An inverse acoustic waveguide problem in the time domain

We consider the problem of locating an obstacle in a waveguide from time domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface located inside the waveguide away from the obstacle, where the scattered field is measured on the same surface. From this multi-static scattering data we wish to determine the position and shape of an obstacle in the waveguide. To deal with this inverse problem, we adapt and analyze the time domain linear sampling method. This involves proving new time domain estimates for the forward problem, as well as analyzing several time domain operators arising in the inversion scheme. We also implement the inversion algorithm and provide numerical results in two-dimensions using synthetic data.