A novel time-domain iterative method for a three-dimensional inverse acoustic obstacle scattering problem

This paper concerns the three-dimensional forward and inverse acoustic obstacle scattering problem in the time domain. For the forward problem, a retarded potential formulation discretized by convolution quadrature and Galerkin methods is introduced. By introducing the retarded boundary integral defined on a homothetic surface, we propose a novel time-domain convolution quadrature based iterative method to reconstruct both the shape and location of a rigid obstacle. The retarded integral in the time domain is reformulated into a system of integrals in the s-domain. The resulting s-domain integrals are very fast to compute, as they only involve non-singular integrals over the homothetic surfaces. Moreover, the Fr\'echet derivative with respect to the boundary can be derived straightforwardly. We also prove that the scattered field generated by the homothetic surface converges to the exact field in the time domain. To improve the stability of the inversion algorithm, an incremental truncation technique is proposed, and numerical experiments confirm the effectiveness and robustness of our method.