预条件方程组及其应用
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摘要
建立了与一般Toeplitz方程组ANX =B所对应的ω循环型预条件方程组PN[ω]Xω =B .通过采用不同的准则构造预条件矩阵 ,可以得到不同的预条件方程组 ,计算出合理的ω值 .理论分析和实际计算证明了该方法所得到的近似计算结果优于普通Fourier变换方法 (ω =1)的分析结果
We establish the ω-circulant preconditioned equations set P NX ω=B corresponding to the general Toeplitz equation set A NX=B.By choosing different criteria for constructing preconditioners,we obtain different preconditioned equation sets P NX ω=B and an optimal value for ω.Theoretical analysis and practical examples prove that the inversed results by the use of this method are much better than those of the traditional Fourier method.
We establish the ω-circulant preconditioned equations set P NX ω=B corresponding to the general Toeplitz equation set A NX=B.By choosing different criteria for constructing preconditioners,we obtain different preconditioned equation sets P NX ω=B and an optimal value for ω.Theoretical analysis and practical examples prove that the inversed results by the use of this method are much better than those of the traditional Fourier method.
引文
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[7] 罗明秋,刘 洪,李幼铭.基于螺旋线上谱因式分解的地震波场隐式辛算法.地球物理学报,2001,44(3):379～388LuoMQ ,LiuH ,LiYM .Seismicwavemodelingwithimplicitsym plecticmethodbasedonspectralfactorizationonhelix.ChineseJ.Geo phys.(inChinese),2001,44(3):379～388
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[13] ChanR ,MNg.ToeplitzpreconditionersforHermitianToeplitzpys tems.LinearAlgebraAppl.,1993,(190):181～208
[14] ChanR ,YeungM ,Circulantpreconditionersconstructedfromkernels.SIAMJ .Numer.Anal.,1992(294):1093～1103
[15] KuT ,KuoC .DesignandanalysisofToeplitzpreconditioners.IEEETrans.OnSignalProcess.,1992,(40):129～141
[16] ChanT .AnoptimalcirculantpreconditionerforToeplitzsystems.SIAMJ .Sci.Stat.Comput.,1988,(9):766～771
[17] OlkinJ.Linearandnonlineardeconvolutionproblems[PH .D .thesis].Houston:RiceUniv.,1986
[18] StrangG ,EdelmanA .TheToeplitz circulanteigenvalueproblemAx=λCx.In:BraggL ,DettmanJ ,eds.OaklandConf.OnPDEandAppl.Math.NewYork:LongmanSci.Tech.,1987
[2] 蒋增荣,曾泳泓,余品能.快速算法.长沙:国防科技大学出版社,1993JiangZR ,ZengYH ,YuPN .FastAlgorithm(inChinese).Chang sha:NationalDefenceScientificUniversityPress,1993
[3] 程乾生编著.地震信号数字处理的数学原理.北京:石油工业出版社,1979ChengQS .MathematicalPrinciplesofDataProcessingforSeismicSig nal(inChinese).Beijing:PetroleumIndustryPress,1979
[4] 杨文采著.地球物理反演的理论与方法.北京:地质出版社,1997YangWC .TheoriesandMethodsofInversioninGeophysics(inChi nese).Beijing:GeologicalPublishingHouse,1997
[5] 梅金顺,刘 洪.ω循环型边界条件.地球物理学报,2003,46(6):835～841MeiJS ,LiuH .ω circulantboundarycondition.ChineseJ .Geophys.(inChinese),2003,46(6):835～841
[6] 梅金顺,刘 洪.Toeplitz方程组的近似计算.地球物理学进展,2003,18(1):128～133MeiJS ,LiuH .ApproximatecomputationofToeplitzsystemsofequations.ProgressinGeophysics(inChinese),2003,18(1):128～133
[7] 罗明秋,刘 洪,李幼铭.基于螺旋线上谱因式分解的地震波场隐式辛算法.地球物理学报,2001,44(3):379～388LuoMQ ,LiuH ,LiYM .Seismicwavemodelingwithimplicitsym plecticmethodbasedonspectralfactorizationonhelix.ChineseJ.Geo phys.(inChinese),2001,44(3):379～388
[8] StrangG .AproposalforToeplitzmatrixcalculations.Stud.Appl.Math.,1986,(74):171～176
[9] AxelssonO ,BarkerV .Finiteelementsolutionofboundaryvalueprob lems.TheoryandComputation.Orlando:AcademicPress,1984
[10] TismenetskyM .AdecompositionofToeplitzmatricesandoptimalcir culantpreconditioning.LinearAlgebraAppl.,1991,(154156):105～121
[11] TyrtyshnikovE .Optimalandsuperoptimalcirculantpreconditioners.SIAMJ.MatrixAnal.Appl.,1992,(13):459～473
[12] ChanR .CirculantpreconditionersforHermitianToeplitzsystems.SIAMJ.MatrixAnal.Appl.,1989,(10):542～550
[13] ChanR ,MNg.ToeplitzpreconditionersforHermitianToeplitzpys tems.LinearAlgebraAppl.,1993,(190):181～208
[14] ChanR ,YeungM ,Circulantpreconditionersconstructedfromkernels.SIAMJ .Numer.Anal.,1992(294):1093～1103
[15] KuT ,KuoC .DesignandanalysisofToeplitzpreconditioners.IEEETrans.OnSignalProcess.,1992,(40):129～141
[16] ChanT .AnoptimalcirculantpreconditionerforToeplitzsystems.SIAMJ .Sci.Stat.Comput.,1988,(9):766～771
[17] OlkinJ.Linearandnonlineardeconvolutionproblems[PH .D .thesis].Houston:RiceUniv.,1986
[18] StrangG ,EdelmanA .TheToeplitz circulanteigenvalueproblemAx=λCx.In:BraggL ,DettmanJ ,eds.OaklandConf.OnPDEandAppl.Math.NewYork:LongmanSci.Tech.,1987
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