{"ref":"master","ref_type":"branch","master_branch":"master","description":"We present a ray-based finite element method (ray-FEM) by learning basis adap- tive to the underlying high-frequency Helmholtz equation in smooth media. Based on the geometric optics ansatz of the wave field, we learn local dominant ray direc- tions by probing the medium using low-frequency waves with the same source. Once local ray directions are extracted, they are incorporated into the finite element basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve approximations for both local ray directions and the high frequency wave field iteratively. The method requires a fixed number of grid points per wave- length to represent the wave field and achieves an asymptotic convergence as the frequency ω → ∞ without the pollution effect. A fast solver is developed for the resulting linear system with an empirical complexity O(ωd) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.","pusher_type":"user"}
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