On regularizing effects of MINRES and MR-II for large scale symmetric discrete ill-posed problems
On regularizing effects of MINRES and MR-II for large scale symmetric discrete ill-posed problems
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MINRES 和 MR-II 对大规模对称离散不适定问题的正则化效果
DOI：
10.1016/j.cam.2017.02.008
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发表时间：
2015-03
影响因子：
2.4
通讯作者：
Jia Zhongxiao
中科院分区：
文献类型：
--
作者：
Huang Yi;Jia Zhongxiao
对于大规模对称离散不适定问题，MinRes和MR-II是常用的迭代正则化求解器。如果一个正则解至少与截断奇异值分解(TSVD)方法得到的最佳正则解一样精确，我们称它为最佳可能解。本文分析了它们的正则化效应，得到了如下结果：(I)得到了MinRes正则解的过滤奇异值分解表达式；(Ii)在投影问题中使用显式正则化的混合MinRes需要计算给定病态问题的最佳可能正则解；(Iii)在MinRes半收敛之前，MinRes的第k次迭代比MR-II迭代的第(k−1)次更精确，但MR-II的正则化效果总体上好于MinRes；(4)得到了k维Krylov子空间与k维主特征空间之间的2-范数距离的界。结果表明，对于严重和中度不适定问题，MR-II比对于轻度不适定问题具有更好的正则化效果，并且需要一个混合的MR-II来获得轻度不适定问题的最佳可能正则解；(V)推导了MR-II所基于的对称Lanczos过程生成的条目的界，表明它们的衰减速度有多快。数值实验证实了我们的断言。实验表明，MR-II的正则化效果比我们的理论更强，足以获得严重和中度不适定问题的最好可能的正则化解。
For large scale symmetric discrete ill-posed problems, MINRES and MR-II are often used iterative regularization solvers. We call a regularized solution best possible if it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition (TSVD) method. In this paper, we analyze their regularizing effects and establish the following results:(i) the filtered SVD expression are derived for the regularized solutions by MINRES;(ii) a hybrid MINRES that uses explicit regularization within projected problems is needed to compute a best possible regularized solution to a given ill-posed problem;(iii) the k th iterate by MINRES is more accurate than the (k− 1) th iterate by MR-II until the semi-convergence of MINRES, but MR-II has globally better regularizing effects than MINRES;(iv) bounds are obtained for the 2-norm distance between an underlying k-dimensional Krylov subspace and the k-dimensional dominant eigenspace. They show that MR-II has better regularizing effects for severely and moderately ill-posed problems than for mildly ill-posed problems, and a hybrid MR-II is needed to get a best possible regularized solution for mildly ill-posed problems;(v) bounds are derived for the entries generated by the symmetric Lanczos process that MR-II is based on, showing how fast they decay. Numerical experiments confirm our assertions. Stronger than our theory, the regularizing effects of MR-II are experimentally shown to be good enough to obtain best possible regularized solutions for severely and moderately ill-posed problems.
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DOI：
10.5860/choice.34-1602
发表时间：
1996
期刊：
--
影响因子：
--
作者：
J. Navarro-Pedreño
通讯作者：
J. Navarro-Pedreño
影响因子：
14.3
作者：
G. Stewart
通讯作者：
G. Stewart
影响因子：
1.5
作者：
Toke Koldborg Jensen;P. Hansen
通讯作者：
Toke Koldborg Jensen;P. Hansen
DOI：
10.1137/s0895479898348623
发表时间：
1999-10
期刊：
SIAM J. Matrix Anal. Appl.
影响因子：
--
作者：
M. Kilmer;G. Stewart
通讯作者：
M. Kilmer;G. Stewart
影响因子：
2.1
作者：
M. Hanke;J. Nagy
通讯作者：
M. Hanke;J. Nagy