Numerical Mathematics and Advanced Applications pp 545-556 | Cite as

# 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.

## Keywords

Observation Point Unstructured Mesh Posteriori Error Estimate Absorb Boundary Condition Adjoint Problem## Preview

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