundsforthecorrespondingestimatoraswellasacomplexityresult.
Wealsoconsidergroundwatertransportproblemsandfocus,inparticular,onarrivaltimesproperly
definedstartingfromparticletrajectoriesdrivenbythestochasticDarcyvelocityandsubjectto
moleculardiffusiontakingplaceatporouslevel.Inthiscase,byusingsuitablePDEswhosesolution
canbelinkedtospecificexpectations(withrespecttoallBrownianmotions)thankstothefamous
Feynman-Kacformula,wecomputestatisticsofsucharrivaltimes,e.g.theirexpectedvalueorthe
probabilityofexitingthephysicaldomaininagiventimehorizon. Wediscussseveralscenarios
andreadaptthemethodologiespreviouslydevelopedinvolvingadaptivesparsegridstochastic
collocationandMonteCarlotypeschemestothiscase.
Keywords:StochasticDarcyProblem,Lognormalrandomfields,MultiLevelMonteCarlo,Control
variate,StochasticCollocation,AdaptiveSparseGrid,Matérncovariance,Groundwaterproblems,
Wellcatchments,Capturezones
i

Contents
Abstract i
Listoffigures v
Listoftables ix
Introduction 1
Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
ForwardUQProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 ThesisOverview 5
1.1 Problemsetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Adaptivesparsegrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 MonteCarloandMultiLevelMonteCarlomethodswithControlVariate . . . . . . . . 10
1.4 ContaminantTransportinGroundwaterFlows . . . . . . . . . . . . . . . . . . . . . . . 12
2 TheDarcyProblem 19
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Problemsetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 FiniteElementApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 TheTruncatedDarcyProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Samplingschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 Karhunen-Loèveexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.2 Fourierexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 SparseGridpolynomialapproximationforrandomPDEsinhighdimensionalprobability
spaces 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Preliminaries:abriefoverviewontheproblem . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Sparsegridapproximationsinfinitedimensions . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Possiblecriteriatoselectthemulti-indexsetI . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 An“apriori”strategy:quasi-optimalsparsegrids . . . . . . . . . . . . . . . . . . 42
3.4.2 An“aposteriori”strategy:adaptivesparsegrids . . . . . . . . . . . . . . . . . . . 45
3.5 Adimension-adaptivesparsegridalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6.1 Referencecase:ν=2.5,σ=1,L =0.5 . . . . . . . . . . . . . . . . . . . . . . . . . 51
c
3.6.2 Permeabilitieswithhighvariability:σ2=4 . . . . . . . . . . . . . . . . . . . . . . 56
3.6.3 ConditionedDarcyproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
iii
Contents
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 MonteCarlo-typeschemeswithControlVariate 61
4.1 Inroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 MonteCarlomethodwithcontrolvariate(MCCV) . . . . . . . . . . . . . . . . . . . . . 63
4.3 ErroranalysisoftheMCCVmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 MultiLevelMonteCarlomethodwithcontrolvariate(MLMCCV) . . . . . . . . . . . . 70
4.5 ErroranalysisoftheMLMCCVmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 TheAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6.1 OptimizationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 ComplexityoftheMultiLevelMonteCarlowithControlVariatemethod . . . . . . . . 79
4.8 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.8.1 Alevel-dependentapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Appendices 93
4.A ProofofLemma4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.B OptimalratesinTheorem4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 ExtensiontoTransportProblem 101
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Problemsetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.1 PureDirichletcase:Γ =∂D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
D
5.2.2 Mixedcase:|Γ |>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
N
5.3 Case1:anundisturbedflowfromlefttoright . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.1 SpatialDiscretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.2 ApproximationofthemeanFPTthroughMLMCCV. . . . . . . . . . . . . . . . . 112
5.3.3 Delineationofcapturezonesforsmalltemporalhorizonbystochasticcolloca-
tiononmeanFPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Case2:aflowinducedbythepresenceofanextractingwell . . . . . . . . . . . . . . . . 119
5.4.1 AMLMCapproachtoapproximateexpectedarrivaltimes . . . . . . . . . . . . . 121
5.4.2 Delineationofcapturezonesforsmalltemporalhorizon . . . . . . . . . . . . . 123
5.4.3 Delineationofcapturezonesforinfinitetemporalhorizon:astreamlineapproach124
5.4.4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Conclusionsandfuturework 131
iv
List of Figures
1.1 Caseν=2.5.AdaptivesparsegriderrorscomputedwithrespecttoareferencesetI(cid:3)
as|QWIm[Q]−QWIm(cid:3)[Q]|fordifferentsetsIadaptivelyconstructed.Workmeasuredas
numberoflinearsolves. QoIconsidered: fluxontheoutlet. D=(0,1)2,Γ ={x =
D 1
0}∪{x =1},Γ ={x =0}∪{x =1},g(x)=1−x . . . . . . . . . . . . . . . . . . . . . . . 9
1 N 2 2 1
1.2 ErrorvsComputationalcostforν=0.5(left)andν=2.5(right). Error=sparsegrid
error+statisticalerror.ComputationalcostevaluatedasestimatedCPUtime. . . . . 13
1.3 Case ν=2.5: on the left the sparse gird approximation of Eω[τ¯σ(cid:3)(y)]; on the right
x
thesparsegriderrorcommittedonsuchquantityintheL2spatialnormversusthe
workevaluatedasnumberofsolvesneededtobuildsuchapproximation.σ(cid:3)=0.01,
L =0.5,p =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
C 0
1.4 VarianceofQ(cid:7)andY(cid:7)(x,ω)=Q(cid:7)(x,ω)−Q(cid:7)−1(x,ω)oneachlevel,forν=0.5anddiffer-
entvaluesof(cid:8).σ(cid:3)=0.01,L =0.5,p =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
C 0
1.5 Caseν=2.5: MLMCapproximationofmean(left)andvariance(right)ofQ(x,ω)=
Eω(cid:3)[1{τσ(cid:3)(ω)≤T}].σ(cid:3)=0,LC =0.5,p0=3,T =0.3,L=5. . . . . . . . . . . . . . . . . . . . . 16
1.6 Caseνx,=0 2.5: MLMCapproximationofmean(left)andvariance(right)ofQ(x,ω)=
Eω(cid:3)[1{τσ(cid:3)(ω)≤T}].σ(cid:3)=0.01,LC =0.5,p0=1,q=0.5,r =0.01,T =0.5. . . . . . . . . . . . 17
x,0
2.1 Decayofλ . Left: radialcovariancefunctionasin(2.5). Right: tensorcovariance
n
functionasin(2.6).Top:ν=2.5.Bottom:ν=0.5.d=2. . . . . . . . . . . . . . . . . . . 30
3.1 Caseν=2.5radialcovariance,adaptivesparsegridserror. . . . . . . . . . . . . . . . . . 54
3.2 Caseν=2.5tensorcovariance,adaptivesparsegridserror. . . . . . . . . . . . . . . . . 54
3.3 Caseν=2.5,quasi-optimalsparsegridserrorobtainedwithtensor(left)andradial
(right)covariancefunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Caseν=2.5,radialcovariancefunction. Interpolationerrorwithdifferentweight
(cid:2)
functionsfortheQuasiOptimal(OPT)andAdaptive profitindicatorsDeltaint(AD-D)
(cid:3)
andWeightedLinf/newpoints(AD-WLNP) cases.work=W ,N∗=60,card(R)=
I,m
10000.Dashedlinesrepresenttheslopes-0.5(black)and-1(orange). . . . . . . . . . . 55
3.5 Caseν=2.5,tensorcovariancefunction. Interpolationerrorwithdifferentweight
(cid:2)
functionsfortheQuasiOptimal(OPT)andAdaptive profitindicatorsDeltaint(AD-D)
(cid:3)
andWeightedLinf/newpoints(AD-WLNP) cases.work=W ,N∗=60,card(R)=
I,m
10000.Dashedlinesrepresenttheslopes-0.5(black)and-1(orange). . . . . . . . . . . 55
3.6 Tensorcovariance:sparsegridserrorfordifferentregularitiesoftheinputrandomfield. 56
3.7 Caseν=2.5,σ=2,adaptivesparsegridserror.. . . . . . . . . . . . . . . . . . . . . . . . 57
3.8 Caseν=2.5,σ=2,adaptivesparsegridserror.. . . . . . . . . . . . . . . . . . . . . . . . 57
3.9 Caseν=2.5,σ=2,quasi-optimalsparsegridserrorobtainedwithtensor(left)and
radial(right)covariancefunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Conditionedcase:ν=2.5radialcovariance,adaptivesparsegridserror.. . . . . . . . . 59
v
ListofFigures
3.11 Conditionedcase:ν=2.5tensorcovariance,adaptivesparsegridserror. . . . . . . . . 59
3.12 Conditionedcase:ν=2.5,σ=2,quasi-optimalsparsegridserrorobtainedwithtensor
(left)andradial(right)covariancefunction. . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Threedifferentregularizationsofthesamerealizationofa.ν=0.5,L =0.5,σ=1. . . 64
(cid:4) c
4.2 VarianceofQ(cid:7)1(ω)= Dκ(x)p(cid:7)(x,ω)dxandY(cid:7)1(ω)=Q(cid:7)1(ω)−Q(cid:7)1−1(ω)oneachlevel,for
ν=0.5anddifferentvaluesof(cid:8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
(cid:4)
4.3 VarianceofQ(cid:7)1(ω)= Dκ(x)p(cid:7)(x,ω)dxandY(cid:7)1(ω)=Q(cid:7)1(ω)−Q(cid:7)1−1(ω)oneachlevel,for
ν=2.5anddifferentvalu(cid:5)esof(cid:8). . . . . . . . . .(cid:6). . . . . . . . . . . . . . . . . . . . . . . 84
4.4 SC error computed as E[(Q1)(cid:8) ]−Q [(Q1)(cid:8) ] 2 versus computational cost, com-
hL W hL
putedasnumberofsparsegridknotsW forν=0.5(left)andν=2.5(right). . . . . . . 85
4.5 DiscretizationerrorcomputedasE[Q1 −Q1 ]2forν=0.5(left)andν=2.5(right). . . 85
h(cid:7) hL
4.6 ErrorvsComputationalcostforν=0.5(left)andν=2.5(right).Error=SparseGrid
error+Statisticalerror,computedasrootmeansquareerror. Computationalcost
estimatedaccordingto(4.18)and(4.19),withC(cid:7)=h(cid:7)−dρandρ=1.05,fittedbyusing
aMatlabbackslashcommand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
(cid:4)
4.7 VarianceofQ2(ω)=− a(x,ω)∂p(cid:7)(x,ω)dx andofY2=Q2−Q2 , forν=0.5and
(cid:7) D ∂x1 (cid:7) (cid:7) (cid:7)−1
different(cid:8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
(cid:4)
4.8 VarianceofQ2(ω)=− a(x,ω)∂p(cid:7)(x,ω)dx andofY2=Q2−Q2 , forν=2.5and
(cid:7) D ∂x1 (cid:7) (cid:7) (cid:7)−1
different(cid:8). . . . . . . . (cid:5). . . . . . . . . . . . . . .(cid:6). . . . . . . . . . . . . . . . . . . . . . . 88
4.9 SC error computed as E[(Q2)(cid:8) ]−Q [(Q2)(cid:8) ] 2 versus computational cost, com-
hL W hL
putedasnumberofsparsegridknotsW forν=0.5(left)andν=2.5(right). . . . . . . 89
4.10 DiscretizationerrorcomputedasE[Q1 −Q1 ]2forν=0.5(left)andν=2.5(right). . . 89
h(cid:7) hL
4.11 ErrorvsComputationalcostforν=0.5(left)andν=2.5(right).Error=SparseGrid
error+Statisticalerror,computedasrootmeansquareerror. Computationalcost
estimatedaccordingto(4.18)and(4.19),withC(cid:7)=h(cid:7)−dρandρ=1.05,fittedbyusing
aMatlabbackslashcommand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
(cid:4)
4.12 VarianceofYLD =QLD−QLD ,withQLD(ω)=− a (x,ω)∂p(cid:7)LD(x,ω)dx forν=0.5
(cid:7) (cid:7) (cid:7)−1 (cid:7) D N(cid:7) ∂x1
anddifferent(cid:8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
(cid:4)
4.13 VarianceofYLD =QLD−QLD ,withQLD(ω)=− a (x,ω)∂p(cid:7)LD(x,ω)dx forν=2.5
(cid:7) (cid:7) (cid:7)−1 (cid:7) D N(cid:7) ∂x1
anddifferent(cid:8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.14 ErrorvsComputationalcostforν=0.5(left)andν=2.5(right).Error=SparseGrid
error+Statisticalerror.Computationalcostestimatedaccordingto(4.18)and(4.19).. 92
5.1 TworealizationsofthemeanFPTthroughtheoutletinthevanishingdiffusioncase
forν=0.5(left)andν=2.5(right).Observethattherandomeventyusedtogenerate
therealizationsisthesame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Pω(cid:3)(τσ(cid:3)(ω)≤T|ω)fordifferentvaluesoft.σ=1,p =2,σ(cid:3)=0,ν=0.5,L =0.5. . . . . 115
x,t 0 c
5.3 Caseν=2.5: ontheleftthesparsegirdapproximationofEω[Q(x,·)]; ontheright
thesparsegriderrorcommittedonsuchquantityintheL2spatialnormversusthe
workevaluatedasnumberofsolvesneededtobuildsuchapproximation.σ(cid:3)=0.01,
L =0.5,p =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C 0
5.4 VarianceofQ(cid:7)andY(cid:7)(x,ω)=Q(cid:7)(x,ω)−Q(cid:7)−1(x,ω)oneachlevel,forν=0.5anddiffer-
entvaluesof(cid:8).σ(cid:3)=0.01,L =0.5,p =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C 0
5.5 VarianceofQ(cid:7)andY(cid:7)(x,ω)=Q(cid:7)(x,ω)−Q(cid:7)−1(x,ω)oneachlevel,forν=2.5anddiffer-
entvaluesof(cid:8).σ(cid:3)=0.01,L =0.5,p =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C 0
vi
ListofFigures
5.6 VarianceofQ(cid:7)andY(cid:7)(x,ω)=Q(cid:7)(x,ω)−Q(cid:7)−1(x,ω)oneachlevel,forν=0.5anddiffer-
entvaluesof(cid:8).σ(cid:3)=0,L =0.5,p =1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C 0
5.7 VarianceofQ(cid:7)andY(cid:7)(x,ω)=Q(cid:7)(x,ω)−Q(cid:7)−1(x,ω)oneachlevel,forν=2.5anddiffer-
entvaluesof(cid:8).σ(cid:3)=0,L =0.5,p =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C 0
5.8 Caseν=2.5: MLMCapproximationofmean(left)andvariance(right)ofQ(x,ω)=
Eω(cid:3)[1{τσ(cid:3)(ω)≤T}].σ(cid:3)=0,LC =0.5,p0=3,T =0.3,L=5. . . . . . . . . . . . . . . . . . . . . 119
x,0
5.1 RealizationsofEω(cid:3)[min{τσ(cid:3)(y,ω(cid:3)),T}|y]forν=2.5inthevanishingdiffusioncase(left)
x,t
andforarelativelylargediffusion(right). Observethattherandomeventyusedto
generatetheDarcyvelocityisthesame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 RealizationofPω(cid:3)(τσ(cid:3)(y)≤T|y)forν=2.5inthevanishingdiffusioncase(left)andfor
x,t
arelativelylargediffusion(right).Observethattherandomeventyusedtogenerate
therealizationsisthesame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3 RealizationsofΨ(x,ω)(left)andΨ (x,ω)(right)correspondingtotherandomeventω .125
R i
5.4 Caseν=2.5: MLMCapproximationofmean(left)andvariance(right)ofQ(x,ω)=
Eω(cid:3)[1{τσ(cid:3)(ω)≤T}].σ(cid:3)=0.01,LC =0.5,p0=1,q=0.5,r =0.01,T =0.5. . . . . . . . . . . . 127
x,0
5.5 MLMCvariancereduction(continuousline)versusMCvariance(dashedline)inthe
caseofleveldependentdiffusion(inblue,labelLD)andfixeddiffusion(inred)for
ν=0.5(right)andν=2.5(left).Thevarianceshavebeencomputedwithrespectthe
L2spatialnorm.Q(x,ω)=Eω(cid:3)[1{τσ(cid:3)(ω)≤T}].σ(cid:3)=0.01,LC =0.5,p0=1,q=0.5,r =0.01. 128
5.6 Caseν=2.5:sparsegridapproximx,tationofthestreamfunctionΨ (x,ω).L =0.5,p =1.129
R C 0
5.7 Case ν = 2.5: MC approximation of mean (left) and variance (right) of Q(x,ω) =
1{τx,0(ω)≤∞}.LC =0.5,p0=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
vii