A numerical reconstruction algorithm for the inverse scattering problem with backscatter data
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This paper is concerned with the inverse scattering problem which aims to determine the coefficient of the Helmholtz equation from multifrequency backscatter data. We propose an efficient numerical algorithm to solve this nonlinear and ill-posed inverse problem without using any advanced a priori knowledge of the solution. To study the algorithm we first eliminate the coefficient from the Helmholtz equation using a change of variables. Then using a truncated Fourier expansion for the wave field we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a quasi-reversibility approach. The cost functional for the quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman weight function. Our numerical study shows that using a method of gradient descent type one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it.
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