|
|
带扰动算子的Landweber迭代在Hanke-Raus准则下的收敛阶分析
|
Abstract:
本文针对带有扰动算子的非线性反问题提出了一种基于Hanke-Raus启发式停止准则的Landweber迭代法，并在一定的假设条件下分析了此迭代法的收敛阶。
In this paper, a Landweber iteration based on the Hanke Raus rule for nonlinear inverse problems with perturbed operators is proposed, and the convergence order of this method is analyzed under certain reasonable assumptions.
| [1] | 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. https://doi.org/10.1007/s002110050158 |
| [2] | Ramlau, R. (1999) A Modified Landweber Method for Inverse Problems. Numerical Functional Analysis and Optimization, 20, 79-98. https://doi.org/10.1080/01630569908816882 |
| [3] | Xu, J., Han, B. and Li, L. (2007) Frozen Landweber Iter-ation for Nonlinear Ill-Posed Problems. Acta Mathematicae Applicatae Sinica, 23, 329-336. https://doi.org/10.1007/s10255-007-0375-2 |
| [4] | Kaltenbacher, B., Neubauer, A. and Scherzer, O. (2008) Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Walter de Gruyter, Berlin. https://doi.org/10.1515/9783110208276 |
| [5] | Kaltenbacher, B., Schpfer, F. and Schuster, T. (2009) Iterative Meth-ods for Nonlinear Ill-Posed Problems in Banach Spaces: Convergence and Applications to Parameter Identification Prob-lems. Inverse Problems, 25, Article ID: 065003. https://doi.org/10.1088/0266-5611/25/6/065003 |
| [6] | Jin, Q. and Wang, W. (2013) Landweber Iteration of Kaczmarz Type with General Non-Smooth Convex Penalty Functionals. In-verse Problems, 29, 1400-1416. https://doi.org/10.1088/0266-5611/29/8/085011 |
| [7] | Jose, J. and Rajan, M. P. (2017) A Simplified Landweber Iteration for Solving Nonlinear Ill-Posed Problems. International Journal of Applied and Computational Mathematics, 3, 1001-1018. https://doi.org/10.1007/s40819-017-0395-4 |
| [8] | Hubmer, S. and Ramlau, R. (2017) Convergence Analysis of a Two-Point Gradient Method for Nonlinear Ill-Posed Problems. Inverse Problems, 33, Article ID: 095004. https://doi.org/10.1088/1361-6420/aa7ac7 |
| [9] | 陈宏, 侯宗义. 算子与右端都为近似的迭代正则化方法[J]. 中国科学(A辑数学物理学天文学技术科学), 1994, 24(8): 808-814. |
| [10] | Jin, Q. (2000) The Analysis of a Discrete Scheme of the Iteratively Regularized Gauss-Newton Method. Inverse Problems, 16, 1457-1476. https://doi.org/10.1088/0266-5611/16/5/319 |
| [11] | 韩波, 刘家琦, 后步风. 非线性不适定算子方程算子与右端项皆有扰动的Landweber迭代法[J]. 计算数学, 2002, 24(4): 479-486. |
| [12] | Werner, F. and Hohage, T. (2012) Convergence Rates in Expectation for Tikhonov-Type Regularization of Inverse Problems with Poisson Data. In-verse Problems, 28, Article ID: 104004.
https://doi.org/10.1088/0266-5611/28/10/104004 |
| [13] | Hohage, T. and Miller, P. (2019) Optimal Convergence Rates for Sparsity Promoting Wavelet-Regularization in Besov Spaces. Inverse Problems, 35, Article ID: 065005. https://doi.org/10.1088/1361-6420/ab0b15 |
| [14] | Weidling, F., Sprung, B. and Hohage, T. (2020) Optimal Conver-gence Rates for Tikhonov Regularization in Besov Spaces. SIAM Journal on Numerical Analysis, 58, 21-47. https://doi.org/10.1137/18M1178098 |
| [15] | Lu, S. and Flemming, J. (2012) Convergence Rate Analysis of Tikhonov Regularization for Nonlinear Ill-Posed Problems with Noisy Operators. Inverse Problems, 28, Article ID: 104003.
https://doi.org/10.1088/0266-5611/28/10/104003 |
| [16] | Burger, M., Korolev, Y. and Rasch, J. (2019) Convergence Rates and Structure of Solutions of Inverse Problems with Imperfect Forward Models. Inverse Problems, 35, Article ID: 024006. https://doi.org/10.1088/1361-6420/aaf6f5 |
| [17] | Hohage, T. and Werner, F. (2022) Error Estimates for Var-iational Regularization of Inverse Problems with General Noise Models for Data and Operator. Electronic Transactions on Numerical Analysis, 57, 127-152.
https://doi.org/10.1553/etna_vol57s127 |
| [18] | Hanke, M. and Raus, T. (1996) A General Heuristic for Choosing the Regularization Parameter in Ill-Posed Problems. SIAM Journal on Scientific Computing, 17, 956-972. https://doi.org/10.1137/0917062 |
| [19] | Jin, B. and Lorenz, D.A. (2011) Heuristic Parameter-Choice Rules for Con-vex Variational Regularization Based on Error Estimates. SIAM Journal on Numerical Analysis, 48, 1208-1229. https://doi.org/10.1137/100784369 |
| [20] | Jin, Q. (2016) Hanke-Raus Heuristic Rule for Variational Regularization in Banach Spaces. Inverse Problems, 32, Article ID: 085008. https://doi.org/10.1088/0266-5611/32/8/085008 |
| [21] | Zhang, Z. and Jin, Q. (2018) Heuristic Rule for Non-Stationary Iterated Tikhonov Regularization in Banach Spaces. Inverse Problems, 34, Article ID: 115002. https://doi.org/10.1088/1361-6420/aad918 |
| [22] | Jin, Q. and Wang, W. (2018) Analysis of the Iteratively Regular-ized Gauss-Newton Method under a Heuristic Rule. Inverse Problems, 34, Article ID: 035001. https://doi.org/10.1088/1361-6420/aaa0fb |
| [23] | Fu, Z., Jin, Q. Zhang, Z., Han, B. and Che, Y. (2020) Analysis of a Heuristic Rule for the IRGNM in Banach Spaces with Convex Regularization Terms. Inverse Problems, 36, Article ID: 075002.
https://doi.org/10.1088/1361-6420/ab8448 |
| [24] | Hubmer, S., Sherina, E., Kindermann, S., et al. (2022) A Numeri-cal Comparison of Some Heuristic Stopping Rules for Nonlinear Landweber Iteration. Electronic Transactions on Nu-merical Analysis, 57, 216-241.
https://doi.org/10.1553/etna_vol57s216 |
| [25] | Real, R.R. (2021) Convergence Results for Variational Regularization and Landweber Iteration under Heuristic Rules. Ph.D. Thesis, Australian National University, Canberra. |
