Abstract: We present an adaptive hybrid finite element/difference methods for an inverse scattering problem modelling acoustic wave propagation of light in the two-dimensional photonic crystal to control the optical properties of the material. The objective is to reconstruct unknown parameters from measured in space and time wave reflection data on the boundary of the crystal. The inverse problem is formulated as an optimal control problem, where we solve the equations of optimality expressing stationarity of an associated Lagrangian by a quasi-Newton method: in each step we compute the gradient by solving a forward and an adjoint wave equation. We show possibilities of computational inverse scattering using adaptive error control.
First we present an a posteriori error estimate for the error in the Lagrangian and formulate a corresponding adaptive method, where the finite element mesh is refined from residual feed-back. Then we derive an a posteriori error for the error in the reconstructed parameters by solving and associated dual linearized problem for the Hessian of the Lagrangian. The performance of the computational inverse scattering using adaptive error control is illustrated in numerical examples.