An Adaptive Hybrid FEM/FDM Method for an Inverse Scattering Problem in Scanning Acoustic Microscopy

Abstract. Scanning acoustic microscopy based on focused ultrasound waves is a promising new tool in medical imaging. In this work we apply an adaptive hybrid FEM/FDM (finite element methods/finite difference methods) method to an inverse scattering problem for the time-dependent acoustic wave equation, where one seeks to reconstruct an unknown sound velocity c(x) from a single measurement of wave-reflection data on a small part of the boundary, e.g., to detect pathological defects in bone. Typically, this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium. The inverse problem is formulated as an optimal control problem, where we use an adjoint method to solve the equations of optimality expressing stationarity of an associated augmented Lagrangian by a quasi-Newton method. To treat the problem of multiple minima of the objective function, the optimization procedure is first performed on a coarse grid to smooth the high frequency error, generating a starting point for optimization steps on successively refined meshes. Local refinement based on the results of previous steps will improve computational efficiency of the method. As the main result then, an a posteriori error estimate is proved for the error in the Lagrangian, and a corresponding adaptive method is formulated, where the finite element mesh is refined from residual feedback. The performance of the adaptive hybrid method and the usefulness of the a posteriori error estimator for problems with limited boundary data are illustrated in three dimensional numerical examples.