Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a $\ell^1$-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.Comment: 18 pages, 3 figure

GPU Accelerated Three Dimensional Unstructured Geometric Multi-Grid Solver

Consider a set of points P in three dimensional euclidean space. For each point p, the neighbourhood N(p), is defined as the set of points in P, which are voronoi neighbours. Each point in P represents a variable and its value is dependent on the value of its neighbourhood. Its value is given by the sum of the values of points in its neighbourhood scaled by predefined constants. The constants depend on the spacing between the points. The problem is to solve all the variables. Such representations arise naturally in solving flow equations in Computational Fluid Dynamics with domains represented using unstructured meshes. The problem reduces to solve a system of linear equations. In this work geometric multigrid method is implemented for solving the problem faster. Solving this problem on very large input is a time consuming process. The inputs considered here are having size of the order of millions. Graphics Processing Units(GPU) are dedicated parallel processors which serves both as a programmable graphics processor and a scalable parallel computing platform. The parallelization of this problem for GPUs is not straight forward because of the irregularity. The CFD problem used for experiment is the steady and unsteady heat transfer problem in 3D unstructured meshes.The combination of multigrid algorithm and GPU implementation for the steady problem on a 1.6 million mesh gives 1630 times speed up compared to non-multigrid CPU implementation

Estudo de parâmetros do método Multigrid geométrico para equações 2D em CFD e volumes finitos

Orientador : Prof. Dr. Carlos Henrique MarchiCoorientadores : Prof. Dr. Marcio Augusto Villela Pinto, Prof. Dr. Luciano Kiyoshi ArakiTese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Mecânica. Defesa: Curitiba, 26/02/2013Inclui referênciasÁrea de concentração: Fenômenos de transporte e mecânica dos sólidosResumo: A influencia de alguns parametros do metodo multigrid geometrico sobre o tempo de CPU para tres diferentes modelos matematicos bidimensionais do escopo da CFD (Computational Fluid Dynamics) e investigada. Os modelos matematicos sao: a equacao de Laplace, a equacao de Adveccao-Difusao e as Equacoes de Burgers. Os parametros em estudo sao: numero de iteracoes internas do solver (ƒË); numero de malhas (L); numero de incognitas (N); solvers e operadores de prolongacao. O multigrid e empregado com esquema FAS (Full Approximation Scheme) e tecnica FMG (Full Multigrid) com ciclo V e razao de engrossamento r = 2. As equacoes diferenciais sao discretizadas pelo Metodo dos Volumes Finitos (MVF) em geometrias simples e malhas bidimensionais uniformes por direcao, com aproximacoes de 2a ordem CDS e correcao adiada. As condicoes de contorno, do tipo Dirichlet, sao aplicadas mediante a tecnica de volumes ficticios. Os sistemas de equacoes algebricas sao resolvidos com o emprego do solver Gauss-Seidel Lexicografico (GS-Lex) e, no caso do problema de Burgers, tambem com o emprego do Gauss-Seidel red-black (GS-RB). Verificou-se principalmente que: o esquema FAS-FMG e cerca de duas vezes mais rapido do que o FAS padrao; que o numero de equacoes ou complexidade do problema nao interfere na eficiencia do multigrid; que o operador de prolongacao bilinear e o mais eficiente para interpolar as solucoes entre os niveis do FMG. Palavras-chave: Dinamica dos fluidos computacional. Multigrid. Volumes finitos. Metodos numericos. Equacoes de Burgers.This work investigates the influence of some parameters from the Multigrid Geometric method over CPU processing time for three different mathematical bidimensional methods that make up the Computational Fluid Dynamics scope. These mathematical models are: Laplace equation, Advection-Diffusion equation and Burgers' equations. In order to achieve the main target, which consists on optimize the employed algorithms to solve the problems above, the computational time minimization is sought through parameters modifications at the algorithms. The considered parameters are: number of solver's internal iteration (v); number of grids (L); number of incognites (N); solvers and prolongation operators. The multigrid is employed besides FAS (Full Approximation Scheme) and FMG (Full Multigrid) technique, with V cycle and coarsening ratio r = 2. The differential equations discretization is made by the Finite Volume Method (MVF) over simple geometries and direction uniform bidimensional grids, with second order CDS and delayed correction. The Dirichlet type boundary conditions are applied through fictitious volume technique. The system of algebraic equations are solved by the Gauss-Seidel Lexicographic (GS-Lex) solver and, at the Burgers problem, the Gauss-Seidel red-black (GS-RB) is also employed. The main results that should be emphasized are: the FAS-FMG scheme is about twice faster than the standard FAS; the multigrid efficiency ate not affected by the number of equations or complexity of the problem; the bilinear prolongation operator is the most efficient to interpolate the solution among the FMG levels. Keywords: computational fluid dynamics. Multigrid. Finite volume method. Numerical methods. Burgers' Equation