17 Pages
English

Optimization and Engineering c Kluwer Academic Publishers Manufactured in The Netherlands

-

Gain access to the library to view online
Learn more

Description

Niveau: Supérieur, Doctorat, Bac+8
Optimization and Engineering, 1, 323–339, 2000 c ° 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology F. VEERS E INRIA Rhone-Alpes, Monbonnot Saint Martin, France email: D. AUROUX⁄ Ecole Normale Superieure, Lyon, France email: M. FISHER European Centre for Medium-Range Weather Forecasts, Reading, UK Received December 17, 1999; Revised September 27, 2000 Abstract. This paper uses two simple variational data assimilation problems with the 1D viscous Burgers' equation on a periodic domain to investigate the impact of various diagonal-preconditioner update and scaling strategies, both on the limited-memory BFGS (Broyden, Fletcher, Goldfarb and Shanno) inverse Hessian ap- proximation and on the minimization performance. These simple problems share some characteristics with the large-scale variational data assimilation problems commonly dealt with in meteorology and oceanography. The update formulae studied are those proposed by Gilbert and Lemarechal (Math. Prog., vol. 45, pp. 407– 435, 1989) and the quasi-Cauchy formula of Zhu et al. (SIAM J. Optim., vol. 9, pp. 1192–1204, 1999). Which information should be used for updating the diagonal preconditioner, the one to be forgotten or the most recent one, is considered first.

  • diagonal preconditioners

  • scale variational

  • limited-memory inverse

  • limited-memory bfgs

  • hessian matrix

  • memory bfgs

  • formulae studied

  • preconditioner update


Subjects

Informations

Published by
Reads 44
Language English
OptimizationandEngineering,1,323–339,2000°c2000KluwerAcademicPublishers.ManufacturedinTheNetherlands.Limited-MemoryBFGSDiagonalPreconditionersforaDataAssimilationProbleminMeteorologyF.VEERSE´INRIARhoˆne-Alpes,MonbonnotSaintMartin,Franceemail:fabrice.veerse@imag.frD.AUROUXEcoleNormaleSupe´rieure,Lyon,Franceemail:didier.auroux@ens-lyon.frM.FISHEREuropeanCentreforMedium-RangeWeatherForecasts,Reading,UKReceivedDecember17,1999;RevisedSeptember27,2000Abstract.Thispaperusestwosimplevariationaldataassimilationproblemswiththe1DviscousBurgers’equationonaperiodicdomaintoinvestigatetheimpactofvariousdiagonal-preconditionerupdateandscalingstrategies,bothonthelimited-memoryBFGS(Broyden,Fletcher,GoldfarbandShanno)inverseHessianap-proximationandontheminimizationperformance.Thesesimpleproblemssharesomecharacteristicswiththelarge-scalevariationaldataassimilationproblemscommonlydealtwithinmeteorologyandoceanography.TheupdateformulaestudiedarethoseproposedbyGilbertandLemare´chal(Math.Prog.,vol.45,pp.407–435,1989)andthequasi-CauchyformulaofZhuetal.(SIAMJ.Optim.,vol.9,pp.1192–1204,1999).Whichinformationshouldbeusedforupdatingthediagonalpreconditioner,theonetobeforgottenorthemostrecentone,isconsideredfirst.Then,followingtheformerauthors,ascalingofthediagonalpreconditionerisintroducedforthecorrespondingformulaeinordertoimprovetheminimizationperformance.ThelargenegativeimpactofsuchascalingonthequalityoftheL-BFGSinverseHessianapproximationledustoproposeanalternateupdatingandscalingstrategy,thatprovidesagoodinverseHessianapproximationandgivesthebestminimizationperformancefortheproblemsconsidered.WiththisapproachthequalityoftheinverseHessianapproximationimprovessteadilyduringtheminimizationprocess.Moreover,thisqualityandtheL-BFGSminimizationperformanceimproveswhentheamountofstoredinformationisincreased.Keywords:quasi-Newton,limited-memoryBFGS,diagonalupdating,quasi-Cauchy,large-scaleminimization1.IntroductionNumericalsimulationsinmeteorologyandoceanographyconsistessentiallyinintegratingadiscretizedversionofthesystemofpartialderivativeequations(PDEs)modelingtheevolutionoftheatmosphereand/ortheocean.ThissystemofPDEscomprisesadynamicalcorederivedfromNavier-Stokesequationsusingrelevantapproximations,anequationofstateforthefluidofinterest,anequationrepresentingthefirstlawofthermodynamics,andphysicalparameterizationsmodelingsubgrid-scaleprocesses(convection,radiation,Correspondingauthor.