Hybrid FEM/FDM method for an inverse scattering problem
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.
KeywordsObservation Point Unstructured Mesh Posteriori Error Estimate Absorb Boundary Condition Adjoint Problem
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