Adaptive finite element/difference methods for time-dependent inverse scattering problems

Doktorsavhandling, 2003

In this thesis we develop adaptive hybrid finite element/difference methods for inverse time-domain acoustic and elastic scattering, where we seek to find the location and form of a (small) unknown object inside a large homogeneous body from measured wave-reflection data. We formulate the inverse problem as an optimal control problem, where we seek to reconstruct unknown material coefficients with best least squares wave fit to data. We solve the equations of optimality expressing stationarity of an associated Lagrangian by a quasi-Newton method, where in each step we compute the gradient by solving a forward and an adjoint wave equation. In the first part of the thesis we develop an explicit hybrid finite element/difference method for time-dependent acoustic wave propagation in two and three space dimensions, combining the flexibility of finite elements with the efficiency of finite differences. In the second part we then apply this method to inverse acoustic scattering in two and three space dimensions. We prove an a posteriori error estimate for the error in the Lagrangian and we formulate a corresponding adaptive method, where the finite element mesh covering the object is refined from residual feed-back. The forward and inverse solvers are implemented in object-oriented form in C++. We demonstrate the performance of the adaptive hybrid method for inverse scattering in several examples. In the third and fourth parts we extend to elastic scattering.