On the generalization of wavelet diagonal preconditioning to the Helmholtz equation

TL;DR

This paper introduces a wavelet-based preconditioning technique for the Helmholtz equation that effectively separates wave components, leading to frequency-independent convergence in numerical solutions.

Contribution

It extends wavelet diagonal preconditioning to multi-dimensional Helmholtz problems using ray theory, achieving optimal iteration counts independent of frequency.

Findings

Number of iterations is small and frequency-independent.

Method is demonstrated through 2-D numerical experiments.

Preconditioning improves convergence for smoothly varying coefficients.

Abstract

We present a preconditioning method for the multi-dimensional Helmholtz equation with smoothly varying coefficient. The method is based on a frame of functions, that approximately separates components associated with different singular values of the operator. For the small singular values, corresponding to propagating waves, the frame functions are constructed using ray theory. A series of 2-D numerical experiments demonstrates that the number of iterations required for convergence is small and independent of the frequency. In this sense the method is optimal.