Free distribution parallel sparse solvers. application to lambda modes equation

IADIS International Conference Applied Computing 2006

FREE DISTRIBUTION PARALLEL SPARSE SOLVERS. APPLICATION TO LAMBDA MODES EQUATION
Omar Flores Sánchez
Universidad Politécnica de Valencia, Departamento de Sistemas y Computación--DSIC Instituto Tecnológico de Tuxtepec, Departamento de Sistemas y Computación--DSC Av. Dr. Victor Bravo Ahuja s/n, Col. 5 de Mayo, A.P. 69, C.P.68300, Tuxtepec,Oaxaca, México oflores@dsic.upv.es

Vicente E. Vidal Gimeno
Universidad Politécnica de Valencia, Departamento de Sistemas y Computación--DSIC Camino de Vera s/n 46022, Valencia, España vvidal@dsic.upv.es

ABSTRACT This paper discusses how High Performance Computing can help to solve Engineering problems, where is necessary to reduce the execution time spent in the solution of sparse linearsystems. We have chosen two free-distribution numerical parallel libraries called PETSc (Portable, Extensible Toolkit for Scientific Computation) and pARMS (Parallel Algebraic Recursive Multilevel Solver). Both libraries have been applied to a realistic test case of Nuclear Engineering where is necessary to solve efficiently very-large sparse linear systems to study steady-state neutron diffusionprocesses. Numerical experiments have shown the effectiveness of using parallel and distributed computing. KEYWORDS Parallel Computing, Krylov Subspace Methods, Very-Large Sparse Linear Systems, Lambda Modes Equation.

1. INTRODUCTION
Physical phenomena are often modelled by equations that relate several partial derivatives of physical quantities, such as forces, momentums, velocities, energy,temperature, etc. These equations rarely have a closed-form (explicit) solution. The Partial Differential Equations (PDEs) constitute the biggest source of sparse matrix problems. The typical way to solve such equations is to discretize them, i.e., to approximate them by equations that involve a finite number of unknowns. The matrices that arise from these discretizations are generally large andsparse, i.e., they have very few nonzero entries. There are several different ways to discretize a Partial Differential Equation. The simplest method uses finite difference approximations for the partial differential operators. The Finite Element Method replaces the original function by a function which has some degree of smoothness over the global domain, but which is piecewise polynomial on simplecells, such as small triangles or rectangles. In between these two methods, there are a few conservative schemes called Finite Volume Methods, which attempt to emulate continuous conservation laws of physics. Traditionally, direct methods [Duff1986][Demmel2003] have been used for solving linear systems of equations due to their robustness and predictable behaviour. However, iterative methods[Saad1996] have shown a good competency when they are combined with preconditioning techniques and Krylov subspace iterations, giving rise to efficient and simple general purpose procedures. The main objective of this work is to show the advantages of using High Performance Computing tools such as numerical parallel libraries and PCs clusters to accelerate computing processes in Engineering problems. Inparticular, we have studied the PETSc and pARMS parallel libraries applied to the solution of the linear systems of equations related to the lambda modes equation that appears in stability and security analysis of nuclear reactors. In order to reach our objective, we have made numerical experiments with both

115

IADIS International Conference Applied Computing 2006

libraries. They areanalyzed and evaluated from speedup and efficiency [Kumar1994] points of view. The rest of this paper is organized as follows. Section 2 gives a general overview of the methods and preconditioners contained into PETSc and pARMS. The realistic test case is covered in Section 3. Section 4 is devoted to parallel numerical experiments. Some conclusions are drawn in Section 5.

2. SPARSE SOLVERS
At...