A DDM double sweep preconditioner for the Helmholtz equation with matrix probing of the DtN map

We describe the structure of a fast solver for the Helmholtz equation in the optimized Schwarz framework, based on a preconditioner that leverages impedance-matching boundary conditions on subdomains. In the case of a simple 2D waveguide numerical example, the method requires no more than 4 GMRES iterations, independently of the frequency and the number of subdomains. The challenge remains to make each iteration fast: we give a partial answer to this question by showing how the Dirichlet to Neumann (DtN) map is accurately approximated in a compressed form via the recently introduced notion of matrix probing.