Stochastic convergence of regularized solutions and their finite element approximations to inverse source problems

TL;DR

This paper analyzes the stochastic convergence and finite element approximation of regularized solutions to inverse source problems governed by PDEs, providing error estimates without source conditions and practical parameter guidance.

Contribution

It introduces novel error estimates for regularized solutions without source conditions and links these to finite element approximations considering noise and discretization parameters.

Findings

Error estimates depend explicitly on noise level, regularization, mesh size, and time step.

Numerical experiments confirm the theoretical convergence rates.

An iterative algorithm for optimal regularization parameter selection is proposed.

Abstract

In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and establish the stochastic convergence and optimal finite element convergence rates of these solutions, under pointwise measurement data with random noise. Unlike most existing regularization theories, the regularization error estimates are derived without any source conditions, while the error estimates of finite element solutions show their explicit dependence on the noise level, regularization parameter, mesh size, and time step size, which can guide practical choices among these key parameters in real applications. The error estimates also suggest an iterative algorithm for determining an optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the analytical results.