Identification of matrix diffusion coefficients in a parabolic PDE

TL;DR

This paper extends the linearisation approach for scalar diffusion coefficients to matrix-valued coefficients in parabolic PDEs, employing Tikhonov regularization, error analysis, and Galerkin methods for stable inverse problem solutions.

Contribution

It introduces a novel method for identifying matrix-valued diffusion coefficients in parabolic PDEs using linearisation, regularization, and finite-dimensional approximation techniques.

Findings

Proved uniqueness of the inverse problem solution under certain data conditions.

Derived error estimates for noisy data using Tikhonov regularization.

Implemented Galerkin method for finite-dimensional approximation with optimal convergence rates.

Abstract

We consider an inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE. In 2006, Cao and Pereverzev, used a \textit{natural linearisation} method for identifying a scalar valued diffusion coefficient in a parabolic PDE. In this paper, we make use of that idea for identifying a matrix valued coefficient, namely, using the notion of a weak solution for a parabolic PDE, we transform our non-linear inverse problem into a problem of solving an ill-posed operator equation where the operator depending on the data is linear. For the purpose of obtaining stable approximate solutions, Tikhonov regularization is employed, and error estimates under noisy data are derived. We have also showed the uniqueness of the solution of the inverse problem under some assumptions on the data and obtained explicit representation of adjoint of the linear operator involved. For the obtaining error estimates in the finite dimensional setting, Galerkin method is used, by defining orthogonal projections on the space of matrices with entries from $L^2(\O)$, by making use of standard orthogonal projections on $L^2(\O)$. For choosing the regularizing parameter, we used the adaptive technique, so that we have an order optimal rate of convergence. Finally, for the relaxed noisy data, we described a procedure for obtaining a smoothed version so as to obtain the error estimates.