Numerical recovery of the piecewise constant leading coefficient of an elliptic equation

TL;DR

This paper introduces a numerical algorithm to reconstruct a piecewise constant leading coefficient in an elliptic equation by transforming the inverse problem into a shape optimization task, utilizing finite element methods and level set techniques.

Contribution

It presents a novel shape-based reconstruction algorithm for piecewise constant coefficients in elliptic problems using level set and finite element methods.

Findings

Algorithm successfully reconstructs coefficients from simulated data.

Demonstrates effectiveness of shape optimization approach.

Utilizes open-source FEniCS platform for implementation.

Abstract

We propose a numerical algorithm for the reconstruction of a piecewise constant leading coefficient of an elliptic problem. The inverse problem is reduced to a shape reconstruction problem. The proposed algorithm is based on the minimization of a cost functional where a control function is the right-hand side of an auxiliary elliptic equation for a level set representation of unknown shape. The numerical implementation is based on the finite element method and the open-source computing platform FEniCS. The performance of the algorithm is demonstrated on computationally simulated data.