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1、JOURNALOFCOMPUTATIONALPHYSICS132,384±408(1997)ARTICLENO.CP965647AFourier±WaveletMonteCarloMethodforFractalRandomFieldsFrankW.ElliottJr.,DavidJ.Horntrop,andAndrewJ.MajdaCourantInstituteofMathematicalSciences,251MercerStreet,NewYork,NewYork10012ReceivedAugust2,1996
2、;revisedDecember23,1996k[v(x)2v(y)]2l5Cux2yu2H,(1.1)HAnewhierarchicalmethodfortheMonteCarlosimulationofrandom®eldscalledtheFourier±waveletmethodisdevelopedandwhere0,H,1istheHurstexponentandk?ldenotesappliedtoisotropicGaussianrandom®eldswithpowerlawspectraltheexpe
3、ctedvalue.densityfunctions.ThistechniqueisbasedupontheorthogonalHerewedevelopanewMonteCarlomethodbasedupondecompositionoftheFourierstochasticintegralrepresentationofthe®eldusingwavelets.TheMeyerwaveletisusedherebecauseawaveletexpansionoftheFourierspacerepresentat
4、ionofitsrapiddecaypropertiesallowforaverycompactrepresentationthefractalrandom®eldsin(1.1).Thismethodiscapableofthe®eld.TheFourier±waveletmethodisshowntobestraightfor-ofgeneratingavelocity®eldwiththeKolmogoroffspec-wardtoimplement,giventhenatureofthenecessaryprec
5、omputa-trum(H5Adin(1.1))overmany(10to15)decadesoftionsandtherun-timecalculations,andyieldscomparableresultsscalingbehaviorcomparabletothephysicalspacemulti-withscalingbehavioroverasmanydecadesasthephysicalspacemultiwaveletmethodsdevelopedrecentlybytwooftheauthors
6、.waveletalgorithmdevelopedbytwooftheauthorsinre-However,theFourier±waveletmethoddevelopedhereismore¯ex-centwork[5±7].However,sincetheFourier±waveletibleand,inparticular,appliestoanisotropicspectrageneratedmethoddevelopedbelowisaspectralmethod,itismuchthroughsolut
7、ionsofdifferentialequations.Simulationresultsusingmore¯exibleforgeneratingrandom®eldswithanisotropicthisnewtechniqueandthewell-knownnonhierarchicalsimulationspectraand,inparticular,spectrageneratedthroughthetechnique,therandomizationmethod,aregivenandcomparedforb
8、othasimpleshearlayermodelproblemaswellasatwo-dimen-solutionsofpartialdifferentialequations.sionalisotropicGaussianrandom®eld.TheFourier±waveletAwidevarietyofco
