离散不适定问题的扩展迭代正则化方法

The Enriched Iterative Regularization Method for Discrete Ill-Posed Problems

摘要: Arnoldi-Tikhonov方法是求解大规模离散不适定问题的一种常用子空间迭代正则化方法，其由Arnoldi算法构建低维Krylov子空间，再对低维问题用Tikhonov正则化从而获得正则化解。但由于低维子空间信息缺失，正则化解的有时效果欠佳。为了改进正则化效果，本文通过增加一个含有特定先验信息的低维子空间来扩展Krylov子空间，提出了求解大规模离散不适定问题的一种扩展子空间迭代正则化方法。该方法通过扩展Arnoldi算法构建扩展子空间，并融合Tikhonov正则化，从而获得更优正则化解。针对经典算例，将所提算法与Arnoldi-Tikhonov算法进行了数值实验和性态比较，数值结果验证了所提算法的有效性。

Abstract: Arnoldi-Tikhonov method is one of the often used Krylov subspace iterative regularization methods for solving large scale discrete ill-posed problems. It generates the Krylov subspace through Arnoldi process and gets the regularized solution through applying Tikhonov regularization method to the projected small problem. However, due to the information deficiency of the dimension reduced subspace, the regularized solution by Arnoldi-Tikhonv method sometimes is not as good as expected. In order to improve the regularized solution, an enriched subspace iterative regularization method is proposed in this paper. The proposed new method enriches the Krylov subspace by adding a special subspace that holds some specific prior information. Numerical experiments are carried out. The numerical results show that the proposed method is more effective than the Arnoldi-Tikhonov method.