Iterative methods for the interior inverse scattering problem with single point source

This paper investigates the time-harmonic acoustic interior inverse scattering problem, focusing on cavity shape reconstruction from the measured scattered fields due to a single point source. We rigorously derive an explicit analytical form of the Fréchet derivative for the near-field operator, which can be applied to design iterative schemes for the nonlinear inverse problem. In the numerical simulation, the Levenberg-Marquardt and Gauss-Newton methods are employed to solve the inverse problem. Numerical experiments demonstrate that the proposed Fréchet derivative-based iterative framework can effectively reconstruct several cavity shapes from noisy scattered field measurements generated by a single point source.