Multigrid Smoothers for Ultraparallel Computing

This paper investigates the performance of a parallel Newton, first-order system least-squares (FOSLS) finite-element method with local adaptive refinement and algebraic multigrid (AMG) applied to incompressible, resistive magnetohydrodynamics. In particular, an island coalescence test problem is studied that models magnetic reconnection using a reduced two-dimensional (2D) model of a tokamak fusion reactor. The results show that, using an appropriate temporal and spatial resolution, these methods are capable of resolving the physical instabilities accurately at small computational cost. The time-dependent, nonlinear system of PDEs is solved using work equivalent to about 50-60 simple relaxation sweeps (Gauss-Seidel iterations) per time step. Experiments show that, unless the time step is sufficiently small, nonphysical numerical instabilities may occur. Further, decreasing the time step size does not proportionally increase the cost of the computation, because AMG convergence is improved. In addition, an effective implementation of the methods in parallel keeps load balancing issues to a minimum. Various quantities, such as the reconnection rate and the "sloshing" effect of the plasma instability, are measured to confirm that the correct physics is reproduced. Introduction.The island coalescence problem for studying fast magnetic reconnection in a plasma has been studied extensively (e.g., in [9,11,34,41,42,45,52]). Many numerical algorithms have been implemented to simulate this problem using various types of physical and mathematical models [1,15,22,24,25,35,37,38,41,43,44,46,47,49,50]. The aim of this paper is to extend the results of [2, 3, 4] and show that by using the first-order system least-squares (FOSLS) finite-element method along with nested iteration (NI), algebraic multigrid (AMG), and an efficiency-based adaptive local refinement scheme (ACE) in parallel, the relevant physics of the reconnection is modeled with a low amount of computational cost. Preliminary results are obtained in [2,3,4] using an incompressible resistive magnetohydrodynamics (MHD) model with the above methods. However, limitations with the computational resources used exposed several deficiencies in the method. Namely, at high Lundquist numbers (low resistivity), time integration was not resolving the magnetic instabilities, resulting in numerical oscillations in the current density peak. These, in turn, affected the performance of the solvers and the convergence of the discrete methods. With an efficient implementation of the method on computers with a distributed

Section: Island Coalescencementioning

confidence: 99%

Island Coalescence Using Parallel First-Order System Least Squares on Incompressible Resistive Magnetohydrodynamics

Adler¹,

Brezina²,

Manteuffel³

et al. 2013

“…This in turn reduces the quality of the prolongation, and the resulting algorithms typically require more iterations to converge. For more information about parallel AMG see [4,16,1,2,3] and references therein.…”

Section: 2)mentioning

confidence: 99%

Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation

Treister¹,

Yavneh²

2015

Algebraic multigrid (AMG) methods are known to be efficient in solving linear systems arising from the discretization of partial differential equations and other related problems. These methods employ a hierarchy of representations of the problem on successively coarser meshes. The coarse-grid operators are usually defined by (Petrov-)Galerkin coarsening, which is a projection of the original operator using the restriction and prolongation transfer operators. Therefore, these transfer operators determine the sparsity pattern and operator complexity of the multigrid hierarchy. In many scenarios the multigrid operators tend to become much denser as the coarsening progresses. Such behavior is especially problematic in parallel AMG computations, where it imposes an expensive communication overhead. In this work we present a new algebraic technique for controlling the sparsity pattern of the operators in the AMG hierarchy, independently of the choice of the restriction and prolongation. Our method is based on the aggregation multigrid framework, and it "sparsifies" smoothed aggregation operators while preserving their right and left near nullspaces. Numerical experiments for problems of convection-diffusion and diffusion with discontinuous coefficients demonstrate the efficiency and potential of this approach.

“…The recently proposed L1-Jacobi smoother [2] does not neither require a larger storage. It appears to be well suited for definite and semi-definite Maxwell problems [30].…”

Section: Matrix-free Smoothersmentioning

confidence: 99%

“…For elliptic problems, their robustness and effectiveness have been clearly demonstrated as well as their good parallel efficiency on a large number of cores [2]. For this type of PDE similar approaches where the unstructured mesh is automatically refined [26] within a multigrid solver is reported in [6,21] where parallel performances on huge problems are presented.…”

Section: Introductionmentioning

confidence: 99%

A Parallel Full Geometric Multigrid Solver for Time Harmonic Maxwell Problems

Chanaud¹,

Giraud²,

Goudin³

et al. 2014

The design of scalable parallel simulation codes for complex phenomena is challenging. For simulations that rely on PDE solution on complex 3D geometries, all the components ranging from the initialization phases to the ultimate linear system solution must be efficiently parallelized. In this paper we address the solution of 3D harmonic Maxwell's equations using a parallel geometric full multigrid scheme where only a coarse mesh, tied to geometry, has to be supplied by an external mesh generator. The electromagnetic problem is solved on a finer mesh that respects the discretization rules tied to wavelength. The mesh hierarchy is built in parallel using an automatic mesh refinement technique and parallel matrix-free calculations are performed to process all the finer meshes in the multigrid hierarchy. Only the linear system on the coarse mesh is solved by a parallel sparse direct solver. We illustrate the numerical robustness of the proposed scheme on large 3D complex geometries and assess the parallel scalability of the implementation on large problems with up to 1.3 billion unknowns on a few hundred cores.Key-words: Geometric multigrid, preconditioning, 3D Maxwell/Helmholtz, parallel computing. * Inria, CNRS (LaBRI UMR 5800) and Université de Bordeaux † CEA/CESTA, CS60001, 33116 Le Barp, FranceUn solveur full-multigrille géométrique pour le problème de Maxwell harmonique Résumé :Le développement de codes de calcul passant réellement à l'échelle pour la simulation de phénomènes complexes est un challenge. Pour des simulations basées sur des EDP définies sur des géométries complexes, toutes les composantes du calcul, qui vont souvent des phases d'initialisation à la phase ultime de résolution d'un système linéaire, doivent être parallélisées efficacement. Dans ce rapport, nous considérons la résolution des équations de Maxwell tridimensionnelles par un schéma multigrille géométrique pour lequel seul le maillage le plus grossier, qui capture correctement la géométrie, est fourni par un générateur de maillages externe au code. Le problème d'électromagnétisme est résolu sur un maillage plus fin qui satisfait les contraintes liant le pas de maillage et la fréquence étudiée. La hiérarchie de maillages est construite en parallèle via une technique de raffinement et des calculs sans matrice sont mis en oeuvre sur l'ensemble des maillages plus fins dans la hiérarchie. Seul le système linéaire défini sur le maillage grossier est construit et résolu par une méthode directe parallèle. Nous illustrons la robustesse du solveur obtenu par la résolution de problèmes définis sur des géométries 3D complexes et démontrons le passage à l'échelle de l'implantation parallèle sur des problèmes de grande taille allant jusqu'à 1.3 milliards d'inconnues.