Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Abstract

We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based upon the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free.Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realise full approximation storage (FAS) within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realise a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as enabler for full multigrid (FMG) cycling-the grid unfolds throughout the computation-allows us to reduce the cost per unknown per order of accuracy.The present paper primary contributes towards software realisation and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR and vectorisation-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

ExaHyPE: An engine for parallel dynamically adaptive simulations of wave problems

“…This will open new user communities to the engine. First feasibility studies for this [69,70,71] already exist.…”

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

“…This method is implemented in efficient software packages. 21,59 To solve the linear system (9) using the shifted Laplacian approach, one introduces a shifted system by adding a complex negative mass matrix to (9), as follows:…”

“…[13][14][15] In recent years, there has been a great effort to develop efficient solvers for systems arising from (1), using several different approaches to tackle the problem. One of the most common approaches is the shifted Laplacian multigrid preconditioner, 12,13,[16][17][18][19][20][21][22] which modifies the equation by adding complex values to the diagonal of the matrix. The modified system is then solved using a multigrid method and is used as a preconditioner for the nonshifted system.…”

Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

“…Dissertations, as they have not appeared in peer-reviewed journals, are put in brackets. Techniques [40] without HTMG, [42] BPX [42] Multiplicative ([54]) ( [53])) ( [53])) [12] for unknowns instead of stencils ( [53]) ( [54]),( [53]) [23] for unknowns in SPH 2 Previous work and shortcomings of present approach Peano [55,57] serves as code base to realize our single-touch tree traversals. All implementation ideas however apply to other spacetree software, too.…”

“…Single-touch additive multigrid solvers for spacetrees with rediscretization are subject of discussion in [40], though the discussion lacks details on the handling of dynamic adaptivity. The FAS and HTMG combination is explored in [54], and detailed for additive multigrid and BPX in [42]. A combined, concise presentation for additive, BPX and multiplicative solvers is new (cmp.…”