On iterative methods for solving ill-posed problems modeled by PDE's

TL;DR

This paper analyzes iterative methods for solving ill-posed inverse PDE problems, providing convergence proofs, spectral analysis, and convergence rate estimates with noisy data considerations.

Contribution

It offers alternative convergence proofs using spectral theory and extends analysis to problems with noisy data, enhancing understanding of these iterative methods.

Findings

Convergence of the iterative methods is established using spectral theory.

The methods are shown to be non-expansive operators with specific functional properties.

Convergence rate estimates are derived for noisy data scenarios.

Abstract

We investigate the iterative methods proposed by Maz'ya and Kozlov (see [KM1], [KM2]) for solving ill-posed inverse problems modeled by partial differential equations. We consider linear evolutionary problems of elliptic, hyperbolic and parabolic types. Each iteration of the analyzed methods consists in the solution of a well posed problem (boundary value problem or initial value problem respectively). The iterations are described as powers of affine operators, as in [KM2]. We give alternative convergence proofs for the algorithms by using spectral theory and the fact that the linear parts of these affine operators are non-expansive with additional functional analytical properties (see [Le1,2]). Also problems with noisy data are considered and estimates for the convergence rate are obtained under a priori regularity assumptions on the problem data.