MFEM: A modular finite element methods library

Anderson, Robert; Andrej, Julian; Barker, Andrew T.; Bramwell, Jamie A.; Camier, Jean-Sylvain; Červený, Jakub; Dobrev, Veselin; Dudouit, Yohann; Fisher, Aaron; Kolev, Tzanio; Pazner, Will; Stowell, Mark; Tomov, Vladimir; Akkerman, I.; Dahm, Johann; Medina, David; Zampini, Stefano

doi:10.1016/j.camwa.2020.06.009

Cited by 293 publications

(206 citation statements)

References 49 publications

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“…The outer CG iteration is stopped at relative residual reduction of 10 −12 . The implementation uses the MFEM package 30,31 and is fully parallel, but the results below focus on scalability with respect to p and not parallel scalability. The parameters for BoomerAMG are the defaults in the MFEM interface to hypre, so that we use HMIS coarsening with one level of aggressive coarsening, and a one‐sweep ℓ 1 ‐Gauss–Seidel smoother.…”

Section: Numerical Resultsmentioning

confidence: 99%

Matrix‐free preconditioning for high‐orderH(curl) discretizations

Barker

Kolev

2020

Numerical Linear Algebra App

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The greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators has received significant attention in the literature in the context of Poisson‐type problems, but preconditioners and solvers for inverting more general operators are not very well‐developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high‐order H1 finite element problems on the same mesh, efficient H(curl) preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor‐product basis functions on an unstructured mesh of quadrilaterals. Our approach uses a sparsified H1 solver constructed on a low‐order mesh of the nodal points of the underlying high‐order space, and we show that the resulting H(curl) preconditioner is effective at very high polynomial orders for two‐dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimal O(pd + 1) floating point operation count and optimal O(pd) memory transfer requirements, outperforming existing Maxwell preconditioners in the high‐order regime.

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Section: Numerical Resultsmentioning

confidence: 99%

Matrix‐free preconditioning for high‐orderH(curl) discretizations

Barker

Kolev

2020

Numerical Linear Algebra App

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“…The concept of matrix-free evaluation with sum factorization has been widely adopted by now, like in the deal.II [1], DUNE [5,40,60], Firedrake [63], mfem [2], Nek5000 [28] or Nektar++ [13] projects. These fast evaluation techniques are directly applicable to explicit time stepping schemes, as we have demonstrated for wave propagation in [42,53,[65][66][67][68] and the compressible Navier-Stokes equations [24].…”

Section: Implementation Of Sum Factorization In the Dealii Librarymentioning

confidence: 99%

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

Arndt

Fehn

Kanschat

et al. 2020

Lecture Notes in Computational Science and Engineering

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This text presents contributions to efficient high-order finite element solvers in the context of the project ExaDG, part of the DFG priority program 1648 Software for Exascale Computing (SPPEXA). The main algorithmic components are the matrix-free evaluation of finite element and discontinuous Galerkin operators with sum factorization to reach a high node-level performance and parallel scalability, a massively parallel multigrid framework, and efficient multigrid smoothers. The algorithms have been applied in a computational fluid dynamics context. The software contributions of the project have led to a speedup by a factor 3 − 4 depending on the hardware. Our implementations are available via the deal.II finite element library.

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“…First, a random initialization of n current dipoles with the current values in the range 0.1-0.9 µA [22] was done. Then, we carried out a calculation of EEG and ECoG data as a numerical solution to the FP (1), discretized with the Finite Element Method (FEM) [2]. The resulting system was solved with preconditioned conjugate gradient method [8].…”

Section: Methodsmentioning

confidence: 99%

Deep Learning for Non-Invasive Cortical Potential Imaging

Razorenova

Yavich

Malovichko

et al. 2020

Preprint

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0000−0002−3386−6914] , Nikolay Yavich 1[0000−0002−8913−7710] , Mikhail Malovichko 1[0000−0002−7618−0998] , Maxim Fedorov 1[0000−0003−3901−3565] , Nikolay Koshev 1[0000−0001−7241−3304] , and Dmitry V. Dylov 1[0000−0003−2251−3221]Abstract. Electroencephalography (EEG) is a well-established non-invasive technique to measure the brain activity, albeit with a limited spatial resolution. Variations in electric conductivity between different tissues distort the electric fields generated by cortical sources, resulting in a smeared potential measurements on the scalp. One needs to solve an ill-posed inverse problem to recover the original neural activity. In this article, we present a generic methodology of recovering the cortical potentials from the EEG measurement by introducing a new inverse-problem solver based on deep convolutional neural networks (CNN) in a U-Net configuration. The solver was trained on a paired dataset from a synthetic head conductivity model by solving the diffusion equations with the finite element method (FEM). This is the first method that provides robust translation of EEG data to the cortex surface using deep learning. Supplying a fast and accurate interpretation of the tracked EEG signal, the proposed approach is a candidate for future non-invasive brain-computer interface devices.

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MFEM: A modular finite element methods library

Cited by 293 publications

References 49 publications

Matrix‐free preconditioning for high‐orderH(curl) discretizations

Matrix‐free preconditioning for high‐orderH(curl) discretizations

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

Deep Learning for Non-Invasive Cortical Potential Imaging

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