2024/07/03 by Wei Gong, Ziyi Zhang, Gong, Wei +1 · 1 citation
Engineering · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Structural Health Monitoring Techniques #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.2407.02909
openalex publication_date 2024/07/03 · openalex created_date 2024/07/06 · openalex updated_date 2026/07/28
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.