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A PARALLEL AUXILIARY GRID algebraic multigrid method FOR GRAPHIC PROCESSING UNITS

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第3期35卷 C263-C283页

作者： Wang, Lu Hu, Xiaozhe Cohen, Jonathan Xu, Jinchao Penn State Univ Dept Math University Pk PA 16802 USA Penn State Univ Ctr Computat Math & Applicat University Pk PA 16802 USA NVIDIA Res Ann Arbor MI 48105 USA

In this paper, we develop a new parallel auxiliary grid algebraic multigrid (AMG) method to leverage the power of graphic processing units (GPUs). In the construction of the hierarchical coarse grid, we use a simple and fixed coarsening procedure based on a region quadtree generated from an auxiliary grid. This allows us to explicitly control the sparsity patterns and operator complexities of the AMG solver. This feature provides (nearly) optimal load balancing and predictable communication patterns on shape regular grids, which makes our new algorithm suitable for parallel computing, especially on GPUs. We also design a parallel smoother based on the special coloring of the quadtree to accelerate the convergence rate and improve the parallel performance of this solver. Based on the CUDA toolkit [NVIDIA CUDA Programming Guide, NVIDIA Corp., 2010], we implemented our new parallel auxiliary grid AMG method on GPUs and the numerical results of this implementation demonstrate the efficiency of our new method for (nearly) isotropic problems. The results achieve an average speedup of over 4 on quasi-uniform grids and 2 on shape regular grids when compared to the AMG implementation in CUSP [M. Garland and N. Bell, CUSP: Generic Parallel Algorithms for Sparse Matrix and Graph Computations, http://***/(2010)].

关键词： algebraic multigrid method aggregation nonlinear AMLI-cycle auxiliary grid GPU parallel algorithm

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Investigation of algebraic multigrid method for Magnetic Field Analysis of an Electric Machine With a Laminated Core

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IEEE TRANSACTIONS ON MAGNETICS 2009年第3期45卷 992-995页

作者： Miwa, Masahiko Yamada, Takashi Wakao, Shinji JRI Solut Ltd Engn Technol Div Osaka 5420081 Japan Waseda Univ Dept Elect Engn & Biosci Tokyo 1698555 Japan

In this paper, we describe the applicability of algebraic multigrid method (AMG method) as a fast solution for magnetic field analysis of an electric machine with a laminated core. In an experimental analysis, we have found that there are some cases where the calculation speed can be improved by using the AMG method for a fine mesh that models a laminated iron core exactly. Therefore we investigated the convergence characteristic of the standard Incomplete Cholesky Conjugate Gradient method (standard ICCG method) and the Conjugate Gradient method preconditioned with AMG method (AMGCG method) for the simple and actual analysis models. The numerical results show that the AMGCG method provides fast solutions for laminated mesh models for which the standard ICCG method does not work well in many analysis cases.

关键词： algebraic multigrid method finite element methods iterative methods laminated core

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An algebraic multigrid method for a class of elliptic differential systems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2003年第1期25卷 302-323页

作者： Borzì, A Borzì, G Karl Franzens Univ Graz Inst Math A-8010 Graz Austria Univ Messina Dept Math I-98166 Messina Italy

The development, numerical investigation, and analysis of an algebraic multigrid (AMG) method for the numerical solution of a representative class of elliptic differential systems are presented. Our AMG scheme is based on the AMG strategy of Brandt, Ruge, and Stuben. However, the present approach may be implemented within other AMG schemes. By means of numerical experiments with model problems arising from some application areas, it is demonstrated that the computational performance of the proposed AMG scheme is comparable to that of AMG applied to a single scalar equation. An AMG convergence theory is presented that extends convergence results for AMG methods for scalar problems to the case of AMG applied to elliptic systems. Some details of our AMG implementation are given.

关键词： systems of elliptic partial differential equations algebraic multigrid method optimal control problems diffusion-convection-reaction problems

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A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticity

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2009年第7期16卷 535-559页

作者： Xiao, Yingxiong Shu, Shi Zha, Tuyan Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Peoples R China Xiangtan Univ Civil Engn & Mech Coll Xiangtan 411105 Peoples R China Hunan Inst Engn Dept Math & Phys Xiangtan 411104 Peoples R China

In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright (C) 2008 John Wiley & Sons, Ltd.

关键词： algebraic multigrid method higher-order elements plane strain problem unstructured grid

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An algebraic multigrid method for finite element systems on criss-cross grids

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2006年第1-3期25卷 287-304页

作者： Shu, Shi Xu, Jinchao Yang, Ying Yu, Haiyuan Xiangtan Univ Inst Computat & Appl Math Xiangtan Peoples R China Penn State Univ Dept Math University Pk PA 16802 USA Penn State Univ Ctr Computat Math & Applicat University Pk PA 16802 USA Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math & Sci Engn Comp Beijing 100864 Peoples R China

In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on criss-cross grids and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element systems that can be reduced to a smaller system by first eliminating certain degrees of freedoms. The algebraic multigrid method that we construct is analogous to many other algebraic multigrid methods for more complicated problems such as unstructured grids, but, because of the specialty of our problem, we are able to provide a rigorous convergence analysis to our algebraic multigrid method.

关键词： algebraic multigrid method finite element criss-cross grids convergence analysis

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A fast solver for FEM analyses using the parallelized algebraic multigrid method

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IEEE TRANSACTIONS ON MAGNETICS 2002年第2期38卷 369-372页

作者： Mifune, T Iwashita, T Shimasaki, M Kyoto Univ Dept Elect Engn Kyoto 6068501 Japan Kyoto Univ Data Proc Ctr Kyoto 6068501 Japan

The algebraic multigrid (AMG) method is an efficient solver for linear systems arising in finite element analyses. The AMG method is applicable at a matrix level, different from the geometric multigrid solvers. This paper proposes a combination of the parallel processing technique and the AMG method as a fast solver for electromagnetic field analyses. While the AMG method consists of a setup phase and a solution phase, parallel processing of the former phase is difficult. We present the use of long-range interpolation instead of the conventional direct interpolation for improvement of the parallel efficiency of the AMG setup phase. A magnetostatic analysis and an eddy-current analysis show the solver performance. The numerical results show that parallelized AMG is a fast solver and has sufficient scalability, as compared with the conventional solver.

关键词： algebraic multigrid method finite element methods long-range interpolation parallel processing

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Strategies with algebraic multigrid method for Coupled Systems

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2024年第1期45卷 251-261页

作者： Konshin, I. Terekhov, K. Vassilevski, Yu. Russian Acad Sci Fed Res Ctr Comp Sci & Control Moscow 119333 Russia Russian Acad Sci Marchuk Inst Numer Math Moscow 119333 Russia Sechenov First Moscow State Med Univ Moscow 119991 Russia Sirius Univ Sci & Technol Soci 354340 Russia

We consider strategies for algebraic multigrid-based solution of coupled systems arising from the anisotropic diffusion problem, the two-phase filtration problem, the Stokes and Navier-Stokes equations for incompressible fluid. We consider specific strategies for each problem: the direct application of the algebraic multigrid (AMG), the constrained pressure residual method with AMG preconditioner for the pressure block, the Bramble-Pasciak conjugate-gradient and biconjugate gradient stabilized methods with AMG preconditioner for the elliptic part of the (linearized) discrete operator. The strategies reveal an elliptic part of each system, which is efficiently addressed by the AMG method. As a result, the presented methods demonstrate linear complexity with respect to the size of the coupled problems.

关键词： algebraic multigrid method CPR method Bramble-Pasciak method mimetic finite difference Stokes equations Navier-Stokes equations

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Stationary Stokes solver for single-phase flow in porous media: A blastingly fast solution based on algebraic multigrid method using GPU

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ADVANCES IN WATER RESOURCES 2023年 171卷

作者： Evstigneev, Nickolay M. Ryabkov, Oleg I. Gerke, Kirill M. Russian Acad Sci Fed Res Ctr Comp Sci & Control Moscow Russia Russian Acad Sci Schmidt Inst Phys Earth Moscow Russia

The paper is focused on high efficiency Stokes solver that is applied to the incompressible flow in porous media. Computational domains are represented by binarized 3D computed tomography voxel models. The solution procedure is constructed around a fast algebraic multigrid solver that utilizes the power of graphics processing units (GPUs). In order to minimize memory footprint and accelerate the solver, a simple MAC-type staggered finite difference discretization is used and a coupled Stokes saddle-point type system is solved directly on a GPU. The MAC discretization is discussed, taking particular care of internal solid boundaries around pores. The method includes topological domain analysis on a GPU, which removes isolated volumes with no flow in the domain (based on the connected component labeling), depending on the boundary conditions. We consider various types of boundary conditions and efficient parallel strategies for the GPUs, including fast matrix assembly and residual regularization. Our method is extensively benchmarked both against analytical solutions and applied problems from digital rock physics in terms of computational wall time, precision, approximation order, and convergence. We demonstrate that it takes up to 5-23 s on a modern GPU card to obtain a solution with 1.10-6 error residuals for 3D geometries with 300-4503 voxels and a porosity range of 5-37%.

关键词： Stokes flow Porous media algebraic multigrid method General purpose GPU Krylov subspace solvers Permeability Permeability tensor

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Learning-based local weighted least squares for algebraic multigrid method

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JOURNAL OF COMPUTATIONAL PHYSICS 2023年第1期493卷

作者： Wang, Fan Gu, Xiang Sun, Jian Xu, Zongben Xi An Jiao Tong Univ Sch Math & Stat 28 Xianning West Rd Xian 710049 Shaanxi Peoples R China

algebraic multigrid (AMG) is an effective iterative algorithm for solving large-scale linear systems. One challenge of constructing the AMG algorithm is the determination of the prolongation operator, which affects the convergence rate of AMG and is problemdependent. In this paper, we propose a new Learning-based Local Weighted Least Squares (L-LWLS) method to construct the prolongation operator of AMG. Specifically, we construct the prolongation operator by solving the LWLS model with learned spatially-varying weights. We use the gradient descent algorithm to optimize the model with a learned initialization of the solution. Then the constructed prolongation operator is further corrected by a learned correction function to improve the convergence rate of AMG. We conduct experiments on solving graph Laplacian linear systems, diffusion partial differential equations, and Helmholtz equations. Experiments show that the proposed method can construct a better prolongation operator leading to a faster convergence rate than the compared methods, including the classical AMG, the smoothed aggregation AMG, the bootstrap AMG, and the learning-based AMG method. The results show that the proposed method can generalize well to different parameter distributions and problem sizes, i.e., the number of variables in the linear system.

关键词： algebraic multigrid method Prolongation operator Local weighted least squares model Deep learning

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An efficient algebraic multigrid method for second-order elliptic equation on polygonal domains

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2021年第0期

作者： Li, Ming Honghe Univ Dept Math & Stat Mengzi 661100 Yunnan Peoples R China

Based on a coarsening strategy of adjacency matrix, a new algebraic prolongation operator is developed for standard V-cycle multigrid method to accelerate the whole process. An efficient algebraic multigrid (EAMG) method is proposed for solving large-scale linear system, which arise from finite element (FE) discretization of second-order elliptic boundary value problem. Numerical experiments on polygonal domains are conducted to demonstrate that the EAMG computation is more efficient than standard method.

关键词： algebraic multigrid method finite element method polygonal domain prolongation operator

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