Stochastic Convergence Analysis of Inverse Potential Problem

TL;DR

This paper analyzes the stochastic convergence of a regularized finite element method for an inverse potential problem, providing error bounds and an adaptive algorithm for parameter selection, supported by numerical experiments.

Contribution

It offers a stochastic $L^2$ convergence analysis for the inverse potential problem with explicit error bounds and introduces an adaptive algorithm for regularization parameter selection.

Findings

Error bounds depend on regularization, sample size, and mesh size.

The adaptive algorithm effectively determines regularization parameters.

Numerical experiments validate the theoretical convergence results.

Abstract

In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a least squares formulation with an $H^1(\Omega)$ penalty on the potential in order to obtain a numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic $L^2(\Omega)$ convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter $\gamma$, the number $n$ of observation points and the mesh size $h$. These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are also provided to complement the theoretical results.