A novel shape optimization approach for source identification in elliptic equations

TL;DR

This paper introduces a new shape optimization method for identifying sources in elliptic equations, combining shape derivatives and level-set techniques, with demonstrated efficiency through numerical experiments.

Contribution

It develops a shape optimization framework for source identification in elliptic PDEs, reducing the problem to a coupled elliptic system and implementing a gradient descent algorithm.

Findings

Effective source support recovery demonstrated in numerical tests

Reduction of the inverse problem to a shape optimization problem

Efficient gradient descent algorithm using level-set method

Abstract

In this paper, we propose a novel shape optimization approach for the source identification of elliptic equations. This identification problem arises from two application backgrounds: actuator placement in PDE-constrained optimal controls and the regularized least-squares formulation of source identifications. The optimization problem seeks both the source strength and its support. By eliminating the variable associated with the source strength, we reduce the problem to a shape optimization problem for a coupled elliptic system, known as the first-order optimality system. As a model problem, we derive the shape derivative for the regularized least-squares formulation of the inverse source problem and propose a gradient descent shape optimization algorithm, implemented using the level-set method. Several numerical experiments are presented to demonstrate the efficiency of our proposed algorithms.