不适定问题的正则化方法综述
A Review of Regularization Methods for Ill-Posed Problems
摘要: 不适定问题或称反问题的研究从20世纪末成为国际上的热点问题，也是现代数学家广为关注的研究领域。随着生产和科学技术的发展，离散不适定问题在自动控制、图像处理、地球物理等诸多领域都有广泛的应用。而反问题求解面临的一个本质性困难是不适定性，求解不适定问题的普遍方法是：用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解，这种方法称为正则化方法。如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容。当前，最为流行的正则化方法是基于变分原理的Tikhonov正则化及其改进方法。
Abstract: The study of ill-posed problems or inverse problems has become a hot topic in the world since the end of the 20th century, and it is also a research field widely concerned by modern mathematicians. With the development of production and scientific technology, discrete ill-posed problems have been widely used in many fields, such as automatic control, image processing, geophysics, etc. An essential difficulty in solving the inverse problem is illness. The general method for solving ill-posed problems is to approximate the solution of the original problem with the solution of the well-posed problem “adjacent” to the original ill-posed problem. This method is called regularization method. How to establish an effective regularization method is an important content of ill-posed problems in the field of inverse problems. At present, the most popular regularization methods are Tikhonov regularization based on variational principle and its improved method.
文章引用：杨佳乐. 不适定问题的正则化方法综述[J]. 理论数学, 2025, 15(1): 382-390. https://doi.org/10.12677/pm.2025.151039

参考文献

[1] 黄光远, 刘小军. 数学物理反问题[M]. 济南: 山东科学技术出版社, 1993.
[2] 王彦飞. 反演问题的计算方法及其应用[M]. 北京: 高等教育出版社, 2007.
[3] 肖庭延, 于渗根, 王彦飞. 反问题数值解法[M]. 北京: 科学出版社, 2003.
[4] 刘继军. 不适定问题的正则化方法及应用[M]. 北京: 科学出版社, 2005.
[5] Keller, J.B. (1976) Inverse Problems. The American Mathematical Monthly, 83, 107-118. [Google Scholar] [CrossRef] 
[6] Hadamard, J. (1923) Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press.
[7] Hansen, P.C. (1990) The Discrete Picard Condition for Discrete Ill-Posed Problems. BIT Numerical Mathematics, 30, 658-672. [Google Scholar] [CrossRef] 
[8] Hansen, P.C. (1998) Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics. [Google Scholar] [CrossRef] 
[9] Tihonov, A.N. (1963) Solution of Incorrectly Formulated Problems and the Regularization Method. Doklady Akademii Nauk SSSR, 151, 501-504.
[10] Tikhonov, A.N. and Arsenin, V.Y. (1977) Solutions of Ill-Posed Problems. Winstion.
[11] Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th Edition, Johns Hopkins University Press.
[12] Hansen, P.C. (1990) Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank. SIAM Journal on Scientific and Statistical Computing, 11, 503-518. [Google Scholar] [CrossRef] 
[13] Hansen, P.C. (1992) Numerical Tools for Analysis and Solution of Fredholm Integral Equations of the First Kind. Inverse Problems, 8, 849-872. [Google Scholar] [CrossRef] 
[14] Rust, B.W. and Rust, B.W. (1998) Truncating the Singular Value Decomposition for Ill-Posed Problems. Report NISTIR 6131, Mathematical and Computational Sciences Division, NIST.
[15] Hansen, P.C. (2010) Discrete Inverse Problems: Insight and Algorithms. Society for Industrial and Applied Mathematics. [Google Scholar] [CrossRef] 
[16] Hansen, P.C. (1998) Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics: Numerical Aspects of Linear Inversion. [Google Scholar] [CrossRef] 
[17] 蔡大用. 数值代数[M]. 北京: 清华大学出版社, 1987.
[18] 李庆扬, 易大义, 王能超. 现代数值分析[M]. 北京: 高等教育出版社, 1995.
[19] Landweber, L. (1951) An Iteration Formula for Fredholm Integral Equations of the First Kind. American Journal of Mathematics, 73, 615-624. [Google Scholar] [CrossRef] 
[20] 冈萨雷斯. 数字图像处理[M]. 第2版. 阮秋琦, 译. 北京: 电子工业出版社, 2003: 1-24, 175-220.
[21] 余成波. 数字图像处理及MATLAB实现[M]. 重庆: 重庆大学出版社, 2003.
[22] 秦贝贝, 毛一敏, 王艳梅. MATLAB在数字图像处理中的应用[J]. 无线互联科技, 2018, 15(12): 135-136.
[23] Elfving, T., Hansen, P.C. and Nikazad, T. (2014) Semi-Convergence Properties of Kaczmarz’s Method. Inverse Problems, 30, Article ID: 055007. [Google Scholar] [CrossRef] 
[24] 戴华. 求解大规模矩阵问题的Krylov子空间方法[J]. 南京航空航天大学学报, 2001, 33(2): 139-145.
[25] Paige, C.C. and Saunders, M.A. (1982) LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. ACM Transactions on Mathematical Software, 8, 43-71. [Google Scholar] [CrossRef] 
[26] Björck, Å. (1988) A Bidiagonalization Algorithm for Solving Large and Sparse Ill-Posed Systems of Linear Equations. BIT, 28, 659-670. [Google Scholar] [CrossRef] 
[27] Saad, Y. (2003) Iterative Methods for Sparse Linear Systems. SIAM.
[28] Fischer, B., Hanke, M. and Hochbruck, M. (1996) A Note on Conjugate-Gradient Type Methods for Indefinite and/or Inconsistent Linear Systems. Numerical Algorithms, 11, 181-187. [Google Scholar] [CrossRef] 
[29] Hanke, M. (2001) On Lanczos Based Methods for the Regularization of Discrete Ill-Posed Problems. BIT Numerical Mathematics, 41, 1008-1018. [Google Scholar] [CrossRef] 
[30] Hanke, M. and Nagy, J.G. (1996) Restoration of Atmospherically Blurred Images by Symmetric Indefinite Conjugate Gradient Techniques. Inverse Problems, 12, 157-173. [Google Scholar] [CrossRef] 
[31] Dykes, L., Marcellán, F. and Reichel, L. (2014) The Structure of Iterative Methods for Symmetric Linear Discrete Ill-Posed Problems. BIT Numerical Mathematics, 54, 129-145. [Google Scholar] [CrossRef] 
[32] Huang, Y. and Jia, Z. (2017) On Regularizing Effects of MINRES and MR-II for Large Scale Symmetric Discrete Ill-Posed Problems. Journal of Computational and Applied Mathematics, 320, 145-163. [Google Scholar] [CrossRef] 
[33] Jia, Z. (2020) Approximation Accuracy of the Krylov Subspaces for Linear Discrete Ill-Posed Problems. Journal of Computational and Applied Mathematics, 374, 112786. [Google Scholar] [CrossRef] 
[34] Jia, Z. (2020) The Low Rank Approximations and Ritz Values in LSQR for Linear Discrete Ill-Posed Problem. Inverse Problems, 36, Article ID: 045013. [Google Scholar] [CrossRef] 
[35] Jia, Z. (2020) Regularization Properties of LSQR for Linear Discrete Ill-Posed Problems in the Multiple Singular Value Case and Best, near Best and General Low Rank Approximations. Inverse Problems, 36, Article ID: 085009. [Google Scholar] [CrossRef] 
[36] O’Leary, D.P. and Simmons, J.A. (1981) A Bidiagonalization-Regularization Procedure for Large Scale Discretizations of Ill-Posed Problems. SIAM Journal on Scientific and Statistical Computing, 2, 474-489. [Google Scholar] [CrossRef] 
[37] Renaut, R.A., Vatankhah, S. and Ardestani, V.E. (2017) Hybrid and Iteratively Reweighted Regularization by Unbiased Predictive Risk and Weighted GCV for Projected Systems. SIAM Journal on Scientific Computing, 39, B221-B243. [Google Scholar] [CrossRef] 
[38] Van Loan, C.F. (1976) Generalizing the Singular Value Decomposition. SIAM Journal on Numerical Analysis, 13, 76-83. [Google Scholar] [CrossRef] 
[39] Hansen, P.C. (1989) Regularization, GSVD and Trucated GSVD. BIT, 29, 491-504. [Google Scholar] [CrossRef] 
[40] Eldén, L. (1982) A Weighted Pseudoinverse, Generalized Singular Values, and Constrained Least Squares Problems. BIT, 22, 487-502. [Google Scholar] [CrossRef] 
[41] Chung, J. and Saibaba, A.K. (2017) Generalized Hybrid Iterative Methods for Large-Scale Bayesian Inverse Problems. SIAM Journal on Scientific Computing, 39, S24-S46. [Google Scholar] [CrossRef] 
[42] Zha, H. (1996) Computing the Generalized Singular Values/Vectors of Large Sparse or Structured Matrix Pairs. Numerische Mathematik, 72, 391-417. [Google Scholar] [CrossRef] 
[43] Kilmer, M.E., Hansen, P.C. and Español, M.I. (2007) A Projection‐Based Approach to General‐Form Tikhonov Regularization. SIAM Journal on Scientific Computing, 29, 315-330. [Google Scholar] [CrossRef] 
[44] Jia, Z. and Yang, Y. (2020) A Joint Bidiagonalization Based Iterative Algorithm for Large Scale General-Form Tikhonov Regularization. Applied Numerical Mathematics, 157, 159-177. [Google Scholar] [CrossRef] 

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