Iterative methods for nonsymmetric systems on MIMD machines

A wide variety of physical phenomena arising within many scientific disciplines can be described by systems of coupled partial differential equations (PDEs). The numerical approximation of these PDEs often involves the solution of a system of algebraic equations (possibly nonlinear) which are typically large, sparse and nonsymmetric. The increasing computational demands required by the solution of such complex scientific applications has motivated the current direction toward large-scale parallel computers. We, therefore, consider solution techniques of representative systems of equations on large scale MIMD machines. Our primary emphasis in this study is the evaluation of iterative methods for the solution of nonsymmetric systems. In particular, we discuss two Krylov subspace methods, the conjugate gradient squared algorithm (CGS) and the generalized minimum residual method (GMRES), along with the multigrid algorithm (MG) on massively parallel MIMD architectures. The focus of this evaluation considers the performance of various algorithm and implementation variations over a broad selection of problems using a parallel machine

Aztec user`s guide. Version 1

Aztec is an iterative library that greatly simplifies the parallelization process when solving the linear systems of equations Ax = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. Aztec is intended as a software tool for users who want to avoid cumbersome parallel programming details but who have large sparse linear systems which require an efficiently utilized parallel processing system. A collection of data transformation tools are provided that allow for easy creation of distributed sparse unstructured matrices for parallel solution. Once the distributed matrix is created, computation can be performed on any of the parallel machines running Aztec: nCUBE 2, IBM SP2 and Intel Paragon, MPI platforms as well as standard serial and vector platforms. Aztec includes a number of Krylov iterative methods such as conjugate gradient (CG), generalized minimum residual (GMRES) and stabilized biconjugate gradient (BICGSTAB) to solve systems of equations. These Krylov methods are used in conjunction with various preconditioners such as polynomial or domain decomposition methods using LU or incomplete LU factorizations within subdomains. Although the matrix A can be general, the package has been designed for matrices arising from the approximation of partial differential equations (PDEs). In particular, the Aztec package is oriented toward systems arising from PDE applications