Directional Preconditioner for High Frequency Obstacle Scattering

TL;DR

This paper introduces a directional preconditioner for boundary integral equations in high frequency obstacle scattering, significantly improving iterative solution efficiency and frequency independence.

Contribution

The paper develops a novel data-sparse, hierarchical preconditioner that transforms boundary integral operators into sparse systems with efficient approximate inverses.

Findings

Reduces iteration counts in high frequency scattering problems

Achieves near frequency-independent convergence rates

Demonstrates effectiveness through numerical experiments

Abstract

The boundary integral method is an efficient approach for solving time-harmonic obstacle scattering problems by a bounded scatterer. This paper presents the directional preconditioner for the iterative solution of linear systems of the boundary integral method. This new preconditioner builds a data-sparse approximation of the integral operator, transforms it into a sparse linear system, and computes an approximate inverse with efficient sparse and hierarchical linear algebra algorithms. This preconditioner is efficient and results in small and almost frequency-independent iteration counts when combined with standard iterative solvers. Numerical results are provided to demonstrate the effectiveness of the new preconditioner.