Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations

TL;DR

This paper analyzes how discretization errors affect solutions of ill-posed operator equations and proposes a combined approach using Arnoldi process and Tikhonov regularization with error bounds.

Contribution

It introduces an error estimation framework for Arnoldi-Tikhonov regularization applied to discretized ill-posed problems, emphasizing discretization impact.

Findings

Error bounds for discretization and regularization errors

Effectiveness of reduced systems via Arnoldi process

Validation through computed examples

Abstract

Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.