The temporal domain derivative in inverse acoustic obstacle scattering

This work describes and analyzes the domain derivative for a time-dependent acoustic scattering problem. We study the nonlinear operator that maps a sound-soft scattering object to the solution of the time-dependent wave equation evaluated at a finite number of points away from the obstacle. The Fr\'echet derivative of this operator with respect to variations of the scatterer coincides with point evaluations of the temporal domain derivative. The latter is the solution to another time-dependent scattering problem, for which a well-posedness result is shown under sufficient temporal regularity of the incoming wave. Applying convolution quadrature to this scattering problem gives a stable and provably convergent semi-discretization in time, provided that the incoming wave is sufficient regular. Using the discrete domain derivative in a Gauss--Newton method, we describe an efficient algorithm to reconstruct the boundary of an unknown scattering object from time domain measurements in a few points away from the boundary. Numerical examples for the acoustic wave equation in two dimensions demonstrate the performance of the method.