An accelerated, high-order accurate direct solver for the Lippmann-Schwinger equation for acoustic scattering in the plane

Abstract: An efficient direct solver for solving the Lippmann-Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem with $N$ degrees of freedom, the solver constructs an approximate inverse in $\mathcal{O}(N^{3/2})$ operations and then, given an incident field, can compute the scattered field in $\mathcal{O}(N \log N)$ operations. The solver is based on a previously published direct solver for integral equations that relies on rank-deficiencies in the off-diagonal blocks; specifically, the so-called Hierarchically Block Separable format is used. The particular solver described here has been reformulated in a way that improves numerical stability and robustness, and exploits the particular structure of the kernel in the Lippmann-Schwinger equation to accelerate the computation of an approximate inverse. The solver is coupled with a Nystr\"om discretization on a regular square grid, using a quadrature method developed by Ran Duan and Vladimir Rokhlin that attains high-order accuracy despite the singularity in the kernel of the integral equation. A particularly efficient solver is obtained when the direct solver is run at four digits of accuracy, and is used as a preconditioner to GMRES, with each forwards application of the integral operators accelerated by the FFT. Extensive numerical experiments are presented that illustrate the high performance of the method in challenging environments. Using the $10^{\rm th}$-order accurate version of the Duan-Rokhlin quadrature rule, the scheme is capable of solving problems on domains that are over 500 wavelengths wide to residual error below $10^{-10}$ in a couple of hours on a workstation, using 26M degrees of freedom.