A Two-Level Domain-Decomposition Preconditioner for the Time-Harmonic Maxwell’s Equations

Bonazzoli, Marcella; Dolean, Victorita; Graham, Ivan G.; Spence, Euan A.; Tournier, Pierre Henri

doi:10.1007/978-3-319-93873-8_12

Cited by 9 publications

(13 citation statements)

References 6 publications

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“…We emphasize that (2.16a) is independent of ω and leads to a symmetric and positive definite linear system, which can be solved efficiently and in parallel with standard numerical (multigrid, domain decomposition, etc.) methods [18,6].…”

Section: 2mentioning

confidence: 99%

“…The elliptic problem (2.16a) in the CMCG algorithm is solved by HPDDM using a two-level overlapping Schwarz DD preconditioner, where the coarse space is built using Generalized Eigenproblems in the Overlap (GenEO) [28]. The Ge-nEO approach has proved effective in producing highly scalable preconditioners for solving various elliptic problems [6,28].…”

Section: Parallel Computationsmentioning

confidence: 99%

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Parallel controllability methods for the Helmholtz equation

Grote

Nataf

Tang

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures.

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Section: 2mentioning

confidence: 99%

Section: Parallel Computationsmentioning

confidence: 99%

Parallel controllability methods for the Helmholtz equation

Grote

Nataf

Tang

et al. 2020

Computer Methods in Applied Mechanics and Engineering

Self Cite

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“…We emphasize that (2.21a) is independent of ω and leads to a symmetric and positive definite linear system, which can be solved efficiently and in parallel with standard numerical (multigrid, domain decomposition, etc.) methods [5,7].…”

Section: Bypx Tqmentioning

confidence: 99%

“…Indeed, the performance of standard numerical methods for Laplace/Poisson-type problems quickly deteriorates when applied to the Helmholtz equation, in fact increasingly so at higher frequency, which thus remains notoriously difficult to solve [2]. Despite the recent development of various preconditioners to accelerate the convergence of standard iterative methods [3][4][5][6][7], the numerical solution of the Helmholtz equation in three-dimensional heterogeneous media remains a formidable challenge.…”

Section: Introductionmentioning

confidence: 99%

On controllability methods for the Helmholtz equation

Grote

Tang

2019

Journal of Computational and Applied Mathematics

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When the Helmholtz equation is discretized by standard finite difference or finite element methods, the resulting linear system is highly indefinite and thus notoriously difficult to solve, in fact increasingly so at higher frequency. The exact controllability approach [1] instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period, the controllability method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. The original energy functional used to penalize the departure from periodicity is strictly convex only for sound-soft scattering problems. To extend the controllability approach to general boundaryvalue problems governed by the Helmholtz equation, new penalty functionals are proposed, which are numerically efficient. Numerical experiments for wave scattering from sound-soft and sound-hard obstacles, inclusions, but also for wave propagation in closed wave guides illustrate the usefulness of the resulting controllability methods.Date: May 2, 2018.(CM) transforms the problem back to the time domain, where it seeks a periodic solution yp¨, tq with (known) period T " 2π{ω of the corresponding time-dependent wave equation. The unknown initial conditions, v 0 " yp¨, 0q and v 1 " y t p¨, 0q, that yield the desired periodic solution are then determined by minimizing a convex cost functional, J 1 pv 0 , v 1 q, which penalizes the departure from periodicity. Akin to a shooting method, the controllability approach iteratively solves the least-squares optimization problem with a standard conjugate gradient (CG) iteration [10]. Each CG iteration then requires the solution of a forward and a backward wave equation together with the solution of a symmetric and positive definite linear system independent of ω, both easily solved using standard numerical methods. Hence, the CM-CG approach solely relies on standard numerical algorithms, which are not only robust with respect to ω but also easy to parallelize. In [11], Bardos and Rauch proved the uniqueness of the minimizer for sound-soft exterior Helmholtz problems. They also proposed an alternative functional, J 8 pv 0 , v 1 q, which is unconditionally coercive even for trapping obstacles. Later Koyama proved convergence of the CM-CG method based on J 1 for sound-soft wave scattering from a disk [12].The CM-CG method in [1,8] relied on a piecewise linear finite element (FE) discretization in space and the second-order leapfrog scheme in time. Low-order FE discretizations, however, are notoriously prone to the pollution effect [13]. Moreover, local mesh refinement imposes a severe CFL stability constraint on explicit time integration, as the maximal time-step is dictated by the smallest element in the mesh. Recently, Heikkola et al. [14,15...

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“…Here, we focus on the second category because we target large computational domains (several hundred of millions of unknowns) with a limited number of reciprocal sources (from few hundreds to few thousands). Our method relies on a finiteelement discretization on a tetrahedral mesh with Lagrange elements of order 3 whose dispersion properties are improved compared to the 27-points finite difference scheme from Operto et al (2007Operto et al ( , 2014; Gosselin-Cliche and Giroux (2014), the Krylov subspace GMRES solver (Saad, 2003) and a Schwarz two-level domain decomposition preconditioner (Graham et al, 2017;Bonazzoli et al, 2019). Compared to the well-known preconditioner based upon shifted Laplacian (Erlangga, 2008), it is less sensitive to the shift (added attenuation) and can be used without it.…”

Section: Introductionmentioning

confidence: 99%

Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

Dolean

Tournier

Jolivet

et al. 2020

SEG Technical Program Expanded Abstracts 2020

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Frequency-domain full-waveform inversion (FWI) is suitable for long-offset stationary-recording acquisition, since reliable subsurface models can be reconstructed with a few frequencies and attenuation is easily implemented without computational overhead. In the frequency domain, wave modeling is a Helmholtz-type boundary-value problem which requires to solve a large and sparse system of linear equations per frequency with multiple right-hand sides (sources). This system can be solved with direct or iterative methods. While the former are suitable for FWI application on 3D dense OBC acquisitions covering spatial domains of moderate size, the later should be the approach of choice for sparse node acquisitions covering large domains (more than 50 millions of unknowns). Fast convergence of iterative solvers for Helmholtz problems remains however challenging in high frequency regime due to the non definiteness of the Helmholtz operator, on one side and on the discretization constraints in order to minimize the dispersion error for a given frequency, on the other side, hence requiring efficient preconditioners. In this study, we use the Krylov subspace GMRES iterative solver combined with a two-level domain-decomposition preconditioner. Discretization relies on continuous Lagrange finite elements of order 3 on unstructured tetrahedral meshes to comply with complex geometries and adapt the size of the elements to the local wavelength (h-adaptivity). We assess the accuracy, the convergence and the scalability of our method with the acoustic 3D SEG/EAGE Overthrust model up to a frequency of 20 Hz.

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A Two-Level Domain-Decomposition Preconditioner for the Time-Harmonic Maxwell’s Equations

Cited by 9 publications

References 6 publications

Parallel controllability methods for the Helmholtz equation

Parallel controllability methods for the Helmholtz equation

On controllability methods for the Helmholtz equation

Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

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