Reduced Basis Methods for Partial Differential Equations.pdf

Reduced Basis Methods for Partial Differential Equations.pdf

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ISBN978-3-319-15430-5MathematicsAlfioQuarteroniAndreaManzoniFedericoNegriQuarteroni?

ReducedBasisMethodsforPartialDifferentialEquations192?

ReducedBasisMethodsforPartialDifferentialEquationsAnIntroductionThisbookprovidesabasicintroductiontoreducedbasis（RB）methodsforproblemsinvolvingtherepeatedsolutionofpartialdifferentialequations（PDEs）arisingfromengineeringandappliedsciences,suchasPDEsde-pendingonseveralparametersandPDE-constrainedoptimization.ThebookpresentsageneralmathematicalformulationofRBmethods,an-alyzestheirfundamentaltheoreticalproperties,discussestherelatedalgo-rithmicandimplementationaspects,andhighlightstheirbuilt-inalgebraicandgeometricstructures.Morespecifically,theauthorsdiscussalternativestrategiesforconstruct-ingaccurateRBspacesusinggreedyalgorithmsandproperorthogonaldecompositiontechniques,investigatetheirapproximationpropertiesandanalyzeoffline-onlinedecompositionstrategiesaimedatthereductionofcomputationalcomplexity.Furthermore,theycarryoutbothaprioriandaposteriorierroranalysis.ThewholemathematicalpresentationismademorestimulatingbytheuseofrepresentativeexamplesofapplicativeinterestinthecontextofbothlinearandnonlinearPDEs.Moreover,theinclusionofmanypseudocodesallowsthereadertoeasilyimplementthealgorithmsillustratedthrough-outthetext.Thebookwillbeidealforupperundergraduatestudentsand,moregenerally,peopleinterestedinscientificcomputing.UNITEXTUNITEXTReducedBasisMethodsforPartialDifferentialEquationsAnIntroduction9783319154305UNITEXTLaMatematicaperil3+2Volume92Editor-in-chiefA.QuarteroniSerieseditorsL.AmbrosioP.BiscariC.CilibertoM.LedouxW.J.RunggaldierMoreinformationaboutthisseriesathttp:

/BasisMethodsforPartialDifferentialEquationsAnIntroductionAlfioQuarteroniAndreaManzoniEcolePolytechniqueFdraledeLausanneEcolePolytechniqueFdraledeLausanneLausanne,SwitzerlandLausanne,SwitzerlandFedericoNegriEcolePolytechniqueFdraledeLausanneLausanne,SwitzerlandISSN2038-5722ISSN2038-5757（electronic）UNITEXTLaMatematicaperil3+2ISBN978-3-319-15430-5ISBN978-3-319-15431-2（eBook）DOI10.1007/978-3-319-15431-2LibraryofCongressControlNumber:

2015930287SpringerChamHeidelbergNewYorkDordrechtLondonSpringerInternationalPublishingSwitzerland2016Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillus-trations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareex-emptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeenmade.Coverillustration:

Thecoverfiguredisplaysasetofreducedbasisfunctionsforanadvection-diffusion-reactionboundaryvalueprobleminarectangularcomputationaldomain.CoverDesign:

SimonaColombo,GiochidiGrafica,Milano,ItalySpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia（）PrefaceReducedbasis（RB）methodsrepresentaveryefficientapproachforthenumericalapproximationofproblemsinvolvingtherepeatedsolutionofdifferentialequationsarisingfromengineeringandappliedsciences.Noteworthyexamplesincludepartialdifferentialequations（PDEs）dependingonseveralparameters,PDE-constrainedoptimization,andoptimalcontrolandinverseproblems.Inallthesecases,reducingtheseverecomputationalcomplexityiscrucial.Withthisinmind,overthepastfourdecades,reduced-ordermodels（ROMs）havebeendevelopedaimingatreplacingtheoriginallarge-dimensionnumericalproblem（typ-icallycalledhigh-fidelityapproximation）byareducedproblemofsubstantiallysmallerdimension.Strategiestogeneratethereducedproblemfromthehigh-fidelityonecanbemanifold,dependingonthecontext.ThestrategyadoptedinRBmethodsconsistsintheprojectionofthehigh-fidelityproblemuponasubspacemadeofspeciallyselectedbasisfunctions,representingasetofhigh-fidelitysolutionscorrespondingtosuitablychosenparameters.Pioneer-ingworksinthisareadatebacktothelate1970s（e.g.,B.O.Almrothetal.5,6,D.Nagy193,A.K.NoorandJ.M.Peters201,202,203,204andaddresslinearandnonlinearstructuralanalysisproblems.ThefirsttheoreticalanalysisofRBmethodsinconnectionwiththeuseofthecontinuationmethodforparametrizedequationswaspresentedbyJ.P.FinkandW.C.Rheinboldt109,110inthemid1980s.Ex-tensionstoproblemsinfluiddynamicsareprimarilyduetothecontributionsofPeterson210andGunzburger124inthelate1980s.Themethodwassetonamoregeneralandsoundmathematicalgroundintheearly2000sthankstotheseminalworkofA.T.Patera,Y.Madayandcoauthors214,255.TheirworkhasledtoadecisiveimprovementinthecomputationalaspectsofRBmethodsowingtoanefficientcriterionfortheselectionofthebasisfunctions,asystematicsplittingofthecomputationalprocedureintoanoffline（parameter-independent）andanonline（parameter-dependent）phase,andtheuseofaposteriorierrorestimatesthatguaranteecertifiednumericalsolutionsforthereducedproblem.ThesehavebecometheessentialconstituentsoftheRBmethodsnowmostwidelyused.Often,theyarealsoembeddedintomoregeneralreduced-ordermodels.vviPrefaceRBmethodshavewitnessedaspectaculareffervescenceinthepastdecade.Addi-tionalachievementsduringthattimerelatetothetreatmentofnonlinearand/orpara-metricallynonaffineproblemsbytheso-calledempiricalinterpolationmethodanditsseveralextensions.ThishassubstantiallyimprovedRBmethods,makingpossi-bletheirapplicationtoabroadvarietyofcomplexproblemssuchastime-dependentproblems,optimalcontrolanddesignproblems,andreal-timecomputing.ThisisthefirsttextbooktoprovideabasicmathematicalintroductiontoRBmethods.WepresentageneralformulationofRBmethods,analyzetheirfundamen-taltheoreticalproperties,anddiscusstheiralgorithmicandimplementationaspects,highlightingtheirbuilt-inalgebraicandgeometricstructures.Morespecifically,wecarryoutbothaprioriandaposteriorierroranalysis,formulatestrategiesfortheconstructionofaccuratereducedbasisspaces,andanalyzeoffline-onlinedecom-positionstrategiestoensurethereductionofcomputationalcomplexity.Theentiremathematicaldiscussionismademorestimulatingbytheuseofseveralrepresen-tativeexamplesofapplicativeinterest,inthecontextofbothlinearandnonlinearPDEs.TheauthorsaregratefultoCharbelFarhat,YvonMaday,andAnthonyPateraforbeingsourceofinspirationandformanyfruitfuldiscussionsonreduced-ordermodels.WealsoacknowledgeDavidAmsallem,LucaDede,SimoneDeparisandToniLassilaforthegreatamountoftimethattheyhavespentwiththeauthorstalkingaboutdifferentsubjectscoveredinthisbookand,lastbutnotleast,GianluigiRozzaforhavingintroducedthelasttwoauthorstothesubject.Inaddition,specialthanksareduetoFrancescaBonadeiandFrancescaFerrariofSpringerItaliafortheirinvaluablehelpinthepreparationofthemanuscript.Lausanne,SwitzerlandAlfioQuarteroniMay2015AndreaManzoniFedericoNegriContents1Introduction.11.1NumericalSimulationandBeyond.21.2TheNeedforReduction.41.3ReducedBasisMethodsforPDEsataGlance.51.4AccuracyandComputationalEfficiencyofRBMethods.71.5ContentoftheBook.82RepresentativeProblems:

Analysisand（High-Fidelity）Approximation.112.1FourProblems.112.1.1Advection-Diffusion-ReactionEquation.122.1.2LinearElasticityEquations.122.1.3StokesEquations.132.1.4Navier-StokesEquations.132.2FormulationandAnalysisofVariationalProblems.142.2.1StronglyCoerciveProblems.142.2.2WeaklyCoercive（orInf-SupStable）Problems.162.2.3Saddle-PointProblems.172.3AnalysisofThree（outofFour）Problems.202.3.1Advection-Diffusion-ReactionEquation.202.3.2LinearElasticityEquations.222.3.3StokesEquations.222.4OntheNumericalApproximationofVariationalProblems.232.4.1StronglyCoerciveProblems.232.4.2AlgebraicFormof（Ph1）.252.4.3ComputationoftheDiscreteCoercivityConstant.262.4.4WeaklyCoerciveProblems.272.4.5AlgebraicFormof（Ph2）.292.4.6ComputationoftheDiscreteInf-SupConstant.302.4.7Saddle-PointProblems.312.4.8AlgebraicFormof（Ph3）.33viiviiiContents2.5FiniteElementSpaces.332.6Exercises.353RBMethods:

BasicPrinciples,BasicProperties.393.1ParametrizedPDEs:

FormulationandAssumptions.393.2High-FidelityDiscretizationTechniques.413.3ReducedBasisMethods.433.3.1GalerkinRBMethod.453.3.2Least-SquaresRBMethod.483.4AlgebraicFormofGalerkinandLeast-SquaresRBProblems.513.4.1GalerkinRBCase.513.4.2Least-SquaresRBCase.543.5ReductionofComputationalComplexity:

Offline/OnlineDecomposition.553.6APosterioriErrorEstimation.563.6.1ARelationshipbetweenErrorandResidual.573.6.2ErrorBound.593.7Practical（andEfficient）ComputationofErrorBounds.603.7.1ComputingtheNormoftheResidual.613.7.2ComputingtheStabilityFactorbytheSuccessiveConstraintMethod.623.7.3ComputingtheStabilityFactorbyInterpolatoryRadialBasisFunctions.653.8AnIllustrativeNumericalExample.673.9Exercises.714OntheAlgebraicandGeometricStr