Learning Dominant Wave Directions For Plane Wave Methods For High-Frequency Helmholtz Equations

摘要

We present a ray-based finite element method (ray-FEM) by learning basisadaptive to the underlying high-frequency Helmholtz equation in smooth media.Based on the geometric optics ansatz of the wave field, we learn local dominantray directions by probing the medium using low-frequency waves with the samesource. Once local ray directions are extracted, they are incorporated into thefinite element basis to solve the high-frequency Helmholtz equation. Thisprocess can be continued to further improve approximations for both local raydirections and the high frequency wave field iteratively. The method requires afixed number of grid points per wavelength to represent the wave field andachieves an asymptotic convergence as the frequency $\omega\rightarrow \infty$without the pollution effect. A fast solver is developed for the resultinglinear system with an empirical complexity $\mathcal{O}(\omega^d)$ up to apoly-logarithmic factor. Numerical examples in 2D are presented to corroboratethe claims.