# Regularization matrices for discrete ill-posed problems in several space-dimensions

## TL;DR

This paper explores the construction of regularization matrices for solving large, ill-conditioned linear systems from discretized integral equations in multiple dimensions, improving stability and accuracy in inverse problems.

## Contribution

It introduces a method to construct penalty matrices via matrix-nearness problems, facilitating better regularization in multi-dimensional ill-posed problems.

## Key findings

- Constructed penalty matrices improve regularization stability.
- Method applies to discretizations of integral equations in several dimensions.
- Enhances the effectiveness of Tikhonov regularization for complex problems.

## Abstract

Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix that defines these problems is very ill-conditioned and generally numerically singular, and the right-hand side, which represents measured data, typically is contaminated by measurement error. Straightforward solution of these problems generally is not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to the error in the right-hand side and to round-off errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix-nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space-dimensions.