On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation

TL;DR

This paper addresses the inverse problem of simultaneously recovering the spatially-dependent conductivity and nonlinear reaction term in a reaction-diffusion equation, providing theoretical guarantees and numerical algorithms.

Contribution

It introduces new uniqueness results and iterative algorithms for recovering both coefficients from various types of overposed data.

Findings

Proved uniqueness of the inverse problem solutions.

Developed convergent iterative reconstruction algorithms.

Demonstrated numerical reconstructions validating the methods.

Abstract

This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $a(x)$ and the nonlinear reaction term $f(u)$ in a reaction-diffusion equation from overposed data. These measurements can consist of: the value of two different solution measurements taken at a later time $T$; time-trace profiles from two solutions; or both final time and time-trace measurements from a single forwards solve data run. We prove both uniqueness results and the convergence of iteration schemes designed to recover these coefficients. The last section of the paper shows numerical reconstructions based on these algorithms.