On the numerical solution of a three-dimensional inverse medium scattering problem

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Abstract
We examine the scattering of time-harmonic acoustic waves in inhomogeneous media. The problem is to recover a spatially varying refractive index in a three-dimensional medium from far-field measurements of scattered waves corresponding to incoming waves from all directions. This problem is exponentially ill-posed and of a large scale since a solution of the direct problem corresponds to solving a partial differential equation in 3 for each incident wave. We construct a preconditioner for the conjugate gradient method applied to the normal equation to solve the regularized linearized operator equation in each Newton step. This reduces the number of operator evaluations dramatically compared to standard regularized Newton methods. Our method can also be applied effectively to other exponentially ill-posed problems, for example, in impedance tomography, heat conduction and obstacle scattering. To solve the direct problems, we use an improved fast solver for the Lippmann–Schwinger equation suggested by Vainikko.