基于自适应BB步长的修正Landweber迭代算法及收敛性分析
Modified Landweber Iterative Algorithm Based on Adaptive BB Step Size and Its Convergence Analysis
DOI: 10.12677/ORF.2022.123088, PDF,   
作者: 邹江艳, 何清龙：贵州大学，数学与统计学院，贵州 贵阳
关键词: 不适定问题；Landweber；正则化；ABB步长；Ill-Posed Problem； Landweber； Regularization； ABB Step
摘要: Landweber迭代法是求解数学物理反问题的一种重要方法，其具有良好的稳定性。然而，Landweber迭代正则化方法收敛速度相对较慢，限制了其在现实问题中的广泛应用。本文基于Landweber迭代正则化方法，引入自适应Barzilai-Borwein (ABB)步长加速技巧，提出了加速的Landweber迭代正则化方法，并给出了方法的收敛性分析。基于椭圆方程约束的参数识别问题，从数值计算角度验证了方法的有效性。实验结果表明相对于Landweber迭代方法，本文提出的Landweber-ABB迭代正则化方法在收敛速度方面具有较好的优势。
Abstract: Landweber iterative method is an important method for solving inverse problems in mathematical physics, which has good stability. However, the convergence speed of Landweber iterative regularization method is relatively slow, which limits its wide application in practical problems. Based on the Landweber iterative regularization method, this paper proposes an accelerated Landweber iterative regularization method by introducing adaptive Barzilai-Borwein (ABB) step acceleration technique and gives the convergence analysis of the method. Based on the parameter identification problem constrained by elliptic equation, the effectiveness of the method is verified from the perspective of numerical calculation. The experimental results demonstrate that the Landweber-ABB iterative regularization method proposed in this paper has better convergence speed than the Landweber iterative method.
文章引用：邹江艳, 何清龙. 基于自适应BB步长的修正Landweber迭代算法及收敛性分析[J]. 运筹与模糊学, 2022, 12(3): 832-842. https://doi.org/10.12677/ORF.2022.123088

参考文献

[1] 韩波, 李莉. 非线性不适定问题的求解方法及其应用[M]. 北京: 科学出版社, 2011: 25-33.
[2] Landweber, L. (1951) An Iteration Formula for Fredholm Integral Equations of the First Kind. American Journal of Mathematics, 73, 615-624. [Google Scholar] [CrossRef] 
[3] Hanke, M., Neubauer, A. and Scherzer, O. (1995) A Convergence Analysis of the Landweber Iteration for Nonlinear Ill-Posed Problems. Numerische Mathematik, 72, 21-37. [Google Scholar] [CrossRef] 
[4] Hubmer, S. and Ramlau, R. (2018) Nesterov’s Accelerated Gradient Method for Nonlinear Ill-Posed Problems with a Locally Convex Residual Functional. Inverse Problems, 34, Article ID: 095003. [Google Scholar] [CrossRef] 
[5] Gao, G., Han, B. and Tong, S. (2021) A Projective Two-Point Gradient Kaczmarz Iteration for Nonlinear Ill-Posed Problems. Inverse Problems, 37, Article ID: 075007. [Google Scholar] [CrossRef] 
[6] Xia, Y., Han, B. and Fu, Z. (2021) An Accelerated Homotopy Perturbation Iteration for Nonlinear Ill-Posed Problems in Banach Spaces with Uniformly Convex Penalty. Inverse Problems, 37, Article ID: 105003. [Google Scholar] [CrossRef] 
[7] Barzilai, J. and Borwein, J.M. (1988) Two-Point Step Size Gradient Methods. IMA Journal of Numerical Analysis, 8, 141-148. [Google Scholar] [CrossRef] 
[8] Dai, Y. and Yuan, Y.X. (2005) Analysis of Monotone Gradient Methods. Journal of Industrial & Management Optimization, 1, 181-192. [Google Scholar] [CrossRef] 
[9] Yuan, Y. (2006) A New Stepsize for the Steepest Descent Method. Journal of Computational Mathematics, 24, 149-156.
[10] Dai, Y.H. and Yuan, Y.X. (2003) Alternate Minimization Gradient Method. IMA Journal of Numerical Analysis, 23, 377-393. [Google Scholar] [CrossRef] 
[11] Dai, Y.H. and Yang, X.Q. (2006) A New Gradient Method with an Optimal Stepsize Property. Computational Optimization and Applications, 33, 73-88. [Google Scholar] [CrossRef] 
[12] Dai, Y.H. (2003) Alternate Step Gradient Method. Optimization, 52, 395-415. [Google Scholar] [CrossRef] 
[13] Zhou, B., Gao, L. and Dai, Y.H. (2006) Gradient Methods with Adaptive Step-Sizes. Computational Optimization and Applications, 35, 69-86. [Google Scholar] [CrossRef] 
[14] 王彦飞, 樊树芳, 汪丽华, Leonov, A.S., Lukyanenko, D.V, Yagola, A.G, 等. 页岩微米孔隙结构重构的非光滑正则化及快速梯度优化算法[J]. 地球物理学报, 2020, 63(5): 2036-2042.
[15] Otmar, S. (1996) A Convergence Analysis of a Method of Steepest Descent and a Two-Step Algorothm for Nonlinear Ill-Posed Problems: A Two-Step Algorothm for Nonlinear Ill-Posed Problems. Numerical Functional Analysis and Optimization, 17, 197-214. [Google Scholar] [CrossRef] 
[16] 王薇. 非线性不适定算子方程的Runge-Kutta型Landweber迭代法研究[D]: [硕士学位论文]. 哈尔滨: 哈尔滨工业大学, 2007.
[17] Kaltenbacher, B., Neubauer, A. and Scherzer, O. (2008) Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter, Berlin, New York. [Google Scholar] [CrossRef] 
[18] Zhu, M. and Chan, T. (2008) An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration. CAM Report 8-34. University of California, Los Angeles.
[19] Jin, Q. and Wang, W. (2013) Landweber Iteration of Kaczmarz Type with General Non-Smooth Convex Penalty Functional. Inverse Problems, 29, Article ID: 085011. [Google Scholar] [CrossRef] 

为你推荐