Sweeping Preconditioner for the Helmholtz Equation: Hierarchical Matrix Representation

TL;DR

The paper presents a novel sweeping preconditioner for the Helmholtz equation that uses hierarchical matrix representation, achieving near-constant iteration counts and linear complexity in 2D and 3D problems.

Contribution

It introduces a new preconditioning method combining domain sweeping and hierarchical matrices, significantly improving efficiency for high-frequency Helmholtz problems.

Findings

Linear complexity in 2D for construction and application

Converges in very few iterations, independent of problem size

Effective extension to 3D problems

Abstract

The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea of this approach is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The GMRES solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.