On inverse problems modeled by PDE's

TL;DR

This paper analyzes iterative methods for solving ill-posed PDE-based reconstruction problems, providing alternative convergence proofs using spectral theory across elliptic, hyperbolic, and parabolic PDEs.

Contribution

It offers new convergence proofs for existing iterative algorithms applied to PDE-based inverse problems, expanding understanding of their theoretical foundations.

Findings

Convergence of the iterative methods is established using spectral theory.

The methods are applicable to elliptic, hyperbolic, and parabolic PDEs.

Alternative proofs enhance the theoretical understanding of these algorithms.

Abstract

We investigate the iterative methods proposed by Maz'ya and Kozlov (see [3], [4]) for solving ill-posed reconstruction problems modeled by PDE's. We consider linear time dependent problems of elliptic, hyperbolic and parabolic types. Each iteration of the analyzed methods consists on the solution of a well posed boundary (or initial) value problem. The iterations are described as powers of affine operators, as in [4]. We give alternative convergence proofs for the algorithms, using spectral theory and some functional analytical results (see [5], [6]).