Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.

A survey of direct methods for sparse linear systems

Some aspects of the parallel and distributed iterative algorithms—a survey

We consider iterative algorithms of the form x := ƒ(x), executed by a parallel or distributed computing system. We focus on asynchronous implementations whereby each processor iterates on a different component of x, at its own pace, using the most recently received (but possibly outdated) information on the remaining components of x. We provide results on the convergence rate of such algorithms and make a comparison with the convergence rate of the corresponding synchronous methods in which the computation proceeds in phases. We also present results on how to terminate asynchronous iterations in finite time with an approximate solution of the computational problem under consideration.

Convergence rate and termination of asynchronous iterative algorithms

Circuit simulation is a very time-consuming and numerically intensive application, especially when the problem size is large as in the case of VLSI circuits. To improve the performance of circuit simulators without sacrificing accuracy, a variety of parallel processing algorithms have been investigated. Research in the field of parallel circuit simulation is surveyed, and the ongoing research in this area at the University of Illinois is described. Both standard and relaxation-based approaches are considered. In particular, the forms of parallelism available within the direct method approach, used in programs such as SPICE2 and SLATE, and within the relaxation-based approaches, such as waveform relaxation, iterated timing analysis, and waveform-relaxation-Newton, are described. The specific implementation issues addressed are primarily related to general-purpose multiprocessors with a shared-memory architecture having a limited number of processors, although many of the comments apply to a number of other architectures. >

Parallel circuit simulation on supercomputers

Reports the parallelization on a shared-memory vector multiprocessor of the computationally intensive components of a circuit simulator-matrix assembly (including device model evaluation) and the unstructured sparse linear system solution. A theoretical model is used to predict the performance of the lock-synchronized parallel matrix assembly, and the results are compared to experimental measurements. Alternate approaches to efficient sparse matrix solution are contrasted, highlighting the impact of the matrix representation/access strategy on achievable performance, and medium-grained approach with superior performance is introduced. The techniques developed have been incorporated into a prototype parallel implementation of the production circuit simulator ADVICE on the Alliant FX/8 multiprocessor. >

Circuit simulation on shared-memory multiprocessors

The authors examine two approaches for reducing parallel sparse matrix solution time: the first based on pivot ordering algorithms for Gaussian elimination, and the second based on relaxation algorithms A pivot ordering algorithm is presented which increases the parallelism of Gaussian elimination compared to the commonly used Markowitz method The minimum number of parallel steps for the solution of a tridiagonal matrix is derived, and it is shown that this optimum is nearly achieved by the ordering heuristics which attempt to maximize parallelism Also presented is an optimality result about Gauss-Jacobi over Gauss-Seidel relaxation on parallel processors >

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Reducing the parallel solution time of sparse circuit matrices using reordered Gaussian elimination and relaxation

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Reducing the parallel solution time of sparse circuit matrices using reordered Gaussian elimination and relaxation