The short wave asymptotic form of the solution for the problem of a point source in an inhomogeneous medium

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

“…This has been justified in [62] for oscillatory initial value problems of hyperbolic equations and further made rigorous in the theory of Fourier integral operators [48]. In practice, the one-term asymptotic expansion (7), namely the so-called geometric optics term, usually yields sufficiently accurate asymptotic solutions [1,2,59,65,66,85,86].…”

“…A more adaptive approach to solve the high-frequency Helmholtz equation is based on the geometric optics ansatz of the wave field (2). In the ansatz, phases and amplitudes are independent of frequency and hence are non-oscillatory and smooth except at a measure zero set, e.g., focus points, caustics, corners in a smooth medium.…”

“…The coefficients {A l } in the asymptotic expansion (6) satisfy a recursive system of transport equations [1,2,86] which are coupled with the eikonal equation. Under the assumption that the medium is smooth and no caustic occurs, one may solve the transport equations to estimate the coefficients {A l } in different formulations [1,2,65].…”

“…However, in the case of the Helmholtz equation they yield either pollution error, 1 inducing oversampled sparse discretizations [4,6] whose associated linear systems can be solved in optimal complexity [28,29,91,104,109], or quasi-optimal sparse discretizations whose associated linear systems are prohibitively expensive to solve [44,101] in the high-frequency regime. 2 Adaptive methods, on the other hand, aim to leverage à priori knowledge of the solution of the Helmholtz equation, such as its known oscillatory behavior. In practice, adaptive methods have mostly focused on adaptivity to the medium, such as polynomial Galerkin methods with hp refinement [3,70,73,96,107,111], specially optimized finite differences [23,45,92,93,102] and finite elements [4,99], enriched finite elements [30][31][32][33], plane wave methods [5,21,42,43,46,69,74], generalized plane wave methods [54,55], locally corrected finite elements [17,38,82], and discretizations with specially chosen basis functions [7,8,…”

“…Indeed, phase-based methods [41,49,78] are instances of fully adaptive discretizations. These methods use (2) to build an approximation space by modulating a polynomial basis with an oscillatory component using the phase functions, which need to be computed beforehand.…”