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.

翻译：本文针对椭圆方程的源识别问题提出了一种新型形状优化方法。该识别问题源于两个应用背景：偏微分方程约束最优控制中的执行器配置，以及源识别问题的正则化最小二乘公式。该优化问题同时寻求源强度及其支撑集。通过消除与源强度相关的变量，我们将问题简化为一个耦合椭圆系统的形状优化问题，即一阶最优性系统。作为模型问题，我们推导了逆源问题正则化最小二乘公式的形状导数，并提出了一种梯度下降形状优化算法，该算法采用水平集方法实现。多个数值实验展示了所提算法的有效性。