Numerical Reconstruction of Diffusion and Potential Coefficients from Two Observations: Decoupled Recovery and Error Estimates

TL;DR

This paper presents a decoupled numerical method for reconstructing diffusion and potential coefficients in elliptic/parabolic equations from two measurements, with error estimates and numerical validation.

Contribution

The paper introduces a sequential, decoupled algorithm for reconstructing both coefficients with rigorous error estimates and practical guidance for regularization and discretization.

Findings

Successful numerical reconstruction of coefficients.

Error estimates guide parameter choices.

Numerical experiments confirm theoretical results.

Abstract

The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output least-square method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in $L^2(\Omega)$ for the recovered diffusion and potential coefficients. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in $L^2(\Omega)$ for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results.