  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.
