正则化预处理迭代算法在频率域声波模拟中的应用
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摘要
高精度及高效频率域声波数值模拟的关键在于高效求解声波方程经离散化后得到的大型稀疏线性方程组.该方程组系数矩阵具有很强的稀疏性,非对称性和非正定性等特征,常用的迭代算法难以准确、高效地求解.为了改善数值模拟迭代算法的收敛性与稳定性,在算法基础上添加预条件算子是求解该类方程的常用方案.本文基于以上思路,引入正则化技术来构造合适的预条件算子,提出正则化预条件迭代算法,以加速求解方程组.通过包含有均匀介质和高非均匀度介质(Marmousi)模型的数值模拟实验结果表明:与单独使用迭代算法相比,本文提出的正则化预条件迭代算法在计算量方面仅多了一次矩阵-矢量相乘,内存消耗未增加;同时,基于该算法的数值模拟结果能够满足精度要求,较单独使用迭代法能够有效改善收敛性质,加快收敛速度;而且,在二维模型算例下,与LU分解算法相比,基于该算法的内存消耗大幅下降.
One of the keys for efficient numerical modeling of frequency-domain acoustic wave modeling is to solve a large sparse and linear system,which the coefficient matrix have thecharacter of non-symmetric and non-positive.To improve the convergence property and robustness of numerical modeling,we have developed a new regularized preconditioning iterative method that enforced the sparsity of solutions in frequency-domain.The construction of preconditioner requires the regularization for large sparse original problem firstly,which is called regularized system.Then we regarded the approximate iterative solution of the regularized system as initial value of original problem.The numerical computational experiments including the synthetic complex model indicate that the extra computation of the method is almost negligible.By comparison with the storage space of LU decomposition,the regularized preconditioning iterative algorithm is dramatically decreased.Besides,effectiveness and convergence rates of regularized preconditioning iterative solvers are greatly improved by regularized preprocessing.
One of the keys for efficient numerical modeling of frequency-domain acoustic wave modeling is to solve a large sparse and linear system,which the coefficient matrix have thecharacter of non-symmetric and non-positive.To improve the convergence property and robustness of numerical modeling,we have developed a new regularized preconditioning iterative method that enforced the sparsity of solutions in frequency-domain.The construction of preconditioner requires the regularization for large sparse original problem firstly,which is called regularized system.Then we regarded the approximate iterative solution of the regularized system as initial value of original problem.The numerical computational experiments including the synthetic complex model indicate that the extra computation of the method is almost negligible.By comparison with the storage space of LU decomposition,the regularized preconditioning iterative algorithm is dramatically decreased.Besides,effectiveness and convergence rates of regularized preconditioning iterative solvers are greatly improved by regularized preprocessing.
引文
Al-Attar D,Woodhouse J H,Deuss A.2012.Calculation of normal mode spectra in laterally heterogeneous earth models using an iterative direct solution method.Geophysical Journal International,189(2):1038-1046,doi:10.1111/j.1365-246X.2012.05406.x.
Alford R M,Kelly K R,Boore D M.1974.Accuracy of finitedifference modeling of the acoustic wave equation.Geophysics,39(6):834-842.
Benson M,Krettmann J,Wright M.2007.Parallel algorithms for the solution of certain large sparse linear systems.International Journal of Computer Mathematics,16(4):245-260.
Benzi M,Tuma M.1998.A sparse approximate inverse preconditioner for nonsymmetric linear systems.SIAM Journal on Scientific Computing,19(3):968-994.
Benzi M,Haws J C,Tuma M.2000.Preconditioning Highly Indefinite and Nonsymmetric Matrices.Society for Industrial and Applied Mathematics,22(4):1333-1353.
Broyden C G.1965.A class of methods for solving nonlinear simultaneous equations.Mathematics of Computation,19(92):577-593.
Broyden C G.1970.The convergence of a class of double rank minimization algorithms,Part I′.Journal of Applied Mathematics,6(1):76-90.
Clapp R G,Biondi B,Claerbout J F.2004.Incorporating geologic information into reflection tomography.Geophysics,69(2):533-546.
Cui Y,Wang Y F.2015.Tikhonov regularization and gradient descent algorithms for tomography using first-arrival seismic traveltimes.Chinese Journal of Geophysics(in Chinese),58(4):1367-1377,doi:10.6038/cjg20150423.
Dablain M A.1986.The application of high-order differencing to the scalar wave equation.Geophysics,51(1):54-66.
Davidon and William C.1991.Variable metric method for minimization.SIAM Journal on Optimization,1(1):1-17.
Dong L G,Li P M.2004.Dispersive problem in seismic wave propagation numerical modeling.Natural Gas Industry(in Chinese),24(6):53-56.
Fletcher R.1970.A new approach to variable metric algorithms.The Computer Journal,13(3):317-322.
Grote M,Simon H D.1992.Parallel preconditioning and approximate inverses on the connection machine.∥Proc.Sixth SIAMConference on Parallel Processing for Scientific Computing.Williamsburg,VA:Philadelphia,PA,519-523.
Hustedt B,Operto S,Virieux J.2004.Mixed-grid and staggeredgrid finite-difference methods for frequency-domain acoustic wave modelling.Geophysical Journal International,157(3):1269-1296.
Jo C H,Shin C,Suh J H.1996.An optimal 9-point,finite-difference,frequency-space,2Dscalar wave extrapolator.Geophysics,61(2):529-537.
Leary D.1993.Why Broyden's Nonsymmetric Method Terminates On Linear Equations.University of Maryland at College Park.
Lin S L.2005.Studying the algorithm for solving Ill-conditioned linear system of equations[Master′s thesis](in Chinese).Hangzhou:Zhejiang University.
Lysmer J,Drake L A.1972.A finite element method for seismology.∥Bolt B A ed.Methods in Computational Physics,volume 11:Seismology:Surface Waves and Earth Oscillations.New York:Academic Press Inc.,181-216.
Magnus.R.Hestence and Stiefel.1952.Methods of Conjugate Gradients for solving linear systems.Journal of Research of the National Bureau of Standards,49(6):409-436.
Marfurt K J.1984.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations.Geophysics,49(4):533-549.
Marfurt K J,Shin C S.1989.The future of iterative modeling in geophysical exploration.∥Eisner E ed.Handbook of Geophysical Exploration:I Seismic Exploration,volume 21:Supercomputers in seismic exploration.New York:Pergamon Press,203-228.
Meijerink J A,Van der Vorst H.1977.An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix.Mathematics of Computation,31(137):148-162.
Operto S,Virieux J,Amestoy P,et al.2007.3Dfinite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver:A feasibility study.Geophysics,72(5):SM195-SM211.
Pratt R G,Worthington M H.1990.Inverse theory applied to multi-source cross-hole tomography:Part I:Acoustic waveequation method.Geophysical Prospecting,27(6):287-310.
Rafiei A.2014.Left-looking version of AINV preconditioner with complete pivoting strategy.Linear Algebra and its Applications,445:103-126.
Rui P L.2007.Fast iterative techniques for electromagnetic scattering analysis[Ph.D.thesis](in Chinese).Nanjing:Nanjing University of Science and Technology.
Saad Y.1994.ILUT:A dual threshold incomplete LU factorization.Numerical Linear Algebra with Applications,1(4):387-402.
Saad Y.2003.Iterative Methods for Sparse Linear System.Philadelphia:Society for industrial and mathematics.
Tang X D,Liu H,Zhang H.2015.Frequency-space domain acoustic modeling based on an average-derivative and GPU implementation.Chinese Journal of Geophysics(in Chinese),58(4):1341-1354,doi:10.6038/cjg20150421.
Tikhonov A N.1943.On the stability of inverse problems.Dolk.akad.nauk Sssr,39(5):176-179.
Tikhonov A N,Arsenin V Y.1977.Solutions of Ill-posed Problems.Wiley.
Wang W,Han B,Tang J P.2013.Regularization method with sparsity constraints for seismic waveform inversion.Chinese Journal of Geophysics(in Chinese),56(1):289-297,doi:10.6038/cjg20130130.
Wong Y S.1988.Preconditioned gradient type methods applied to nonsymmetric linear systems.International Journal of Computer Mathematics,23(2):141-165.
崔岩,王彦飞.2015.基于初至波走时层析成像的Tikhonov正则化与梯度优化算法.地球物理学报,58(4):1367-1377,doi:10.6038/cjg20150423.
董良国,李培明.2004.地震波传播数值模拟中的频散问题.天然气工业,24(6):53-56.
林胜良.2005.病态线性方程组解法研究[硕士论文].杭州:浙江大学.
芮平亮.2007.电磁散射分析中的快速迭代求解技术[博士论文].南京:南京理工大学.
唐祥德,刘洪,张衡.2015.基于加权平均导数的频率-空间域正演模拟及GPU实现.地球物理学报,58(4):1341-1354,doi:10.6038/cjg20150421.
王薇,韩波,唐锦萍.2013.地震波形反演的稀疏约束正则化方法.地球物理学报,56(1):289-297,doi:10.6038/cjg20130130.
Alford R M,Kelly K R,Boore D M.1974.Accuracy of finitedifference modeling of the acoustic wave equation.Geophysics,39(6):834-842.
Benson M,Krettmann J,Wright M.2007.Parallel algorithms for the solution of certain large sparse linear systems.International Journal of Computer Mathematics,16(4):245-260.
Benzi M,Tuma M.1998.A sparse approximate inverse preconditioner for nonsymmetric linear systems.SIAM Journal on Scientific Computing,19(3):968-994.
Benzi M,Haws J C,Tuma M.2000.Preconditioning Highly Indefinite and Nonsymmetric Matrices.Society for Industrial and Applied Mathematics,22(4):1333-1353.
Broyden C G.1965.A class of methods for solving nonlinear simultaneous equations.Mathematics of Computation,19(92):577-593.
Broyden C G.1970.The convergence of a class of double rank minimization algorithms,Part I′.Journal of Applied Mathematics,6(1):76-90.
Clapp R G,Biondi B,Claerbout J F.2004.Incorporating geologic information into reflection tomography.Geophysics,69(2):533-546.
Cui Y,Wang Y F.2015.Tikhonov regularization and gradient descent algorithms for tomography using first-arrival seismic traveltimes.Chinese Journal of Geophysics(in Chinese),58(4):1367-1377,doi:10.6038/cjg20150423.
Dablain M A.1986.The application of high-order differencing to the scalar wave equation.Geophysics,51(1):54-66.
Davidon and William C.1991.Variable metric method for minimization.SIAM Journal on Optimization,1(1):1-17.
Dong L G,Li P M.2004.Dispersive problem in seismic wave propagation numerical modeling.Natural Gas Industry(in Chinese),24(6):53-56.
Fletcher R.1970.A new approach to variable metric algorithms.The Computer Journal,13(3):317-322.
Grote M,Simon H D.1992.Parallel preconditioning and approximate inverses on the connection machine.∥Proc.Sixth SIAMConference on Parallel Processing for Scientific Computing.Williamsburg,VA:Philadelphia,PA,519-523.
Hustedt B,Operto S,Virieux J.2004.Mixed-grid and staggeredgrid finite-difference methods for frequency-domain acoustic wave modelling.Geophysical Journal International,157(3):1269-1296.
Jo C H,Shin C,Suh J H.1996.An optimal 9-point,finite-difference,frequency-space,2Dscalar wave extrapolator.Geophysics,61(2):529-537.
Leary D.1993.Why Broyden's Nonsymmetric Method Terminates On Linear Equations.University of Maryland at College Park.
Lin S L.2005.Studying the algorithm for solving Ill-conditioned linear system of equations[Master′s thesis](in Chinese).Hangzhou:Zhejiang University.
Lysmer J,Drake L A.1972.A finite element method for seismology.∥Bolt B A ed.Methods in Computational Physics,volume 11:Seismology:Surface Waves and Earth Oscillations.New York:Academic Press Inc.,181-216.
Magnus.R.Hestence and Stiefel.1952.Methods of Conjugate Gradients for solving linear systems.Journal of Research of the National Bureau of Standards,49(6):409-436.
Marfurt K J.1984.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations.Geophysics,49(4):533-549.
Marfurt K J,Shin C S.1989.The future of iterative modeling in geophysical exploration.∥Eisner E ed.Handbook of Geophysical Exploration:I Seismic Exploration,volume 21:Supercomputers in seismic exploration.New York:Pergamon Press,203-228.
Meijerink J A,Van der Vorst H.1977.An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix.Mathematics of Computation,31(137):148-162.
Operto S,Virieux J,Amestoy P,et al.2007.3Dfinite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver:A feasibility study.Geophysics,72(5):SM195-SM211.
Pratt R G,Worthington M H.1990.Inverse theory applied to multi-source cross-hole tomography:Part I:Acoustic waveequation method.Geophysical Prospecting,27(6):287-310.
Rafiei A.2014.Left-looking version of AINV preconditioner with complete pivoting strategy.Linear Algebra and its Applications,445:103-126.
Rui P L.2007.Fast iterative techniques for electromagnetic scattering analysis[Ph.D.thesis](in Chinese).Nanjing:Nanjing University of Science and Technology.
Saad Y.1994.ILUT:A dual threshold incomplete LU factorization.Numerical Linear Algebra with Applications,1(4):387-402.
Saad Y.2003.Iterative Methods for Sparse Linear System.Philadelphia:Society for industrial and mathematics.
Tang X D,Liu H,Zhang H.2015.Frequency-space domain acoustic modeling based on an average-derivative and GPU implementation.Chinese Journal of Geophysics(in Chinese),58(4):1341-1354,doi:10.6038/cjg20150421.
Tikhonov A N.1943.On the stability of inverse problems.Dolk.akad.nauk Sssr,39(5):176-179.
Tikhonov A N,Arsenin V Y.1977.Solutions of Ill-posed Problems.Wiley.
Wang W,Han B,Tang J P.2013.Regularization method with sparsity constraints for seismic waveform inversion.Chinese Journal of Geophysics(in Chinese),56(1):289-297,doi:10.6038/cjg20130130.
Wong Y S.1988.Preconditioned gradient type methods applied to nonsymmetric linear systems.International Journal of Computer Mathematics,23(2):141-165.
崔岩,王彦飞.2015.基于初至波走时层析成像的Tikhonov正则化与梯度优化算法.地球物理学报,58(4):1367-1377,doi:10.6038/cjg20150423.
董良国,李培明.2004.地震波传播数值模拟中的频散问题.天然气工业,24(6):53-56.
林胜良.2005.病态线性方程组解法研究[硕士论文].杭州:浙江大学.
芮平亮.2007.电磁散射分析中的快速迭代求解技术[博士论文].南京:南京理工大学.
唐祥德,刘洪,张衡.2015.基于加权平均导数的频率-空间域正演模拟及GPU实现.地球物理学报,58(4):1341-1354,doi:10.6038/cjg20150421.
王薇,韩波,唐锦萍.2013.地震波形反演的稀疏约束正则化方法.地球物理学报,56(1):289-297,doi:10.6038/cjg20130130.