Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient

TL;DR

This paper introduces a phase field method for recovering piecewise constant diffusion coefficients in elliptic PDEs, combining regularisation, finite element approximation, and iterative algorithms, with proven convergence and numerical validation.

Contribution

It develops a novel double obstacle phase field approach with Gamma-convergence analysis and finite element discretisation for inverse diffusion coefficient problems.

Findings

Convergence of the phase field approximation to perimeter regularisation.

Finite element discretisation converges to the phase field solution.

Numerical experiments demonstrate the method's effectiveness.

Abstract

We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a quadratic fidelity term to which we add a perimeter regularisation weighted by a parameter sigma. This yields a functional which is optimised over a set of diffusion coefficients subject to a state equation which is the underlying elliptic PDE. In order to derive a problem which is amenable to computation the perimeter functional is relaxed using a gradient energy functional together with an obstacle potential in which there is an interface parameter epsilon. This phase field approach is justified by proving Gamma-convergence to the functional with perimeter regularisation as epsilon tends to zero. The computational approach is based on a finite element approximation. This discretisation is shown to converge in an appropriate way to the solution of the phase field problem. We derive an iterative method which is shown to yield an energy decreasing sequence converging to a discrete critical point. The efficacy of the approach is illustrated with numerical experiments.