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parallelism in multigrid methods: How much is too much?

INTERNATIONAL JOURNAL OF parallel PROGRAMMING 1996年第5期24卷 397-432页

作者： Matheson, LR Tarjan, RE PRINCETON UNIV DEPT COMP SCI PRINCETON NJ 08544 USA

multigrid methods are powerful techniques to accelerate the solution of computationally-intensive problems arising in a broad range of applications. Used in conjunction with iterative processes for solving partial differential equations, multigrid methods speed up iterative methods by moving the computation from the original mesh covering the problem domain through a series of coarser meshes. But this hierarchical structure leaves domain-parallel versions of the standard multigrid algorithms with a deficiency of parallelism on coarser grids. To compensate, several parallel multigrid strategies with more parallelism, but also more work, have been designed. We examine these parallel strategies and compare them to simpler standard algorithms to try to determine which techniques are more efficient and practical. We consider three parallel multigrid strategies: (1) domain-parallel versions of the standard V-cycle and F-cycle algorithms;(2) a multiple coarse grid algorithm, proposed by Fredrickson and McBryan, which generates several coarse grids for each fine grid;and (3) two concurrent methods, the Chan-Tuminaro algorithm and the Gannonvan Rosendale algorithm, which allow computation on all grids simultaneously. We study an elliptic model problem on simple domains, discretized with finite difference techniques on block-structured meshes in two or three dimensions with up to 10(6) or 10(9) points, respectively. We analyze performance using three models of parallel computation: the PRAM and two bridging models. The bridging models reflect the salient characteristics of two kinds of parallel computers: SIMD fine-grain computers, which contain a large number of small (bit-serial) processors, and SPMD medium-grain computers, which have a more modest number of powerful (single chip) processors. Our analysis suggests that the standard algorithms are substantially more efficient than algorithms utilizing either parallel strategy. Both parallel strategies need too much extra wor

关键词： parallel multigrid methods parallel computers models of parallel computation

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parallel multigrid algorithms based on generic approximate sparse inverses: an SMP approach

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JOURNAL OF SUPERCOMPUTING 2014年第2期67卷 384-407页

作者： Filelis-Papadopoulos, Christos K. Gravvanis, George A. Democritus Univ Thrace Sch Engn Dept Elect & Comp Engn GR-67100 Xanthi Greece

New parallel computational techniques are introduced for the parallelization of Generic Approximate Sparse Inverse multigrid methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. parallelization of the Generic Approximate Sparse Inverse Matrix (GenAspI) algorithm is achieved based on a new computational approach, namely "strip," which utilizes the data independence of the rows assigned in each available processor. Additionally, new parallel computational techniques are proposed for the parallelization of a modified multigrid V-Cycle method, based on POSIX Threads, for multicore systems. The modified V-Cycle utilized a parallel PGenAspI Preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) as a coarse solver to ensure better parallel performance of the multigrid method. For parallelization purposes, a replication of the multigrid method function is executed on each processor with different index bands and with proper synchronization points to ensure less thread-creation overhead and to maximize parallel performance. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PGenAspI algorithm and the PGenAspI-MGV method for solving classical two-dimensional boundary value problems on multicore computer systems are presented. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.

关键词： Sparse linear systems Finite differences parallel generic sparse approximate inverse matrix algorithms parallel multigrid methods POSIX threads Multicore systems

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SOLUTION OF THE NEUTRON DIFFUSION EQUATION THROUGH multigrid methods IMPLEMENTED ON A MEMORY-COUPLED 25-PROCESSOR SYSTEM

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parallel COMPUTING 1988年第1-3期8卷 391-398页

作者： FINNEMANN, H BREHM, J MICHEL, E VOLKERT, J UNIV ERLANGEN NURNBERG INST MATH MASCH & DATENVERARBEITUNG 3D-8520 ERLANGENFED REP GER

For the complex task to attain maximum use and maximum security maintenance of nuclear reactors, we simulate the physical behaviour by way of mathematical models. The stationary solution of the Neutron Diffusion Equation as an eigenvalue problem, presented in this paper, results in a large non-linear partial differential equation system (PDE) whose solution is parallelized and implemented on the DIRMU multiprocessor system. We demonstrate that modern algorithms (e.g. multigrid methods) combined with innovative hardware (e.g. vector-multiprocessor systems) can lead to powerful tools which are needed for real-time simulation of physical events. This approach also allows to solve problems which arise in fluid dynamics, aerodynamics, weather forecast, etc. in a reasonable amount of time.

关键词： memory-coupled multiprocessor system parallel multigrid methods

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Semicoarsening multigrid on distributed memory machines

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2000年第5期21卷 1823-1834页

作者： Brown, PN Falgout, RD Jones, JE Lawrence Livermore Natl Lab Ctr Appl Sci Comp Livermore CA 94550 USA

This paper presents the results of a scalability study for a three-dimensional semi-coarsening multigrid solver on a distributed memory computer. In particular, we are interested in the scalability of the solver how the solution time varies as both problem size and number of processors are increased. For an iterative linear solver, scalability involves both algorithmic issues and implementation issues. We examine the scalability of the solver theoretically by constructing a simple parallel model and experimentally by results obtained on an IBM SP. The results are compared with those obtained for other solvers on the same computer.

关键词： parallel multigrid methods diffusion equations scalability studies

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Analysis of a class of parallel multigrid smoothers

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BIT NUMERICAL MATHEMATICS 2004年第4期44卷 813-828页

作者： Xie, DX Univ Wisconsin Dept Math Sci Milwaukee WI 53201 USA

This paper proposes and analyzes a class of multigrid smoothers called the parallel multiplicative (PM) smoother by subspace decomposition techniques. It shows that the well known additive and multiplicative smoothers and the JSOR smoother are special cases of the PM smoother, and their smoothing properties can be obtained directly from the PM analysis. Moreover, numerical results are presented in this paper to show that the JSOR smoother is more robust and effective than the damped Jacobi smoother on current MIMD parallel computers.

关键词： parallel multigrid methods additive and multiplicative smoothers JSOR

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