Some matrix nearness problems suggested by Tikhonov regularization

TL;DR

This paper explores matrix nearness problems related to Tikhonov regularization, leading to new methods that improve the quality of solutions for ill-posed linear problems compared to traditional approaches.

Contribution

It introduces novel regularization methods derived from matrix nearness problems that outperform standard Tikhonov and TSVD techniques.

Findings

New regularization methods with higher solution quality

Methods combine properties of Tikhonov and TSVD

Improved solutions for ill-posed problems

Abstract

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods.