反问题中正则化方法的某些研究
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摘要
反问题是一类由效果表现反求原因原象的数学物理问题。此类问题不仅有着广泛而重要的应用背景,而且其理论还具有鲜明的新颖性与挑战性。迄今,它已发展成为计算数学、应用数学和系统科学等学科交叉的一个热门学科方向。求解数学物理反问题面临的两个本质性的实际困难是:①原始数据可能不属于所讨论问题精确解所对应的数据集合;②近似解的不稳定性,即:原始资料的小的观测误差会导致近似解与真解的严重偏离。因此,求解数学物理反问题常常是不适定的。非线性不适定问题相对线性不适定问题更为复杂,正则化方法是解决这一不适定问题的一类有效的方法。各种选取正则化参数的方法会带来不同的收敛速度,此方面的研究吸引了大量的学者的工作,现已有较多实用的结论。
基于[6]中的正则化同伦连续方法计算全局极小值的优点,本文在该文结果的基础上引入Morozov偏差原理计算迭代过程中的正则化参数,提出了带有后验参数的正则化同伦连续方法。同时在文中考虑了多实验数据下远场测量的反散射问题的计算,并在数值实验中分析和总结了不同扰动下每步迭代的正则化参数,说明了它的选取符合Tikhonov正则化理论。后验选取正则化参数的方法可以克服按经验选取参数的缺点。本文的数值例子表明了此方法在解决带有扰动的多试验数据的反问题时的实用性。
As an important class of mathematical physical problems, inverse problems have developed into a popular research direction. Solving an inverse problem is to determine unknown causes based on observation of their effects. Nowadays, inverse problems have been used in many fields,such as inverse medium scattering, computerized tomography, etc, and the theory and methods are novel and challenging.Two essential difficulties appear frequently. One is the observation data possibly does not belong to the corresponding set to the exact solution, another is that the approximation is not stable. Thus inverse problems are often ill-posed, and most are non-linear. Regularization technique is an effective method of solving ill-posed inverse problems. Since regularized parameter influence the convergent rate, which becomes more and more important, and there are several significant results gotten by many researchers.Based on the merit of computing global minimum with homotopy continuation method in [6], a posterior regularized homotopy continuation method is presented in this thesis. Regularized parameter in the iteration is decided by Morozov discrepancy principle. We realized the computation in inverse medium scattering problems with multi-experimental data of far field pattern. Moreover, the analytical result of that the selected regularized parameter in each iteration accord with Tikhonov regularization theory from a few of numerical experiments is stated. The method of posterior regularized parameter can overcome the shortcomings of prior choose, and the numerical examples show the practicability of our new methods in inverse problems with perturbation data.
基于[6]中的正则化同伦连续方法计算全局极小值的优点,本文在该文结果的基础上引入Morozov偏差原理计算迭代过程中的正则化参数,提出了带有后验参数的正则化同伦连续方法。同时在文中考虑了多实验数据下远场测量的反散射问题的计算,并在数值实验中分析和总结了不同扰动下每步迭代的正则化参数,说明了它的选取符合Tikhonov正则化理论。后验选取正则化参数的方法可以克服按经验选取参数的缺点。本文的数值例子表明了此方法在解决带有扰动的多试验数据的反问题时的实用性。
As an important class of mathematical physical problems, inverse problems have developed into a popular research direction. Solving an inverse problem is to determine unknown causes based on observation of their effects. Nowadays, inverse problems have been used in many fields,such as inverse medium scattering, computerized tomography, etc, and the theory and methods are novel and challenging.Two essential difficulties appear frequently. One is the observation data possibly does not belong to the corresponding set to the exact solution, another is that the approximation is not stable. Thus inverse problems are often ill-posed, and most are non-linear. Regularization technique is an effective method of solving ill-posed inverse problems. Since regularized parameter influence the convergent rate, which becomes more and more important, and there are several significant results gotten by many researchers.Based on the merit of computing global minimum with homotopy continuation method in [6], a posterior regularized homotopy continuation method is presented in this thesis. Regularized parameter in the iteration is decided by Morozov discrepancy principle. We realized the computation in inverse medium scattering problems with multi-experimental data of far field pattern. Moreover, the analytical result of that the selected regularized parameter in each iteration accord with Tikhonov regularization theory from a few of numerical experiments is stated. The method of posterior regularized parameter can overcome the shortcomings of prior choose, and the numerical examples show the practicability of our new methods in inverse problems with perturbation data.
引文
[1] Cheng J, Liu J J, Nakamura G. Recovery of boundaries and types for multiple obsta cles from the far-field pattern. Submit to Quart. Applied Maths.
[2] Cheng J, Liu J J, Nakamura G. Recovery of the shape of an obstacle and the boundary impedance from the far-field pattern. J.of Maths of Kyoto University. 43(i):165-186, 2003.
[3] Colton D L, Kress R. Integral Equation Methods in Scattering Theory. New York:John Willey Sons, Inc.,1983.
[4] Defrise M, De Mol C. A Note on Stopping rules for Iterative Regularization Methods and Filtered SVD, in:P C Sabatier, ed: Inverse Problems: An Interdisciplinary Study, Academic Press, London, Orlando, San Diego, New York, 261-268. 1987
[5] Engl H W: Discrepancy Principles for Tikhonov Regularzation of Ill-Posed Problems Leading to Optimal Convergence Rates, J.Optim.Theory.Appl., 52: 209-215. 1987.
[6] Gang Bao, Jun Liu. Numerical solution of inverse scattering problems with multiexperimental limited aperture data. SIAM J.SCI.COMPUT, 2003, 25:1102-1117.
[7] Gang Bao, Peijun Li. Inverse medium scattering for three-dimensional time harmonic Maxwell equations. Inverse Problems, 2004, 20:LI-LT.
[8] Gang Bao, Yu Chen, Fuming Ma. Regularity and stability for the scattering map of a linearized inverse medium problem. J.Math.Anal.Appl., 2000, 247:255-271.
[9] Gennadi Vainikko. Fast solvers of the Lippmann-Schwinger equation. Direct and inverse problems of mathematical physics. Dordrecht: Kluwer Acad.Publ, 1997.
[10] Gfrerer H. An A-Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates,Mathematics of Computation 49(180), 507-522. 1987.
[11] Habib Ammari, Gang Bao. Analysis of the scattering map of alinearized inverse medium problem for electromagnetic waves. Inverse Problem, 2000, 17: 219-234.
[12] Juan C.Aguilar, Yu Chen. High-Order Corrected Trapezoidal Quadrature Rules for Functions with a Logarithmic Singularity in 2-D. Computers and Mathematics with Applications, 2002, vol.44, No.8-9:1031-1039.
[13] Keller J B. Inverse Problems. American Mathematics Monthly, 1976, 83: 107-118.
[14] Kirsh A. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996.
[15] Kress R. Linear Integral Equations, Springer, Berlin. 1989.
[16] Levenberg K. A Method for the Solution of Certain Nonlinear Problems in Least Squares, Qart.Appl.Math.2, 164-166. 1944.
[17] Marquardt D W. An Algorithm for Least-Square Estimation of Nonlinear Inequalities, SIAM J.Appl.Math. 11, 431-441. 1963.
[18] M.Hanke. A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems, (13): 79-95, 1997.
[19] M.Hanke. Regularizing properties of a truncated Newton-CG algorithm for nonlinear ill-posed problems. Num.Funct.Anal.Optim.,(18): 971-933
[20] M.Hanke, A. Neubauer and O. Scherzer. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numerical Mathematic, (72): 21-37, 1995.
[21] MoreJ J. The Levenberg-Marquardt Algorithm :Implementation and Theory, In Watson G A ed., Lecture Notes in Mathematics 630: Numerical Analysis, Springer-Verlag, Berlin, 105-116. 1978.
[22] Morozov V A. On Regularization of Ill-Posed Problems and selection of Regularization Parameter. J.Comp.Math.Math.Phys., 6(1), 170-175. 1996.
[23] Ortega M, Rhienboldt. Iterative Solution of Nonlinear Equations in Several Variables, Acad.Press, N.Y.London. 1975.
[24] Phillips D L.A Technique for the numerical solution certain integral equations of the first kind,J.Assoc.Comput.Mach., 9: 84-97. 1962.
[25] Plato R. Optimal Algorithms for Linear Ill-Posed Problems Yield Regularization Methods, Numer.Funct.Anal. Optim.11, 111-118. 1990.
[26] Potthast,R. Point sources and multipoles in inverse scattering theroy. Chapman Hall. 2001.
[27] S.N.Chow, J.Mallet-Paret, J.A.Yorke. Finding zeros of maps:Homotopy methods that are constructive with probability one. Math.Comp., 1978, 32: 887-899.
[28] Tikhonov A N. On solving incorrectly posed problems and method of regularization. Dokl.Acad.Nauk USSR,151(3), 1963.
[29] T.Y.Li, Z.Zeng. Homotopy-determinant algorithm solving nonsymmetric eigenvalue problems. Math.Comp., 1992, 59: 381-403.
[30] Wang Yanfei. Iterative Choices of Regularization Parameter with Perturbed Operator and Noisy Data, Report, AMSS, Chinese Academy of Sciences, 2001.
[31] Wang Yanfei. On the Regularity of Trust Region-CG Algorithms:with Application to Image Deconvolution Problem, Science in China 46, 312-325. 2003.
[32] Wang Yanfei, Yuan Yaxiang. On the Pegularity of Trust Region-CG Algorithm for Nonlinear Ill-Posed problems, Proceedings of the Third Asian Mathematical Conference 2000, World Science Publishing Co. Pte Ltd. 562-580, 2002.
[33] Wang Yanfei, Yuan Yaxiang. A Trust Region Algorithm for Steady State Parameter Identification Problem, J.Comput.Math., vol.21(6), 759-772, 2003.
[34] Wang Yanfei, Yuan Yaxiang. On the Regularity of Trust Region Algorithms for Nonlinear Ill-Posed Irlverse Problems. short communications at the International Congress of Mathematicians, China, August 22, 2002.
[35] 肖庭延,于慎根,王彦飞.反问题的数值解法.北京:科学出版社,2003.
[36] Y.Saad, M.H.Schultz. GMRES:A generalized minimal residual algorithm for solving nonsymmetric linear systerms. SIAM J.Sci.Statist.Comput.,1986,7:856-869.
[37] 袁亚湘,文瑜.最优化理论与方法.北京:科学出版社,1997.
[2] Cheng J, Liu J J, Nakamura G. Recovery of the shape of an obstacle and the boundary impedance from the far-field pattern. J.of Maths of Kyoto University. 43(i):165-186, 2003.
[3] Colton D L, Kress R. Integral Equation Methods in Scattering Theory. New York:John Willey Sons, Inc.,1983.
[4] Defrise M, De Mol C. A Note on Stopping rules for Iterative Regularization Methods and Filtered SVD, in:P C Sabatier, ed: Inverse Problems: An Interdisciplinary Study, Academic Press, London, Orlando, San Diego, New York, 261-268. 1987
[5] Engl H W: Discrepancy Principles for Tikhonov Regularzation of Ill-Posed Problems Leading to Optimal Convergence Rates, J.Optim.Theory.Appl., 52: 209-215. 1987.
[6] Gang Bao, Jun Liu. Numerical solution of inverse scattering problems with multiexperimental limited aperture data. SIAM J.SCI.COMPUT, 2003, 25:1102-1117.
[7] Gang Bao, Peijun Li. Inverse medium scattering for three-dimensional time harmonic Maxwell equations. Inverse Problems, 2004, 20:LI-LT.
[8] Gang Bao, Yu Chen, Fuming Ma. Regularity and stability for the scattering map of a linearized inverse medium problem. J.Math.Anal.Appl., 2000, 247:255-271.
[9] Gennadi Vainikko. Fast solvers of the Lippmann-Schwinger equation. Direct and inverse problems of mathematical physics. Dordrecht: Kluwer Acad.Publ, 1997.
[10] Gfrerer H. An A-Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates,Mathematics of Computation 49(180), 507-522. 1987.
[11] Habib Ammari, Gang Bao. Analysis of the scattering map of alinearized inverse medium problem for electromagnetic waves. Inverse Problem, 2000, 17: 219-234.
[12] Juan C.Aguilar, Yu Chen. High-Order Corrected Trapezoidal Quadrature Rules for Functions with a Logarithmic Singularity in 2-D. Computers and Mathematics with Applications, 2002, vol.44, No.8-9:1031-1039.
[13] Keller J B. Inverse Problems. American Mathematics Monthly, 1976, 83: 107-118.
[14] Kirsh A. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996.
[15] Kress R. Linear Integral Equations, Springer, Berlin. 1989.
[16] Levenberg K. A Method for the Solution of Certain Nonlinear Problems in Least Squares, Qart.Appl.Math.2, 164-166. 1944.
[17] Marquardt D W. An Algorithm for Least-Square Estimation of Nonlinear Inequalities, SIAM J.Appl.Math. 11, 431-441. 1963.
[18] M.Hanke. A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems, (13): 79-95, 1997.
[19] M.Hanke. Regularizing properties of a truncated Newton-CG algorithm for nonlinear ill-posed problems. Num.Funct.Anal.Optim.,(18): 971-933
[20] M.Hanke, A. Neubauer and O. Scherzer. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numerical Mathematic, (72): 21-37, 1995.
[21] MoreJ J. The Levenberg-Marquardt Algorithm :Implementation and Theory, In Watson G A ed., Lecture Notes in Mathematics 630: Numerical Analysis, Springer-Verlag, Berlin, 105-116. 1978.
[22] Morozov V A. On Regularization of Ill-Posed Problems and selection of Regularization Parameter. J.Comp.Math.Math.Phys., 6(1), 170-175. 1996.
[23] Ortega M, Rhienboldt. Iterative Solution of Nonlinear Equations in Several Variables, Acad.Press, N.Y.London. 1975.
[24] Phillips D L.A Technique for the numerical solution certain integral equations of the first kind,J.Assoc.Comput.Mach., 9: 84-97. 1962.
[25] Plato R. Optimal Algorithms for Linear Ill-Posed Problems Yield Regularization Methods, Numer.Funct.Anal. Optim.11, 111-118. 1990.
[26] Potthast,R. Point sources and multipoles in inverse scattering theroy. Chapman Hall. 2001.
[27] S.N.Chow, J.Mallet-Paret, J.A.Yorke. Finding zeros of maps:Homotopy methods that are constructive with probability one. Math.Comp., 1978, 32: 887-899.
[28] Tikhonov A N. On solving incorrectly posed problems and method of regularization. Dokl.Acad.Nauk USSR,151(3), 1963.
[29] T.Y.Li, Z.Zeng. Homotopy-determinant algorithm solving nonsymmetric eigenvalue problems. Math.Comp., 1992, 59: 381-403.
[30] Wang Yanfei. Iterative Choices of Regularization Parameter with Perturbed Operator and Noisy Data, Report, AMSS, Chinese Academy of Sciences, 2001.
[31] Wang Yanfei. On the Regularity of Trust Region-CG Algorithms:with Application to Image Deconvolution Problem, Science in China 46, 312-325. 2003.
[32] Wang Yanfei, Yuan Yaxiang. On the Pegularity of Trust Region-CG Algorithm for Nonlinear Ill-Posed problems, Proceedings of the Third Asian Mathematical Conference 2000, World Science Publishing Co. Pte Ltd. 562-580, 2002.
[33] Wang Yanfei, Yuan Yaxiang. A Trust Region Algorithm for Steady State Parameter Identification Problem, J.Comput.Math., vol.21(6), 759-772, 2003.
[34] Wang Yanfei, Yuan Yaxiang. On the Regularity of Trust Region Algorithms for Nonlinear Ill-Posed Irlverse Problems. short communications at the International Congress of Mathematicians, China, August 22, 2002.
[35] 肖庭延,于慎根,王彦飞.反问题的数值解法.北京:科学出版社,2003.
[36] Y.Saad, M.H.Schultz. GMRES:A generalized minimal residual algorithm for solving nonsymmetric linear systerms. SIAM J.Sci.Statist.Comput.,1986,7:856-869.
[37] 袁亚湘,文瑜.最优化理论与方法.北京:科学出版社,1997.