Fast alternating bi-directional preconditioner for the 2D high-frequency Lippmann-Schwinger equation

TL;DR

This paper introduces a fast iterative solver for the high-frequency 2D Lippmann-Schwinger equation, combining a sparsifying preconditioner with a bi-directional domain decomposition approach for efficient computation.

Contribution

The paper develops a novel two-level iterative solver that achieves near-linear complexity for high-frequency wave scattering problems in 2D.

Findings

Complexity scales as O(N log N) for large problems.

Number of iterations depends weakly on frequency.

Solver effectively handles high-frequency wave scattering.

Abstract

This paper presents a fast iterative solver for Lippmann-Schwinger equation for high-frequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner and a domain decomposition approach similar to the method of polarized traces. The iterative solver has two levels, the outer level in which a sparsifying preconditioner for the Lippmann-Schwinger equation is constructed, and the inner level, in which the resulting sparsified system is solved fast using an iterative solver preconditioned with a bi-directional matrix-free variant of the method of polarized traces. The complexity of the construction and application of the preconditioner is $\mathcal{O}(N)$ and $\mathcal{O}(N\log{N})$ respectively, where $N$ is the number of degrees of freedom. Numerical experiments in 2D indicate that the number of iterations in both levels depends weakly on the frequency resulting in method with an overall $\mathcal{O}(N\log{N})$ complexity.