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This paper is concerned with the inverse scattering problem which aims todetermine the spatially distributed dielectric constant coefficient of the 2DHelmholtz equation from multifrequency backscatter data associated with asingle direction of the incident plane wave. We propose a globally convergentconvexification numerical algorithm to solve this nonlinear and ill-posedinverse problem. The key advantage of our method over conventional optimizationapproaches is that it does not require a good first guess about the solution.First, we eliminate the coefficient from the Helmholtz equation using a changeof variables. Next, using a truncated expansion with respect to a specialFourier basis, we approximately reformulate the inverse problem as a system ofquasilinear elliptic PDEs, which can be numerically solved by a weightedquasi-reversibility approach. The cost functional for the weightedquasi-reversibility method is constructed as a Tikhonov-like functional thatinvolves a Carleman Weight Function. Our numerical study shows that, using aversion of the gradient descent method, one can find the minimizer of thisTikhonov-like functional without any advanced \emph{a priori} knowledge aboutit.

Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data