We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient 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, the singular value decomposition, and generalizations of these to two matrices. We consider dense, band and sparse matrices.

This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient–type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell–Boeing collection and from Tim Davis’s collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.

Abstract not found

In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equations by direct and iterative methods. We consider preconditioning techniques for iterative solvers and discuss some of the present research issues in this field. Keywords: linear systems, dense matrices, sparse matrices, tridiagonal systems, parallelism, direct methods, iterative methods, Krylov methods, preconditioning. AMS(MOS) subject classifications: 65F05, 65F50. 1 Introduction Solution methods for systems of linear equations Ax = b; (1) where A is a coefficient matrix of order n and x and b are n-vectors, are usually grouped into two distinct classes: direct methods and iterative methods. However, CCLRC - Rutherford Appleton Laboratory, Oxfordshire, England and CERFACS, Toulouse,...

In this chapter, we give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g. computed via Gaussian elimination) by disallowing certain fill-ins. As opposed to other PDE-based preconditioners such as multigrid and domain decomposition, this class of preconditioners are primarily algebraic in nature and can in principle be applied to any sparse matrices. When applied to PDE problems, they are usually not optimal in the sense that the condition number of the preconditioned system will grow as the mesh size h is reduced, although usually at a slower rate than for the unpreconditioned system. On the other hand, they are often quite robust with respect to other more algebraic features of the problem such as rough and anisotropic coefficients and strong convection terms. We will describe the basic ILU and (modified) MILU preconditioners. Then we will review ...

This paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century. Although iterative methods for solving linear systems find their origin in the early nineteenth century (work by Gauss),the field has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences. The past five decades have been particularly rich in new developments,ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientific and industrial computational models. As in any other scientific area,research in iterative methods has been a journey characterized by a chain of contributions building on each other. It is the aim of this paper not only to sketch the most significant of these contributions during the past century,but also to relate them to one another.

Many loop nests in scientific codes contain a parallelizable outer loop but have an inner loop for which the number of iterations varies between different iterations of the outer loop. When running this kind of loop nest on a SIMD machine, the SIMD-inherent restriction to single program counter common to all processors will cause a performance degradation relative to comparable MIMD implementations. This problem is not due to limited parallelism or bad load balance, it is merely a problem of control flow. This paper presents a loop transformation, which we call loop flattening, that overcomes this limitation by letting each processor advance to the next loop iteration containing useful computation, if there is such an iteration for the given processor. We study a concrete example derived from a molecular dynamics code and compare performance results for flattened and unflattened versions of this kernel on two SIMD machines, the CM-2 and the DECmpp 12000. We then evaluate loop flattenin...

In these notes we will present anoverview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and they include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, CGS, Bi-CGSTAB, QMR, LSQR and GMRES. We will showhow these methods can be derived from simple basic iteration formulas. We will not give convergence proofs, but we will refer for these, as far as available, to litterature. Iterative methods are often used in combination with so-called preconditioning operators (approximations for the inverses of the operator of the system to be solved). Since these preconditioners are not essential in the derivation of the iterative methods, we will not givemuch attention to them in these notes. However, in most of the actual iteration schemes, we have included them in order to facilitate the use of these schemes in actual computations. For the application of the iterative schemes one usually thinks of linear sparse systems, e.g., like those arising in the nite element or nite di erence approximations of (systems of) partial di erential equations. However, the structure of the operators plays no explicit role in any oftheseschemes, and these schemes might also successfully be used to solve certain large dense linear systems. Depending on the situation that might be attractive in terms of numbers of oating point operations. It will turn out that all of the iterative are parallelizable in a straight forward manner. However, especially for computers with a memory hierarchy (i.e., like cache or vector registers), and for distributed memory computers, the performance can often be improved signi cantly through rescheduling of the operations. We will discuss parallel implementations, and occasionally we will report on experimental ndings.

Introduction In these notes we will present an overview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and they include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, LSQR and GMRES. We will show how these methods can be derived from simple basic iteration formulas. We will not give convergence proofs, but we will refer for these, as far as available, to litterature. Iterative methods are often used in combination with so-called preconditioning operators (approximations for the inverses of the operator of the system to be solved). Since these preconditioners are not essential in the derivation of these iterative methods, we will not discuss on them explicitly in these notes. However, in most of the actual iteration schemes, we have included them in order to facilitate the use of these schemes in actual computations. For the application of the iterative schemes

. This report describes a parallelization of the University of Houston Brownian Dynamics (UHBD) program which has two complimentary stages. In the first, an electrostatics problem is solved for a macromolecule in solution. Then "Brownian dynamics" simulates the diffusion of molecules in the solution subject to the electrostatic forces. The most challenging part of the work is the parallel computation of electrostatic forces. Here we describe a parallel preconditioner for conjugate gradients which is easy to implement and scalable. We report experimental results both for this preconditioner and for the calculation of Brownian trajectories. We are finalizing experiments with larger numbers of processors on the Intel Delta machine and on other parallel processors including the KSR1. These results will augment data presented here and will appear in the final paper. 1. Introduction. This report describes a parallelization of the University of Houston Brownian Dynamics (UHBD) program [2] whi...