Numerical study of a parametric parabolic equation and a related inverse boundary value problem

TL;DR

This paper presents a spectral Galerkin method for efficiently solving a parametric diffusion equation and its inverse problem, enabling accurate interior diffusivity reconstruction from boundary data in multiple dimensions.

Contribution

It introduces a spectral Galerkin approach that decouples spatial and parameter discretizations, improving computational efficiency for inverse diffusion problems.

Findings

Successful diffusivity reconstructions in 2D and 3D.

The method's complexity remains unaffected by the parameter approximation.

Efficient evaluation of parametric solutions for inverse problems.

Abstract

We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the non-homogeneous diffusion coefficient in the interior of an object. The method in this paper relies on solving the forward problem for a whole family of diffusivities by using a spectral Galerkin method in the high-dimensional parameter domain. The evaluation of the parametric solution and its derivatives is then completely independent of spatial and temporal discretizations. In case of a quadratic approximation for the parameter dependence and a direct solver for linear least squares problems, we show that the evaluation of the parametric solution does not increase the complexity of any linearized subproblem arising from a Gauss-Newtonian method that is used to minimize a Tikhonov functional. The feasibility of the proposed algorithm is demonstrated by diffusivity reconstructions in two and three spatial dimensions.