Hybrid FEM/FDM method for an inverse scattering problem

Summary

We apply an adaptive hybrid FEM/FDM method to an inverse scattering problem for the time-dependent acoustic wave equation in 2D and 3D, where we seek to reconstruct an unknown sound velocity c(x) from measured wave-reflection data. Typically,this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium.

We use an optimal control approach where we seek a sound velocity c(x) which minimizes the difference between computed and measured output data in a discrete L ₂ norm. We solve the optimization problem by a gradient method where in each step we compute the gradient by solving a forward (direct) and a backward (adjoint) wave propagation problem.

To compute the backward and forward wave propagation problems we use an adaptive hybrid finite element/finite difference method, where we exploit the flexibility of mesh refinement and adaption of the finite element method in a domain covering the object, and the efficiency of a structured mesh finite difference method in the surrounding homogeneous domain. The hybrid scheme can be viewed as a finite element scheme on a partially unstructured mesh which gives a stable coupling of the two methods.

We use an adaptive mesh refinement algorithm to improve the accuracy of the reconstruction and speed up the convergence of the gradient method.