Regularization matrices determined by matrix nearness problems

TL;DR

This paper introduces a novel method for selecting regularization matrices in Tikhonov regularization by solving matrix nearness problems, improving the recovery of important features in solutions to large-scale ill-posed problems.

Contribution

It proposes a new approach to determine regularization matrices with desired properties by solving Frobenius norm matrix nearness problems, enhancing solution quality.

Findings

Regularization matrices improve feature recovery in ill-posed problems

Numerical examples demonstrate the effectiveness of the proposed matrices

The method yields matrices with prescribed null spaces close to initial matrices

Abstract

This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained.