图法及其在Toeplitz矩阵分解中的应用
|
摘要
实现了基于图法的稀疏正定系统的求解 ,并在此基础上实现了具有Toeplitz结构的大型稀疏矩阵的快速LU分解 .在基于波动方程的地震数据处理如地震波场模拟和叠前深度偏移等隐式方法中 ,拉普拉氏算子或亥姆霍兹算子的快速分解是这些方法能否实现的关键 .在螺旋边界条件下 ,这些算子的表示矩阵是具有Toeplitz结构的正定厄密矩阵 ,可以通过本文方法实现快速分解
The solution of a sparse positive definite system of equations based on the graphic method is completed, and on the basis of this algorithm we complete the LU decomposition of a large sparse matrix with the structure of Toeplitz. In the processing methods of seismic data based on wave equations such as implicit methods of modeling of seismic wave field and prestack migration, the rapid decomposition of Laplacian or Helmholtz operator is the key to complete these methods. With a helix boundary condition, the expressing matrix of these operators has the structure of Toeplitz, it can be decomposed rapidly with the method proposed in this article.
The solution of a sparse positive definite system of equations based on the graphic method is completed, and on the basis of this algorithm we complete the LU decomposition of a large sparse matrix with the structure of Toeplitz. In the processing methods of seismic data based on wave equations such as implicit methods of modeling of seismic wave field and prestack migration, the rapid decomposition of Laplacian or Helmholtz operator is the key to complete these methods. With a helix boundary condition, the expressing matrix of these operators has the structure of Toeplitz, it can be decomposed rapidly with the method proposed in this article.
引文
[1] JAGeorge,JWLiu.Theevolutionoftheminimumdegreeorderingalgorithm[J].SIAMReview,1989,31(1):1～19.
[2] J .W .LiuModificationoftheminimum degreealgorithmbymultipleelimination[J].ACMTrans.MathSoftw.1985,11(2):141～153.
[3] LuoMingqiu,GaoHongwei,LiuHong,etal.AcomparisonontheLUdecompositionofLaplacianmatrixwithspectralanddirecteliminationmethods[A].SEG 70thInternationalMeeting,ExpandedAbstracts[C],Calgary,Canada,2000.
[4] RickettJ .Implicit3Ddepthmigrationbywavefieldextrapolationwithhelicalboundarycondition[A].SEG 68thInternationalMeeting,ExpandedAbstracts[C],1998:1124～1127.
[5] 石根华著.数值流形方法与非连续变形分析[M].北京:清华大学出版社,1997,146～167.(ShiGen hua,Numericalmanifoldmethodanddiscontinuousdeformationanalysis[M].Beijing:TsinghuaUniversityPress,1997,146～167.
[2] J .W .LiuModificationoftheminimum degreealgorithmbymultipleelimination[J].ACMTrans.MathSoftw.1985,11(2):141～153.
[3] LuoMingqiu,GaoHongwei,LiuHong,etal.AcomparisonontheLUdecompositionofLaplacianmatrixwithspectralanddirecteliminationmethods[A].SEG 70thInternationalMeeting,ExpandedAbstracts[C],Calgary,Canada,2000.
[4] RickettJ .Implicit3Ddepthmigrationbywavefieldextrapolationwithhelicalboundarycondition[A].SEG 68thInternationalMeeting,ExpandedAbstracts[C],1998:1124～1127.
[5] 石根华著.数值流形方法与非连续变形分析[M].北京:清华大学出版社,1997,146～167.(ShiGen hua,Numericalmanifoldmethodanddiscontinuousdeformationanalysis[M].Beijing:TsinghuaUniversityPress,1997,146～167.