A fast direct solver for quasi-periodic scattering problems

TL;DR

This paper introduces a fast, direct integral equation solver for 2D quasi-periodic scattering problems that efficiently handles complex geometries and high frequencies, outperforming iterative methods especially for multiple incident angles.

Contribution

The authors develop an O(N) complexity direct solver using hierarchical linear algebra for quasi-periodic scattering, robust against ill-conditioning and Wood's anomalies.

Findings

Solver achieves 66x speedup over iterative methods for large problems.

Handles complex obstacles with high accuracy ($10^{-10}$ error).

Significantly faster when multiple incident angles share Bloch phases.

Abstract

We consider the numerical solution of the scattering of time-harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium, in two dimensions. Boundary integral formulations are ideal since they reduce the problem to $N$ unknowns on the obstacle boundary. However, for complex geometries and/or higher frequencies the resulting dense linear system becomes large, ruling out dense direct methods, and often ill-conditioned (despite being 2nd-kind), rendering fast multipole-based iterative schemes also inefficient. We present an integral equation based solver with O(N) complexity, which handles such ill-conditioning, using recent advances in "fast" direct linear algebra to invert hierarchically the isolated obstacle matrix. This is combined with a recent periodizing scheme that is robust for all incident angles, including Wood's anomalies, based upon the free space Green's function kernel. The resulting solver is extremely efficient when multiple incident angles are needed, as occurs in many applications. Our numerical tests include a complicated obstacle several wavelengths in size, with $N=10^5$ and solution error of $10^{-10}$, where the solver is 66 times faster per incident angle than a fast multipole based iterative solution, and 600 times faster when incident angles are chosen to share Bloch phases.