探地雷达反问题的同伦算法研究
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摘要
作为探地雷达数据处理的基础和成像过程的关键环节,探地雷达的数值模拟引发了人们越来越多的关注和兴趣。基于Maxwell方程的地下复杂介质中电磁波传播的正、反演问题,已日益成为探地雷达基础理论研究中的前沿课题。作为代数拓扑学中的一个基本概念,把同伦思想引入到非线性算子方程的求解之中,能够克服传统数值迭代方法容易陷入局部收敛的弱点,放宽对初始猜测选取的严格限制,进而形成高效、实用的大范围收敛方法。有鉴于此,本文将同伦方法引入到探地雷达反问题的求解过程中,开展了一系列兼有计算量小、抗噪能力强、收敛范围广、程序易于实现之特点的数值反演方法研究,实现了探地雷达反问题真正意义上的完全非线性反演。
本文的主要工作包括:1、从电磁场基本理论出发阐述了探地雷达反问题模型的构造过程,在前人工作的基础上对模型增设了吸收边界条件,使之更适于实际计算的要求;2、将大范围收敛的同伦方法引入到地下介质参数识别的反演过程中,构造了大范围收敛且结果稳定的同伦正则化反演算法、同伦参数微分反演算法和同伦自适应方法,并从理论上证明了算法的全局收敛性,通过数值实验验证了算法的有效性;3、将多尺度反演与同伦反演思想进行结合,构造出既具有大范围收敛性质,又能有效降低反演工作量的同伦-小波混合算法和同伦-多重网格混合算法;4、将电磁记录与测井资料相联系,构造了基于同伦和测井约束的系列混合反演算法,提高了探地雷达反演的纵向分辨效果。除绪论之外的各章包括如下内容:
在本文的第2章,通过对Maxwell方程组和吸收边界条件的分析,研究了探地雷达反问题连续模型和相应的有限差分离散模型的构造过程。作为本文的重点章节,在第3章中详细地讨论了同伦算法的原理和同伦反演的基本思路。首先,结合Tikhonov正则化和同伦方法,构造了融合两者优点的同伦正则化算法,算法克服了反问题的不适定性,扩大了初始猜测选取和算法收敛的范围。其次,构造了同伦参数微分反演算法,并从理论上给予了算法全局收敛性的严格证明。在本章的最后,完成了同伦自适应算法的设计:算法借助信赖域方法,通过设置阈值自适应地选择正则化参数,在同伦反演优点的基础上,增强了算法的灵活性和实用性。
在第4、第5章中,基于小波多尺度和多重网格多尺度思想,构造了同伦-小波混合反演算法和同伦-多重网格混合反演算法,将原始反问题转化为一系列嵌套子空间中的反问题序列,在最大尺度上采用大范围收敛的同伦反演方法作为全局搜索的工具,拓宽了搜索范围,保证了最大尺度上求解在全局范围内进行。算法分别利用了小波分析和多重网格两种技术的各自优势,具有全局搜索能力,收敛速度较快,反演效果良好,在多解或目标函数存在大量局部极小点的情况下,仍能保证最大尺度上的全局收敛性。
在第6章中,利用测井约束条件构造了同伦-测井约束混合反演系列算法。算法将地表的横向资料与测井的纵向资料相结合,提高了探地雷达反演的纵向分辨效果。通过引入小波分析和多重网络方法,构造了对应的两种多尺度算法,提高了反演算法的稳定性和抗噪能力,减少了反演计算量。
本文在同伦思想基础上构造的各种反演算法及其应用成果,尽管是针对探地雷达反演这一特定问题而得到的,但由于所构造算法的普遍性和一般性,还可将其推广到其它类型反问题的应用研究中,具有较高的理论价值和实用价值。
As the basis of data processing of GPR (Ground Penetrating Radar) and a key link of the imaging process, the numerical simulation of GPR provokes more and more concern and interest. The forward problem and the inverse problem of the electromagnetic propagation in anisotropy media underground based on Maxwell equation have increasingly been an advanced task in the research of the fundamental theory of GPR. That the idea of homotopy, which is a basic conception of algebraic topology, is used in the solution of nonlinear operator equation will be able to eliminate the weakness that the traditional numerical iteration falls easily into local convergence, broaden the rigorous restrictions on selecting initial guess, and come to a large-scale convergent method which is efficient and pragmatic. This thesis, in the process of the solution to the inverse problems of GPR, draws into homotopy method and makes a series of study of numerical inversion algorithm that has simultaneously less computation load, powerful antinoise capacity, wide-ranging convergence and easy accomplishment of the program; consequently actualizes the fully nonlinear inversion of GPR problem in real sense.
This thesis mainly includes that: (1) inverse problem models used for data interpretation of GPR based on Maxwell equation and absorbing boundary condition have been constructed. (2) wide-ranging convergent homotopy method has been drawn into inversion procedure of the identification to the underground parameter and in the process of it, wide-ranging convergent and steady homotopy regularization inversion algorithm, homotopy parameter differential algorithm and homotopy adaptive inversion algorithm have been constructed, which theoretically proves global convergence of algorithm and effectiveness of algorithm has also been verified by numerical test. (3) homotopy-wavelet hybrid algorithm and homotopy-multigrid hybrid algorithm, which have both wide-ranging convergent character and effective reduction of amount of inversion, have been constructed through combination of the idea of multi-scale inversion with homotopy inversion algorithm. (4) By means of connection of EM record with well-log data, a series of algorithms for homotopy- well-log constraint have been constructed, which improves longitudinal resolving power of inversion of GPR.
Concrete synopses in each Chapter are as the follows:
In Chapter 2, continuous models of inverse problems of GPR and corresponding finite difference discretization models have been constructed by means of ascertaining Maxwell equations and absorbing boundary condition.
In Chapter 3, which is the focal one of this thesis, the principle of homotopy algorithm and the basic train of thought of homotopy inversion are stated at some length. Combining Tikhonov regularization with homotopy method, this thesis constructs homotopy regularization algorithm, which mixes together the advantages of both. The algorithm avoids inelasticity of the inverse problems, and extends initial selection and algorithm convergence. Furthermore, the Chapter constructs homotopy parameter differential inversion algorithm and, in terms of theory, strictly proves global convergence of the algorithm. At length, the Chapter completes the designment of homotopy adaptive algorithm, which can be adaptive to select regularization parameter through the delimitation of threshold, and which greatly improves the flexibility and practicality of the algorithm.
In Chapter 4 and Chapter 5, homotopy-wavelet hybrid inversion algorithm and homotopy-multigrid hybrid inversion algorithm are constructed by utilizing respectively wavelet multi-scale and multigrid multi-scale decomposition, which makes original inverse problems transformed into the inverse problem alignment in a sequence of nested subspace, in the maximum scale adopts wide-ranging convergent homotopy inversion algorithm as a means of global search, makes the search zone bigger and guarantees in the maximum scale the solution can be done in the global limits. The algorithm makes use of the respective advantages of the wavelet analyses and multigrid, and possesses the global search capacity with rapid convergence speed and favorable inversion effect. Global convergence in the maximum scale still can be guaranteed even though large quantities of local minimum exists in multi-solution or target function.
In Chapter 6, homotopy- well-log constraint hybrid inversion algorithm is constructed by making use of the well-log constraint condition. The algorithm combines the lateral data of the surface with the longitudinal data of the well-log, and improves the longitudinal resolving capacity of GPR inversion. In addition, two more multi-scale algorithms correspondent with the algorithm have been constructed through the usage of multigrid and wavelet analyses, which strengthens the stability and antinoise capacity of the inversion algorithm and reduces the computation load of the inverse problem solution.
The inversion algorithm and its application contributed by this thesis solve, to some extent, some puzzles that exist in the process of solution of the numerical value of GPR. These methods are all in generality; they are of certain theoretical significance and widespread practical value thereby.
本文的主要工作包括:1、从电磁场基本理论出发阐述了探地雷达反问题模型的构造过程,在前人工作的基础上对模型增设了吸收边界条件,使之更适于实际计算的要求;2、将大范围收敛的同伦方法引入到地下介质参数识别的反演过程中,构造了大范围收敛且结果稳定的同伦正则化反演算法、同伦参数微分反演算法和同伦自适应方法,并从理论上证明了算法的全局收敛性,通过数值实验验证了算法的有效性;3、将多尺度反演与同伦反演思想进行结合,构造出既具有大范围收敛性质,又能有效降低反演工作量的同伦-小波混合算法和同伦-多重网格混合算法;4、将电磁记录与测井资料相联系,构造了基于同伦和测井约束的系列混合反演算法,提高了探地雷达反演的纵向分辨效果。除绪论之外的各章包括如下内容:
在本文的第2章,通过对Maxwell方程组和吸收边界条件的分析,研究了探地雷达反问题连续模型和相应的有限差分离散模型的构造过程。作为本文的重点章节,在第3章中详细地讨论了同伦算法的原理和同伦反演的基本思路。首先,结合Tikhonov正则化和同伦方法,构造了融合两者优点的同伦正则化算法,算法克服了反问题的不适定性,扩大了初始猜测选取和算法收敛的范围。其次,构造了同伦参数微分反演算法,并从理论上给予了算法全局收敛性的严格证明。在本章的最后,完成了同伦自适应算法的设计:算法借助信赖域方法,通过设置阈值自适应地选择正则化参数,在同伦反演优点的基础上,增强了算法的灵活性和实用性。
在第4、第5章中,基于小波多尺度和多重网格多尺度思想,构造了同伦-小波混合反演算法和同伦-多重网格混合反演算法,将原始反问题转化为一系列嵌套子空间中的反问题序列,在最大尺度上采用大范围收敛的同伦反演方法作为全局搜索的工具,拓宽了搜索范围,保证了最大尺度上求解在全局范围内进行。算法分别利用了小波分析和多重网格两种技术的各自优势,具有全局搜索能力,收敛速度较快,反演效果良好,在多解或目标函数存在大量局部极小点的情况下,仍能保证最大尺度上的全局收敛性。
在第6章中,利用测井约束条件构造了同伦-测井约束混合反演系列算法。算法将地表的横向资料与测井的纵向资料相结合,提高了探地雷达反演的纵向分辨效果。通过引入小波分析和多重网络方法,构造了对应的两种多尺度算法,提高了反演算法的稳定性和抗噪能力,减少了反演计算量。
本文在同伦思想基础上构造的各种反演算法及其应用成果,尽管是针对探地雷达反演这一特定问题而得到的,但由于所构造算法的普遍性和一般性,还可将其推广到其它类型反问题的应用研究中,具有较高的理论价值和实用价值。
As the basis of data processing of GPR (Ground Penetrating Radar) and a key link of the imaging process, the numerical simulation of GPR provokes more and more concern and interest. The forward problem and the inverse problem of the electromagnetic propagation in anisotropy media underground based on Maxwell equation have increasingly been an advanced task in the research of the fundamental theory of GPR. That the idea of homotopy, which is a basic conception of algebraic topology, is used in the solution of nonlinear operator equation will be able to eliminate the weakness that the traditional numerical iteration falls easily into local convergence, broaden the rigorous restrictions on selecting initial guess, and come to a large-scale convergent method which is efficient and pragmatic. This thesis, in the process of the solution to the inverse problems of GPR, draws into homotopy method and makes a series of study of numerical inversion algorithm that has simultaneously less computation load, powerful antinoise capacity, wide-ranging convergence and easy accomplishment of the program; consequently actualizes the fully nonlinear inversion of GPR problem in real sense.
This thesis mainly includes that: (1) inverse problem models used for data interpretation of GPR based on Maxwell equation and absorbing boundary condition have been constructed. (2) wide-ranging convergent homotopy method has been drawn into inversion procedure of the identification to the underground parameter and in the process of it, wide-ranging convergent and steady homotopy regularization inversion algorithm, homotopy parameter differential algorithm and homotopy adaptive inversion algorithm have been constructed, which theoretically proves global convergence of algorithm and effectiveness of algorithm has also been verified by numerical test. (3) homotopy-wavelet hybrid algorithm and homotopy-multigrid hybrid algorithm, which have both wide-ranging convergent character and effective reduction of amount of inversion, have been constructed through combination of the idea of multi-scale inversion with homotopy inversion algorithm. (4) By means of connection of EM record with well-log data, a series of algorithms for homotopy- well-log constraint have been constructed, which improves longitudinal resolving power of inversion of GPR.
Concrete synopses in each Chapter are as the follows:
In Chapter 2, continuous models of inverse problems of GPR and corresponding finite difference discretization models have been constructed by means of ascertaining Maxwell equations and absorbing boundary condition.
In Chapter 3, which is the focal one of this thesis, the principle of homotopy algorithm and the basic train of thought of homotopy inversion are stated at some length. Combining Tikhonov regularization with homotopy method, this thesis constructs homotopy regularization algorithm, which mixes together the advantages of both. The algorithm avoids inelasticity of the inverse problems, and extends initial selection and algorithm convergence. Furthermore, the Chapter constructs homotopy parameter differential inversion algorithm and, in terms of theory, strictly proves global convergence of the algorithm. At length, the Chapter completes the designment of homotopy adaptive algorithm, which can be adaptive to select regularization parameter through the delimitation of threshold, and which greatly improves the flexibility and practicality of the algorithm.
In Chapter 4 and Chapter 5, homotopy-wavelet hybrid inversion algorithm and homotopy-multigrid hybrid inversion algorithm are constructed by utilizing respectively wavelet multi-scale and multigrid multi-scale decomposition, which makes original inverse problems transformed into the inverse problem alignment in a sequence of nested subspace, in the maximum scale adopts wide-ranging convergent homotopy inversion algorithm as a means of global search, makes the search zone bigger and guarantees in the maximum scale the solution can be done in the global limits. The algorithm makes use of the respective advantages of the wavelet analyses and multigrid, and possesses the global search capacity with rapid convergence speed and favorable inversion effect. Global convergence in the maximum scale still can be guaranteed even though large quantities of local minimum exists in multi-solution or target function.
In Chapter 6, homotopy- well-log constraint hybrid inversion algorithm is constructed by making use of the well-log constraint condition. The algorithm combines the lateral data of the surface with the longitudinal data of the well-log, and improves the longitudinal resolving capacity of GPR inversion. In addition, two more multi-scale algorithms correspondent with the algorithm have been constructed through the usage of multigrid and wavelet analyses, which strengthens the stability and antinoise capacity of the inversion algorithm and reduces the computation load of the inverse problem solution.
The inversion algorithm and its application contributed by this thesis solve, to some extent, some puzzles that exist in the process of solution of the numerical value of GPR. These methods are all in generality; they are of certain theoretical significance and widespread practical value thereby.
引文
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61 R. G. Pratt, C. Shin, G. J. Hicks. Gauss-Newton and Full Newton Methods in Frequency-Space Seismic Waveform Inversion. J. Int. Geophys. 1998,133(2):341~362
62 D. S. Tsien, Y. M. Chen. A Numerical Method for Nonlinear Inverse Problems in Fluid Dynamics. Computational Methods in Nonlinear Mechanics, Proc. Inc. Conf. Comput. Mech. Nonliear Mechs. The University of Texas, Austin. 1974:935~945
63 Y. M. Chen. Generalized Pulse-Spectrum Technique. Geophysics. 1985,50: 1664~1675
64 王家映, 刘先洲. 地球物理反演理论. 中国地质大学出版社. 1998:1~181
65 N. M. Shapiro, M. H. Ritzwoller. Monte-Carlo Inversion for a Global Shearvelocity Model of the Crust and Upper Mantle. Geophys. J. Int. 2002, 151(1):88~105
66 Y. X. Pan, Q. F. Xu, H. M. Gao. Research of the Fuzzy Algorithm Optimization Based on Chaos. Control Theory and Applications. 2000, 17(5): 703~706
67 J. Caers. Automatic Histogram and Variogram Reproduction in SimulatedAnnealing Simulation. Math. Geol. 2001, 33(2): 167~190
68 P. J. Ballester, J. N. Carter. Real-Parameter Genetic Algorithms for Finding Multiple Optimal Solutions in Multi-Modal Optimization. Lecture Notes in Computer Science. 2003, 2723(0302-9743):706~717
69 王则柯, 高堂安. 同伦方法引论. 重庆出版社, 1990:1~287
70 L. T. Watson. Globally Convergent Homotopy Methods. Appl. Math. Comput. 1989,31(Spec. Issue):369~396
71 H. B. Keller, D. J. Perozzi. Fast Seismic Ray Tracing. SIAM J. Appl. Math.1983, 43(4): 981~992
72 D. W. Vasco. Singularity and Branching: A Path-Following Formalism for Geophysical Inverse Problems. Geophysics. J. Int. 1994,119:809~830
73 D. W. Vasco. Regularization and Trade-Off Associated with Nonlinear Geophysical Inverse Problems: Penalty Homotopies. Inverse Problems. 1998, 14(4):1033~1052
74 K. P. Bube, R. T. Langan. On a Continuation Approach to Regularization for Cross Well Tomography. 69th Ann. Internat. Mtg., Tulsa, 1999. Soc. Expl. Geophys. 1999:1295~1298
75 M. E. Everett. Homotopy, Polynomial Equations, Gross Boundary Data, and Small Helmholtz Systems. J. Comput. Phys. 1996,124(2):431~441
76 M. D. Jegen, M. E. Everett, A. Schultz. Using Homotopy to Invert Geophysical Data. Geophysics. 2001,66(6):1749~1760
77 韩波, 匡正, 刘家琦. 反演地层电阻率的单调同伦法. 地球物理学报. 1991, 34(4):256~261
78 B. Han, M. L. Zhang, J. Q. Liu. A Widely Convergent Generalized Pulse-Spectrum Technique for the Coefficient Inverse Problems of Differential Equations. Appl. Math. Comput. 1997,81(2):97~112
79 G. F. Feng, B. Han, J. Q. Liu. A Widely Convergent Generalized Pulse-Spectrum Method for 2-D Wave Equation Inversion. Chinese Journal of Geophysics. 2003,46(2):373~380
80 韩波, 付又和. 求解非线性不适定问题的正则化同伦方法. 黑龙江大学自然科学学报, 2005, 22(5): 659~663
81 刘舒考. 同伦神经优化理论及其在地震反演中的应用. 石油地球物理勘探. 1998,33(6):758~768
82 韩华, 章梓茂, 汪越胜. 双相介质波动方程孔隙率反演的同伦法. 力学学报. 2003, (2):235~239
83 张丽琴, 王家映, 严德天. 一维波动方程波阻抗反演的同伦方法. 地球物理学报. 2004,47(6):1111~1117
84 魏培君 , 章梓茂 . 弹性动力学反问题的数值反演方法 . 力学进展 . 2001,(2):172~180
85 T. King. Multilevel Algorithms for Ill-Posed Problems. Numer. Math. 1992,61(3):311~334
86 W. Hackbusch. Iterative Solution of Large Sparse Systems of Equation. Springer, 1994:1~460
87 C. Bunks, F. M. Saleck, S. Zaleski, G. Chavent. Multiscale Seismic Wave-Form Inversion. Geophysics. 1995,60(5):1457~1473
88 Y. M. Chen, F. G. Zhang. Hierarchical Multigrid Strategy for Efficiency Improvement of the GPST Inversion Algorithm. Applied Numerical Mathematics, 1990, 6: 431~446
89 陈维翰. 用多重网格法改进偏微分方程反问题的广义脉冲谱算法. 贵州师范大学学报. 1995,13(4): 1~6
90 钱建良,杨光,刘家琦. 二维弹性波动方程反问题的脉冲谱-多重网格迭代算法. 哈尔滨工业大学学报. 1996,28(1): 6~12
91 B. Kaltenbacher. On the Regularizing Properties of a Full Multigrid Method for Ill-Posed Problems. Inverse Problems. 2001,17(4):767~788
92 O. Scherzer. An Iterative Multi-Level Algorithm for Solving Nonlinear Ill-Posed Problems. Numer. Math. 1998, 80: 579~600
93 O. Seungseok, A. B. Milstein, C. A. Bouman, K. J. Webb. A General Framework for Nonlinear Multigrid Inversion. IEEE Trans. Image Process. 2005,14(1):125~140
94 U. M. Ascher, E. Haber. A Multigrid Method for Distributed Parameter Estimation Problems. Elec. Trans. Numer. Analy. 2003,15:1~17
95 E. Gelman, J. Mandel. On Multilevel Iterative Methods for Optimization Problems. Mathematical Programming. 1990, 48: 1~17
96 J. Liu. A Multiresolution Method for Distributed Parameter Estimation. SIAM J. Sci. Comput. 1993,14(2):389~405
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100 L. Y. Chiao, W. T. Liang. Multiresolution Parameterization for Geophysical Inverse Problems. Geophysics. 2003, 68(1): 199~209
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59 F. Kormendi, M. Dietrich. Nonlinear Waveform Inversion of Plane-Wave Seismograms in Stratified Elastic Media. Geophysics. 1991,56(5):664~674
60 H. Zhao, B. Ursin, L. Amundsen. Frequency-Wavenumber Elastic Inversion of Marine Seismic Data. Geophysics. 1994,59(12):1868~1881
61 R. G. Pratt, C. Shin, G. J. Hicks. Gauss-Newton and Full Newton Methods in Frequency-Space Seismic Waveform Inversion. J. Int. Geophys. 1998,133(2):341~362
62 D. S. Tsien, Y. M. Chen. A Numerical Method for Nonlinear Inverse Problems in Fluid Dynamics. Computational Methods in Nonlinear Mechanics, Proc. Inc. Conf. Comput. Mech. Nonliear Mechs. The University of Texas, Austin. 1974:935~945
63 Y. M. Chen. Generalized Pulse-Spectrum Technique. Geophysics. 1985,50: 1664~1675
64 王家映, 刘先洲. 地球物理反演理论. 中国地质大学出版社. 1998:1~181
65 N. M. Shapiro, M. H. Ritzwoller. Monte-Carlo Inversion for a Global Shearvelocity Model of the Crust and Upper Mantle. Geophys. J. Int. 2002, 151(1):88~105
66 Y. X. Pan, Q. F. Xu, H. M. Gao. Research of the Fuzzy Algorithm Optimization Based on Chaos. Control Theory and Applications. 2000, 17(5): 703~706
67 J. Caers. Automatic Histogram and Variogram Reproduction in SimulatedAnnealing Simulation. Math. Geol. 2001, 33(2): 167~190
68 P. J. Ballester, J. N. Carter. Real-Parameter Genetic Algorithms for Finding Multiple Optimal Solutions in Multi-Modal Optimization. Lecture Notes in Computer Science. 2003, 2723(0302-9743):706~717
69 王则柯, 高堂安. 同伦方法引论. 重庆出版社, 1990:1~287
70 L. T. Watson. Globally Convergent Homotopy Methods. Appl. Math. Comput. 1989,31(Spec. Issue):369~396
71 H. B. Keller, D. J. Perozzi. Fast Seismic Ray Tracing. SIAM J. Appl. Math.1983, 43(4): 981~992
72 D. W. Vasco. Singularity and Branching: A Path-Following Formalism for Geophysical Inverse Problems. Geophysics. J. Int. 1994,119:809~830
73 D. W. Vasco. Regularization and Trade-Off Associated with Nonlinear Geophysical Inverse Problems: Penalty Homotopies. Inverse Problems. 1998, 14(4):1033~1052
74 K. P. Bube, R. T. Langan. On a Continuation Approach to Regularization for Cross Well Tomography. 69th Ann. Internat. Mtg., Tulsa, 1999. Soc. Expl. Geophys. 1999:1295~1298
75 M. E. Everett. Homotopy, Polynomial Equations, Gross Boundary Data, and Small Helmholtz Systems. J. Comput. Phys. 1996,124(2):431~441
76 M. D. Jegen, M. E. Everett, A. Schultz. Using Homotopy to Invert Geophysical Data. Geophysics. 2001,66(6):1749~1760
77 韩波, 匡正, 刘家琦. 反演地层电阻率的单调同伦法. 地球物理学报. 1991, 34(4):256~261
78 B. Han, M. L. Zhang, J. Q. Liu. A Widely Convergent Generalized Pulse-Spectrum Technique for the Coefficient Inverse Problems of Differential Equations. Appl. Math. Comput. 1997,81(2):97~112
79 G. F. Feng, B. Han, J. Q. Liu. A Widely Convergent Generalized Pulse-Spectrum Method for 2-D Wave Equation Inversion. Chinese Journal of Geophysics. 2003,46(2):373~380
80 韩波, 付又和. 求解非线性不适定问题的正则化同伦方法. 黑龙江大学自然科学学报, 2005, 22(5): 659~663
81 刘舒考. 同伦神经优化理论及其在地震反演中的应用. 石油地球物理勘探. 1998,33(6):758~768
82 韩华, 章梓茂, 汪越胜. 双相介质波动方程孔隙率反演的同伦法. 力学学报. 2003, (2):235~239
83 张丽琴, 王家映, 严德天. 一维波动方程波阻抗反演的同伦方法. 地球物理学报. 2004,47(6):1111~1117
84 魏培君 , 章梓茂 . 弹性动力学反问题的数值反演方法 . 力学进展 . 2001,(2):172~180
85 T. King. Multilevel Algorithms for Ill-Posed Problems. Numer. Math. 1992,61(3):311~334
86 W. Hackbusch. Iterative Solution of Large Sparse Systems of Equation. Springer, 1994:1~460
87 C. Bunks, F. M. Saleck, S. Zaleski, G. Chavent. Multiscale Seismic Wave-Form Inversion. Geophysics. 1995,60(5):1457~1473
88 Y. M. Chen, F. G. Zhang. Hierarchical Multigrid Strategy for Efficiency Improvement of the GPST Inversion Algorithm. Applied Numerical Mathematics, 1990, 6: 431~446
89 陈维翰. 用多重网格法改进偏微分方程反问题的广义脉冲谱算法. 贵州师范大学学报. 1995,13(4): 1~6
90 钱建良,杨光,刘家琦. 二维弹性波动方程反问题的脉冲谱-多重网格迭代算法. 哈尔滨工业大学学报. 1996,28(1): 6~12
91 B. Kaltenbacher. On the Regularizing Properties of a Full Multigrid Method for Ill-Posed Problems. Inverse Problems. 2001,17(4):767~788
92 O. Scherzer. An Iterative Multi-Level Algorithm for Solving Nonlinear Ill-Posed Problems. Numer. Math. 1998, 80: 579~600
93 O. Seungseok, A. B. Milstein, C. A. Bouman, K. J. Webb. A General Framework for Nonlinear Multigrid Inversion. IEEE Trans. Image Process. 2005,14(1):125~140
94 U. M. Ascher, E. Haber. A Multigrid Method for Distributed Parameter Estimation Problems. Elec. Trans. Numer. Analy. 2003,15:1~17
95 E. Gelman, J. Mandel. On Multilevel Iterative Methods for Optimization Problems. Mathematical Programming. 1990, 48: 1~17
96 J. Liu. A Multiresolution Method for Distributed Parameter Estimation. SIAM J. Sci. Comput. 1993,14(2):389~405
97 J. Liu, B. Guerrier, C. Benard. A Sensitivity Decomposition for theRegularized Solution of Inverse Heat Conduction Problems by Wavelets. Inverse Problems. 1995, 11(6): 1177~1187
98 A. Rieder. A Wavelet Multilevel Method for Ill-Posed Problems Stabilized By Tikhonov Regularization. J. Numer. Math. 1997, 75(4): 501~522
99 R. Ghanem, F. Romeo. A Wavelet-Based Approach for Model and Parameter Identification of Nonlinear Systems. International Journal of Nonlinear Mechanics. 2001, 36(5): 835~859
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