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International Conference on High Performance Computing in Asia-Pacific Region (HPC Asia)

作者： Nakajima, Kengo Univ Tokyo Informat Technol Ctr Tokyo Japan

ISBN: (纸本)9781450376501

parallel multigrid method is expected to be a useful algorithm in exascale era because of its scalability. It is widely known that overhead of coarse grid solver in parallel multigrid method is significant, if the number of MPI processes is O(10(4)) or larger. The author proposed the hCGA for avoiding such overhead. Recently, the AM-hCGA, further optimized version of the hCGA, was proposed by the author, and its performance was evaluated on the Oakforest-PACS system (OFP) with IHK/McKernel at JCAHPC using up to 2,048 nodes of Intel Xeon Phi (Knights Landing). In the present work, developed method is also implemented to the Oakbridge-CX system (OBCX) at the University of Tokyo using up to 1,024 nodes (2,048 sockets) of Intel Xeon Platinum 8280 (Cascade Lake). Performance in weak and strong scaling are evaluated for application on 3D groundwater flow through heterogeneous porous media (pGW3D-FVM). The hCGA and the AM-hCGA provide excellent performance on both of OFP and OBCX with larger number of nodes. Especially, it achieved excellent performance in strong scaling on OBCX.

关键词： parallel multigrid Multicore/Manycore Clusters Ill-conditioned Problems

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Performance of parallel Approximate Ideal Restriction multigrid for Transport Applications

Performance of Parallel Approximate Ideal Restriction Multig...

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作者： Hanophy, Joshua Thomas Texas A&M University

学位级别：硕士

Algebraic multigrid (AMG) methods have been widely used to solve systems arising from the discretization of elliptic partial differential equations. In serial, AMG algorithms scale linearly with problem size. In parallel, communication costs scale logarithmically with the number of processors. Recently, a classical AMG method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices. AIR has already been shown to be effective for solving the linear systems arising from upwind discontinuous Galerkin (DG) finite element discretization of advection-diffusion problems, including the hyperbolic limit of pure advection. A new parallel version of AIR, pAIR, has been implemented in the hypre library. In this thesis, pAIR is tested for use solving the source iteration equations of the SN approximations to the transport equation. The performance is investigated with various meshes in two and three dimensions. Detailed profiling of parallel performance is also conducted to identify the most important areas for algorithm improvements. An improvement to the Local Ideal Approximate Restriction algorithm is introduced and discussed. Weak scaling results to 4,096 processors are presented. These results show total solve growing logarithmically with the number of processors used. Importantly, this result is shown on both uniform grids and unstructured grids in three dimensions. The unstructured mesh did not include reentrant cells.

关键词： transport multigrid parallel multigrid AMG Approximate Ideal Restriction Thesis

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A parallel multigrid Poisson PDE Solver for Gigapixel Image Editing

A Parallel Multigrid Poisson PDE Solver for Gigapixel Image ...

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8th CCF National Annual Conference on High-Performance Computing (HPC)

作者： Du, Zhenlong Li, Xiaoli Yang, Xiaojian Shen, Kangkang Nanjing Univ Technol Coll Elect & Informat Engn Nanjing 210009 Jiangsu Peoples R China

ISBN: (纸本)9783642415913;9783642415906

With the development of image acquisition technology, gigapixel images are easily produced and widely used in modern society. How to efficiently compile these gigapixel images within gradient domain is the research focus in the community of image processing and computer graphics. To solve Poisson equations involving large-scale unknowns is crucial for gigapixel image editing in gradient domain. Traditional multigrid approach separately performs iteration, restriction and interpolation, bears heavy communication costs between RAM and external memory. In the paper, a parallel multigrid Poisson solver for gigapixel image editing is proposed, which exploits the locality and relevance of memory accessing and updating among the iteration, restriction and interpolation for parallel performing the iteration, restriction and interpolation in the sweeping window. Image stitching experiments show that the presented method exhibits the higher efficiency than the Poisson solver of successive overrelaxation, gauss-seider iteration and traditional multigrid.

关键词： Poisson PDE solver parallel multigrid gigapixel image editing

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A scalable, fully implicit algorithm for the reduced two-field low-β extended MHD model

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JOURNAL OF COMPUTATIONAL PHYSICS 2016年 326卷 763-772页

作者： Chacon, L. Stanier, A. Los Alamos Natl Lab Los Alamos NM 87545 USA

We demonstrate a scalable fully implicit algorithm for the two-field low-beta extended MHD model. This reduced model describes plasma behavior in the presence of strong guide fields, and is of significant practical impact both in nature and in laboratory plasmas. The model displays strong hyperbolic behavior, as manifested by the presence of fast dispersive waves, which make a fully implicit treatment very challenging. In this study, we employ a Jacobian-free Newton-Krylov nonlinear solver, for which we propose a physics-based preconditioner that renders the linearized set of equations suitable for inversion with multigrid methods. As a result, the algorithm is shown to scale both algorithmically (i.e., the iteration count is insensitive to grid refinement and timestep size) and in parallel in a weak-scaling sense, with the wall-clock time scaling weakly with the number of cores for up to 4096 cores. For a 4096 x 4096 mesh, we demonstrate a wall-clock-time speedup of similar to 6700 with respect to explicit algorithms. The model is validated linearly (against linear theory predictions) and nonlinearly (against fully kinetic simulations), demonstrating excellent agreement. Published by Elsevier Inc.

关键词： Implicit algorithms Extended MHD JFNK Physics-based preconditioning parallel multigrid

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Pseudomultigrid Gauss-Seidel method for large scale and high performance computing 7

Pseudomultigrid Gauss-Seidel method for large scale and high...

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7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016

作者： Martynenko, S.I. Volokhov, V.M. Toktaliev, P.D. Institute of Problems of Chemical Physics Russian Academy of Sciences Academician Semenov avenue 1 Chernogolovka Moscow Region142432 Russia

ISBN: (纸本)9786188284401

The paper represents a new approach for development of robust multigrid methods. The approach is based on minimization of the number of problem-dependent components in multigrid algorithm. Robust multigrid Technique (RMT) is application of the essential multigrid principle in single-grid algorithm. Problem-dependent components of the technique are: 1) the number of smoothing iterations;2) unknown ordering (for anisotropic problems);3) underrelaxation parameter (for nonlinear problems);4) stopping criterion of the multigrid iterations. The least number of the problem-dependent components makes it possible to use RMT in black box software. Disadvantage of RMT is extra computational efforts as compared with optimized multigrid. Since coarse grid correction is computed on the single grid, it is possible to avoid the large communication overhead and the processor idleness on the very coarse grids in parallel multigrid. The paper represents main components of RMT, results of numerical experiments, convergence and complexity analysis, estimation of speed-up and efficiency of parallel algorithm.

关键词： multigrid parallel multigrid Robustness Structured grids

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SEGMENTAL REFINEMENT: A multigrid TECHNIQUE FOR DATA LOCALITY

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2016年第4期38卷 C426-C440页

作者： Adams, Mark F. Brown, Jed Knepley, Matt Samtaney, Ravi Lawrence Berkeley Lab Scalable Solvers Grp Berkeley CA 94720 USA Univ Colorado Dept Comp Sci Boulder CO 80309 USA Rice Univ Computat & Appl Math Houston TX 77005 USA King Abdullah Univ Sci & Technol Thuwal 23955 Saudi Arabia

We investigate a domain decomposed multigrid technique, termed segmental refinement, for solving general nonlinear elliptic boundary value problems. We extend the method first proposed in 1994 by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid is eliminated on fine grids, with modest amounts of extra work and storage, while maintaining the asymptotic exactness of full multigrid. We observe an accuracy dependence on the segmental refinement subdomain size, which was not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores on a Cray XC30.

关键词： multigrid parallel multigrid distributed memory multigrid segmental refinement

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Peta-Scale Hierarchical Hybrid multigrid Using Hybrid parallelization

Peta-Scale Hierarchical Hybrid Multigrid Using Hybrid Parall...

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9th International Conference on Large-Scale Scientific Computations (LSSC)

作者： Gmeiner, Bjoern Ruede, Ulrich Univ Erlangen Nurnberg Dept Comp Sci 10 D-91054 Erlangen Germany

ISBN: (纸本)9783662438800;9783662438794

In this article we present a performance study of our finite element package Hierarchical Hybrid Grids (HHG) on current European supercomputers. HHG is designed to close the gap between the flexibility of finite elements and the efficiency of geometric multigrid by using a compromise between structured and unstructured grids. A coarse input finite element mesh is refined in a structured way, resulting in semi-structured meshes. Within this article we compare and analyze the efficiencies of the stencil-based code on those clusters.

关键词： parallel multigrid Performance analysis HHG

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On multigrid methods for the solution of least-squares finite element models for viscous flows

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INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 2012年第1期26卷 45-65页

作者： Ranjan, Rakesh Reddy, J. N. Univ Texas San Antonio Dept Mech Engn San Antonio TX USA Texas A&M Univ Dept Mech Engn College Stn TX 77843 USA

There is a vast literature on least-squares finite element models (LSFEM) applied to fluid dynamics problems. The hp version of the least-squares models is computationally expensive, which necessitates the usage of elegant methods for solving resulting systems of equations. Amongst some of the schemes used for solving large systems of equations is the element-by-element (EBE) solution technique, which has found widespread use in least-squares applications. However, the use of EBE techniques with Jacobi preconditioning leads to very little performance gains as compared to solving a non-preconditioned system. Because of such considerations, the hp version LSFEM solutions are computationally intensive. In this study, we propose to solve the LSFEM systems using the multigrid method, which offers superior convergence rates compared to the EBE-JCG. We demonstrate the superior convergence of the multigrid solver compared to Jacobi preconditioning for the wall-driven cavity and backward facing step problems using the full Navier-Stokes equations. Load balancing issues encountered with multigrid solvers in a parallel environment are resolved elegantly with an element-by-element solution of the coarse grid problem with Jacobi preconditioning.

关键词： multigrid least-squares driven cavity least-squares finite element method backward facing step parallel multigrid

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Reconsidering algorithms for iterative solvers in the multicore era

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INTERNATIONAL JOURNAL OF COMPUTATIONAL SCIENCE AND ENGINEERING 2009年第4期4卷 270-282页

作者： Wallin, Dan Lof, Henrik Hagersten, Erik Holmgren, Sverker Uppsala Univ Dept Informat Technol POB 337 SE-75105 Uppsala Sweden

Efficient solution of computational problems require a match between the algorithm and the underlying architecture. New multicore processors feature low intra-chip communication cost and smaller per-thread caches compared to single-core implementations, indicating that data locality issues are more important than communication overheads. We investigate the impact of these changes on parallel multigrid methods. We present a temporally blocked, naturally ordered, smoother implementation that improves the data locality as much as ten times. compared with the standard red-black algorithm. We present results of the performance of our new algorithm on an SMP system, an UltraSPARC T1 (Niagara) SMT/CMP, and a simulated CMP processor.

关键词： multicore processors data locality parallel multigrid

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A distributed memory parallel implementation of the multigrid method for solving three-dimensional implicit solid mechanics problems

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2004年第8期61卷 1173-1208页

作者： Namazifard, A Parsons, ID Lawrence Livermore Natl Lab Methods Dev Grp Livermore CA 94551 USA Univ Illinois Ctr Simulat Adv Rockets Urbana IL USA

We describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object-based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed- and scaled-size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis. Copyright (C) 2004 John Wiley Sons, Ltd.

关键词： parallel multigrid unstructured finite elements solid mechanics

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