On an inverse problem in the parabolic equation arising from groundwater pollution problem

TL;DR

This paper addresses an inverse problem in a parabolic equation related to groundwater pollution, applying Tikhonov regularization to estimate source terms from data, and compares parameter choice methods through numerical experiments.

Contribution

It introduces a regularization approach with error estimates and compares a priori and a posteriori parameter selection methods for this inverse problem.

Findings

A posteriori parameter choice rule converges faster in certain cases.

Theoretical error bounds are established for the regularized solutions.

Numerical experiments validate the effectiveness of the proposed methods.

Abstract

In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained. We also propose both methods, a priori and a posteriori parameter choice rules. In addition, the proposed methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the a posteriori parameter choice rule method gives a better the convergence speed in comparison with the a priori parameter choice rule method in some specific applications.