Uniqueness and regularization for unknown spacewise lower-order coefficient and source for the heat type equation

TL;DR

This paper establishes conditions for unique identification of unknown coefficients and heat sources in a heat equation, and introduces a regularization method with convergence guarantees, applied to tumor region detection via thermography.

Contribution

It provides a novel theoretical framework for simultaneous unique identification and a regularization approach with convergence analysis for inverse heat problems.

Findings

Proved sufficient conditions for unique identification of unknown parameters.

Developed a regularization method combining Landweber iteration and Tikhonov regularization.

Applied the theory to tumor detection using thermography data.

Abstract

In this contribution we show sufficient conditions for simultaneous unique identification of unknown spacewise coefficients and heat source in a parabolic partial differential equation given additional final time measurements. Our approach is based on density, in suitable spaces, of the corresponding adjoint problem. A second issue of this paper is the regularization approach. The sequence of approximated solution is obtained by coupling the nonlinear Landweber iteration with iterated Tikhonov regularization. We show that the parameter-to-solution map satisfies sufficient conditions to prove stability and convergence of approximated solutions for the identification problem. We use a unified discrepancy principle as the stopping criteria. Finally, we apply the developed theory in the inverse identification problem of unknown parameters (perfusion coefficient, metabolic heat source) for the identification of tumor regions by thermography.