Numerical Recovery of the Diffusion Coefficient in Diffusion Equations from Terminal Measurement

TL;DR

This paper develops a numerical method for recovering a space-dependent diffusion coefficient in diffusion equations from terminal data, providing stability estimates, error analysis, and numerical experiments to validate the approach.

Contribution

It introduces a novel stability estimate for large terminal times and a comprehensive error analysis for the finite element discretization of the inverse problem.

Findings

Established a Hölder type stability estimate for large terminal time T.

Proved convergence rates for the discretized inverse problem.

Validated the theoretical results with numerical experiments.

Abstract

In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting decay behavior of the observation in time, we establish a novel H{\"o}lder type stability estimate for a large terminal time $T$. This is achieved by novel decay estimates of the (fractional) time derivative of the solution. To numerically recover the diffusion coefficient, we employ the standard output least-squares formulation with an $H^1(\Omega)$-seminorm penalty, and discretize the regularized problem by the Galerkin finite element method with continuous piecewise linear finite elements in space and backward Euler convolution quadrature in time. Further, we provide an error analysis of discrete approximations, and prove a convergence rate that matches the stability estimate. The derived $L^2(\Omega)$ error bound depends explicitly on the noise level, regularization parameter and discretization parameter(s), which gives a useful guideline of the \textsl{a priori} choice of discretization parameters with respect to the noise level in practical implementation. The error analysis is achieved using the conditional stability argument and discrete maximum-norm resolvent estimates. Several numerical experiments are also given to illustrate and complement the theoretical analysis.