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Applied Numerical Linear Algebra
 Society for Industrial and Applied Mathematics
, 1997
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate ..."
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Cited by 532 (26 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
The Design of a New Frontal Code for Solving Sparse Unsymmetric Systems
, 1996
"... We describe the design, implementation, and performance of a frontal code for the solution of large sparse unsymmetric systems of linear equations. The resulting software package, MA42, is included in Release 11 of the Harwell Subroutine Library and is intended to supersede the earlier MA32 package. ..."
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Cited by 35 (17 self)
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We describe the design, implementation, and performance of a frontal code for the solution of large sparse unsymmetric systems of linear equations. The resulting software package, MA42, is included in Release 11 of the Harwell Subroutine Library and is intended to supersede the earlier MA32 package. We discuss in detail the extensive use of higher level BLAS kernels within MA42 and illustrate the performance on a range of practical problems on a CRAY YMP, an IBM 3090, and an IBM RISC System/6000. We examine extending the frontal solution scheme to use multiple fronts to allow MA42 to be run in parallel. We indicate some directions for future development. Keywords : sparse unsymmetric linear equations, unsymmetric frontal method, Gaussian elimination, finiteelement equations, level 3 BLAS, parallel processing. AMS(MOS) subject classification : 65F05, 65F50. CR classification system : G.1.3. Computing and Information Systems Department, Atlas Centre, Rutherford Appleton Laborator...
The use of multiple fronts in Gaussian elimination
 in Gaussian Elimination, Proc. Fifth SIAM Conf. on Applied Linear Algebra, SIAM
, 1994
"... We examine a method for extending a frontal solution scheme principally so that parallelism can be exploited in the solution process. We see also that this technique can reduce the amount of work required and enable the solution of very large problems even on uniprocessors. Keywords: sparse matrice ..."
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Cited by 17 (9 self)
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We examine a method for extending a frontal solution scheme principally so that parallelism can be exploited in the solution process. We see also that this technique can reduce the amount of work required and enable the solution of very large problems even on uniprocessors. Keywords: sparse matrices, frontal methods, parallel processing, PVM, domain decomposition. AMS(MOS) subject classifications: 65F05, 65F50. 1 Extended and revised version of paper in Proceedings of the Fifth SIAM Conference on Applied Linear Algebra. Edited by John Lewis. SIAM Press, 567571. 2 Current reports available by anonymous ftp from camelot.cc.rl.ac.uk (internet 130.246.8.61) in the directory "pub/reports". This report is in file dsRAL94040.ps.Z. Central Computing Department Atlas Centre Rutherford Appleton Laboratory Oxon OX11 0QX September 13, 1994. Contents 1 Introduction 1 2 Frontal schemes 1 3 The use of multiple fronts 3 4 The use of MA42 for multiple front algorithms 4 4.1 MA42 code : :...
MA42  A new frontal code for solving sparse unsymmetric systems
, 1993
"... We describe the design, implementation, and performance of a frontal code for the solution of large sparse unsymmetric systems of linear equations. The resulting software package, MA42, is included in Release 11 of the Harwell Subroutine Library and is intended to supersede the earlier MA32 package. ..."
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Cited by 10 (8 self)
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We describe the design, implementation, and performance of a frontal code for the solution of large sparse unsymmetric systems of linear equations. The resulting software package, MA42, is included in Release 11 of the Harwell Subroutine Library and is intended to supersede the earlier MA32 package. We discuss in detail design changes from the earlier code, indicating the way in which they aid clarity, maintainability, and portability. The new design also permits extensive use of higher level BLAS kernels, which aid both modularity and efficiency. We illustrate the performance of our new code on practical problems on a CRAY YMP, an IBM 3090, and an IBM RISC System/6000. We indicate some directions for future development.
On The LU Factorization Of Sequences Of Identically Structured Sparse Matrices Within A Distributed Memory Environment
, 1994
"... : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii CHAPTERS 1 INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Topic Statement : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Overview : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Cited by 10 (1 self)
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: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii CHAPTERS 1 INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Topic Statement : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 BACKGROUND AND RELATED EFFORTS : : : : : : : : : : : : : : 5 2.1 LU Factorization : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Algorithm Stability and Error Analysis : : : : : : : : : : : : : : : 6 2.3 Sparse Matrix Concepts : : : : : : : : : : : : : : : : : : : : : : : 11 2.4 Multifrontal Methods : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.5 Factorization Sequences of Matrices : : : : : : : : : : : : : : : : : 24 2.6 Parallel Matrix Computations : : : : : : : : : : : : : : : : : : : : 28 2.7 Multiprocessor Scheduling : : : : : : : : : : : : : : : : : : : : : : 32 3 IMPLEMENTATION PLATFORM : : : : : : : : : : : : : : : : : : : : 37 3.1 Hardware Ar...
A Parallel Frontal Solver For Large Scale Process Simulation and Optimization
, 1996
"... For the simulation and optimization of largescale chemical processes, the overall computing time is often dominated by the time needed to solve a large sparse system of linear equations. We present here a new parallel frontal solver which can significantly reduce the wallclock time required to solv ..."
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Cited by 7 (5 self)
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For the simulation and optimization of largescale chemical processes, the overall computing time is often dominated by the time needed to solve a large sparse system of linear equations. We present here a new parallel frontal solver which can significantly reduce the wallclock time required to solve these linear equation systems using parallel/vector supercomputers. The algorithm exploits both multiprocessing and vector processing by using a multilevel approach in which frontal elimination is used for the partial factorization of each front. Results on several large scale process simulation and optimization problems are presented. 1 Introduction The solution of realistic, industrialscale simulation and optimization problems is computationally very intense, and may require the use of high performance computing technology to be done in a timely manner. For example, Zitney et al. (1995) described a dynamic simulation problem at Bayer AG requiring 18 hours of CPU time on a CRAY C90 sup...
Parallel frontal solvers for large sparse linear systems
, 2002
"... Many applications in science and engineering give rise to large sparse linear systems of equations that need to be solved as efficiently as possible. As the size of the problems of interest increases, it can become necessary to consider exploiting multiprocessors to solve these systems. We report on ..."
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Cited by 7 (1 self)
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Many applications in science and engineering give rise to large sparse linear systems of equations that need to be solved as efficiently as possible. As the size of the problems of interest increases, it can become necessary to consider exploiting multiprocessors to solve these systems. We report on the design and development of parallel frontal solvers for the numerical solution of large sparse linear systems. Three codes have been developed for the mathematical software library HSL (www.cse.clrc.ac.uk/Activity/HSL). The first is for unsymmetric finiteelement problems; the second is for symmetric positive definite finiteelement problems; and the third is for highly unsymmetric linear systems such as those that arise in chemical process engineering. In each case, the problem is subdivided into a small number of loosely connected subproblems and a frontal method is then applied to each of the subproblems in parallel. We discuss how our software is designed to achieve the goals of portability, ease of use, efficiency, and flexibility, and illustrate the performance on an SGI Origin 2000 using problems arising from real applications.
Parallel Direct Methods For Sparse Linear Systems
, 1997
"... We present an overview of parallel direct methods for solving sparse systems of linear equations, focusing on symmetric positive definite systems. We examine the performance implications of the important differences between dense and sparse systems. Our main emphasis is on parallel implementation of ..."
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Cited by 6 (0 self)
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We present an overview of parallel direct methods for solving sparse systems of linear equations, focusing on symmetric positive definite systems. We examine the performance implications of the important differences between dense and sparse systems. Our main emphasis is on parallel implementation of the numerically intensive factorization process, but we also briefly consider the other major components of direct methods, such as parallel ordering. Introduction In this paper we present a brief overview of parallel direct methods for solving sparse linear systems. Paradoxically, sparse matrix factorization offers additional opportunities for exploiting parallelism beyond those available with dense matrices, yet it is often more difficult to attain good efficiency in the sparse case. We examine both sides of this paradox: the additional parallelism induced by sparsity, and the difficulty in achieving high efficiency in spite of it. We focus on Cholesky factorization, primarily because th...
Direct Methods
, 1998
"... We review current methods for the direct solution of sparse linear equations. We discuss basic concepts such as fillin, sparsity orderings, indirect addressing and compare general sparse codes with codes for dense systems. We examine methods for greatly increasing the efficiency when the matrix is ..."
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Cited by 4 (0 self)
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We review current methods for the direct solution of sparse linear equations. We discuss basic concepts such as fillin, sparsity orderings, indirect addressing and compare general sparse codes with codes for dense systems. We examine methods for greatly increasing the efficiency when the matrix is symmetric positive definite. We consider frontal and multifrontal methods emphasizing how they can take advantage of vectorization, RISC architectures, and parallelism. Some comparisons are made with other techniques and the availability of software for the direct solution of sparse equations is discussed.
Element Resequencing for Use With a Multiple Front Algorithm
, 1995
"... The multiple front algorithm is an extension of the frontal method to allow parallelism to be exploited in the solution process. The finiteelement domain is partitioned into a number of subdomains and a frontal decomposition is performed on each subdomain separately. For a given partitioning of the ..."
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Cited by 2 (1 self)
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The multiple front algorithm is an extension of the frontal method to allow parallelism to be exploited in the solution process. The finiteelement domain is partitioned into a number of subdomains and a frontal decomposition is performed on each subdomain separately. For a given partitioning of the domain, the efficiency of the multiple front algorithm depends on the ordering of the elements within each subdomain. We look at the limitations of existing element reordering algorithms when applied to a subdomain and consider how these limitations may be overcome. Extensive numerical experiments are performed on a range of practical problems and, on the basis of the results, we propose a new element resequencing algorithm for use with a multiple front algorithm. Keywords : sparse matrices, frontal methods, Gaussian elimination, finiteelement equations. AMS(MOS) subject classification : 65F05, 65F50. CR classification system : G.1.3[Numerical Linear Algebra]: linear systems: sparse syst...