A novel shape optimization approach for source identification in elliptic equations
Wei Gong, Ziyi Zhang
TL;DR
The paper addresses identifying both the support $\omega$ and the strength $q$ of a source in elliptic equations from Cauchy data on $\Gamma$. It introduces a regularized least-squares formulation and eliminates $q$ via the first-order optimality system, reducing the problem to a shape optimization on $\omega$. A gradient-descent algorithm based on level-set representations is derived, with a shape derivative and a perimeter-regularization term ensuring existence; numerical experiments show accurate recovery of smooth, polygonal, and topology-changing sources and robustness to noise, with a post-processing step to refine the source strength. The results extend shape optimization techniques to inverse source problems and suggest avenues for extensions to unsteady diffusion, pointwise strength constraints, and more general source models.
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.