The method of polarized traces for the 3D Helmholtz equation

Abstract

We present a fast solver for the 3D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media. The solver is based on the method of polarized traces, coupled with distributed linear algebra libraries and pipelining to obtain an empirical online runtime O(max(1, R/n)N log N ) where N = n 3 is the total number of degrees of freedom and R is the number of right-hand sides. Such a favorable scaling is a prerequisite for large-scale implementations of full waveform inversion (FWI) in frequency domain.Suppose that L, the number of layers in the domain decomposition, scales as L ∼ n, i.e., each layer has a constant thickness in number of grid points * . Each layer is * This hypothesis is critical for obtaining a quasi-linear complexity algorithm and is related to the complexity of solving a quasi-

L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation

“…Depending on the discretization, change in the selection of the sets Γ 1 and Γ 2 may be required. Similar schemes have been applied for many different discretizations such as high-order finite difference methods [89], finite element methods [84], enriched finite element methods [30], discontinuous Galerkin methods [74], and integral representations [87]. For example, for higher-order finite difference methods the stencils centered at the points in Ω 2 (respecting Γ 2 , Γ 1 , or Ω 1 ) cannot involve discretization points in Γ 1 (respecting Ω 1 , Ω 2 , Γ 2 ).…”

“…In particular, using sparse direct solvers on blocks that arise from sufficiently thin layers yields a method with quasi-linear (i.e., linear with poly-logarithmic factors) sequential complexity for a single right-hand side [26,25,86,53,72,16,80,56]. In the presence of many right-hand sides, the layered domain decomposition allows for optimal parallelization [89].…”

“…Thus, we do not anticipate further reduction in complexity, for a single right-hand, due exclusively to modification of the domain decomposition. In the presence of O(n log ω) right-hand-sides, it has been shown, however, that the sequential nature of the sweeps can be mitigated by pipelining multiple right-hand sides [89]. Using pipelining, solutions for all O(n) right-hand-sides can be computed with O(n log ω) parallel complexity, i.e., the average parallel complexity per right-hand-side is O(log ω).…”

A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation

“…For acoustic time-harmonic wave problems, governed by the scalar Helmholtz equation, a recent overview of DDMs can be found in [31]. Among all DDMs, we can highlight Schwarz methods with overlap [12,30,38] or without overlap [4,16,32], FETI algorithms [19,[25][26][27] and the method of polarized traces [65,66], which are eventually combined with preconditioning techniques (see e.g. [18,33,35,58,59,64]).…”

WaveHoltz: Iterative Solution of the Helmholtz Equation via the Wave Equation

“…Chen and Xiang, [9], and Vion and Geuzaine, [34], also considered sweeping domain decomposition method combined with PML and showed that their methods could be used as efficient preconditioners for the Helmholtz equation. The method of polarized traces by Zepeda-Núñez, Demanet and co-authors, [39,38,37], is a two step sweeping preconditioner that compresses the traces of the Greens function in an offline computation and utilizes incomplete Green's formulas to propagate the interface data. See also the recent review by Gander and Zhang [18] for connections between sweeping methods.…”