A New Approach and Error Analysis for Reconstructing the Scattered Wave by the Point Source Method

Source abstract

Consider an inverse scattering problem by an obstacle $D \subset \mathcal{R}^2$ with impedance boundary. We investigate the reconstruction of the scattered field $u^s$ from its far field pattern $u^\infty$ using the point source method. First, by applying the boundary integral equation method, we provide a new approach to the point-source method of Potthast by classical potential theory. This extends the range of the point source method from plane waves to scattering of arbitrary waves. Second, by analyzing the behavior of the Hankel function, we obtain an improved strategy for the choice of the regularizing parameter from which an improved convergence rate (compared to the result of [15]) is achieved for the reconstruction of the scattered wave. Third, numerical implementations are given to test the validity and stability of the inversion method for the impedance obstacle.