Regularization by Discretization in Banach Spaces

TL;DR

This paper extends projection methods for solving ill-posed linear equations from Hilbert to Banach spaces, analyzing their convergence and providing numerical tests for a Volterra integral equation.

Contribution

It introduces and analyzes projection, least squares, and least error methods in Banach spaces, including dimension choice strategies and numerical validation.

Findings

Methods are effective for Banach space problems.

Discrepancy principle and monotone error rule guide dimension selection.

Numerical tests confirm theoretical results.

Abstract

We consider ill-posed linear operator equations with operators acting between Banach spaces. For solution approximation, the methods of choice here are projection methods onto finite dimensional subspaces, thus extending existing results from Hilbert space settings. More precisely, general projection methods, the least squares method and the least error method are analyzed. In order to appropriately choose the dimension of the subspace, we consider a priori and a posteriori choices by the discrepancy principle and by the monotone error rule. Analytical considerations and numerical tests are provided for a collocation method applied to a Volterra integral equation in one space dimension.