Adaptive cross approximation for Tikhonov regularization in general form

TL;DR

This paper presents an adaptive cross approximation method for efficiently handling large, ill-conditioned matrices from integral equations, combined with Tikhonov regularization to improve solution stability and accuracy.

Contribution

It introduces an adaptive cross approximation approach tailored for regularized solutions of integral equations, avoiding full matrix formation and enhancing solution quality.

Findings

Adaptive cross approximation effectively approximates large matrices.

Using a general regularization matrix improves solution quality.

Numerical examples demonstrate significant accuracy improvements.

Abstract

Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful because the matrix determined by adaptive cross approximation is rank-deficient. We remedy this difficulty by using Tikhonov regularization and discuss how a fairly general regularization matrix can be used. Computed examples illustrate that the use of a regularization matrix different from the identity can improve the quality of the computed approximate solutions significantly.