The Numerical Solution of the Three Dimensional Inverse Obstacle Scattering Problem for Point Source Generated Wave Fields

Abstract

This work is concerned with the inverse problem to retrieve the shape of a three dimensional obstacle from near-field measurements of the scattering of time-harmonic point source waves. A numerical method for solving the inverse obstacle scattering problem is presented, which is achieved by solving an ill-posed linear system and a well-posed minimization problem independently. Such a separate numerical treatment for the ill-posedness and nonlinearity of the inverse problem makes the numerical implementation of the proposed method very easy and fast since there involves only the solution of a small scale minimization problem with one unknown function in the nonlinear optimization step for reconstructing the shape of the obstacle. Another attractive feature of the scheme is that it needs only the knowledge of the near-field measurements of the scattered fields due to point source incident fields at a finite number of incidence and observation points distributed over a limited aperture. Some numerical examples based on synthetic near-field data are given, showing that this method produces reasonable reconstructions for low wavenumbers even when the incidence and observation aperture is as small as the possible ~ /4 solid angle.