A probabilistic approach to spectral analysis of Cauchy-type inverse problems: Convergence and stability analysis

TL;DR

This paper provides a detailed convergence and stability analysis of probabilistic numerical methods for solving severely ill-posed Cauchy-type inverse problems involving elliptic PDEs, including explicit error bounds and theoretical guarantees.

Contribution

It introduces a comprehensive probabilistic error analysis and convergence proof for spectral methods applied to Cauchy inverse problems, expanding the theoretical understanding of these techniques.

Findings

Proved convergence of probabilistic spectral methods.

Derived explicit error bounds for the approximations.

Established stability results for the inverse problem solutions.

Abstract

A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential equation (PDE) or a system of elliptic PDEs in a bounded Euclidean domain, subject to incomplete boundary and/or internal conditions, and are usually severely ill-posed. In a very recent paper \cite{CiGrMaI}, a probabilistic numerical framework has been developed by the authors, wherein such inverse problems could be analysed thoroughly by simulating the spectrum of some corresponding direct problem and its singular value decomposition based on stochastic representations and Monte Carlo simulations. Herein a full probabilistic error analysis of the aforementioned methods is provided, whereas the convergence of the corresponding approximations is proved and explicit error bounds are provided. This is achieved by employing tools from several areas such as spectral theory, regularity theory for elliptic measures, stochastic representations, and concentration inequalities.