A Graph-Theoretic Study of the Numerical Solution of Sparse Positive Definite Systems of Linear Equations

Rose, Donald J.

doi:10.1016/b978-1-4832-3187-7.50018-0

Cited by 327 publications

(191 citation statements)

References 22 publications

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“…Besides classical bandwidth-reducing reordering techniques [15], recent work has proposed sophisticated 2-D partitioning schemes with theoretical guarantees on communication volume [18], and traveling salesman-based reordering to create dense block substructure [14].…”

Section: Oski Oski-petsc and Related Workmentioning

confidence: 99%

Optimization of sparse matrix–vector multiplication on emerging multicore platforms

et al. 2009

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We are witnessing a dramatic change in computer architecture due to the multicore paradigm shift, as every electronic device from cell phones to supercomputers confronts parallelism of unprecedented scale. To fully unleash the potential of these systems, the HPC community must develop multicore specific optimization methodologies for important scientific computations. In this work, we examine sparse matrix-vector multiply (SpMV) -one of the most heavily used kernels in scientific computing -across a broad spectrum of multicore designs. Our experimental platform includes the homogeneous AMD dual-core and Intel quad-core designs, the heterogeneous STI Cell, as well as the first scientific study of the highly multithreaded Sun Niagara2. We present several optimization strategies especially effective for the multicore environment, and demonstrate significant performance improvements compared to existing state-of-the-art serial and parallel SpMV implementations. Additionally, we present key insights into the architectural tradeoffs of leading multicore design strategies, in the context of demanding memory-bound numerical algorithms.

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Section: Oski Oski-petsc and Related Workmentioning

confidence: 99%

Optimization of sparse matrix–vector multiplication on emerging multicore platforms

et al. 2009

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“…During Gaussian eliminations of large sparse matrices, new non-zero elements -called fill -can replace original zeros, thus increasing storage requirements, the time needed for the elimination, and the time needed to solve the system after the elimination. The problem of finding the right elimination ordering minimizing the amount of fill elements can be expressed as the Minimum Fill-in problem on graphs [21]. Besides sparse matrix computations, applications of Minimum Fill-in can be found in database management, artificial intelligence, and the theory of Bayesian statistics.…”

Section: That Ismentioning

confidence: 99%

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

2013

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The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k 3/2 ) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log k) on planar graphs and O( √ k log k) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs. ACM Subject Classification G.2.2 Graph Theory -Graph Algorithms Keywords and phrases IntroductionA graph is chordal (or triangulated) if every cycle of length at least four has a chord, i.e. an edge between nonadjacent vertices of the cycle. In the Minimum Fill-in problem (also known as Minimum Triangulation and Chordal Graph Completion) the task is to check if at most k edges can be added to a graph such that the resulting graph is chordal. That isMinimum Fill-in Input: A graph G = (V, E) and a non-negative integer k.This is a classical computational problem motivated by, and named after, a fundamental issue arising in sparse matrix computations. During Gaussian eliminations of large sparse matrices, new non-zero elements -called fill -can replace original zeros, thus increasing storage requirements, the time needed for the elimination, and the time needed to solve the system after the elimination. The problem of finding the right elimination ordering * The research of Fedor V.

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“…Minimum updating matrix ordering. We present a graph model [20,21] for describing the factorization process as a series of node eliminations. The graph model is invaluable in providing an insight into the minimum discarded fill ordering.…”

Section: Minimum Updating Matrix Orderingmentioning

confidence: 99%

Preconditioned conjugate gradient methods for the incompressible Navier‐Stokes equations

Chin

D'Azevedo

Forsyth

et al. 1992

Numerical Methods in Fluids

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Abstract. A robust technique for solving primitive-variable formulations of the incompressibleNavier-Stokes equations is to use Newton iteration for the fully-implicit nonlinear equations. A direct sparse matrix method can be used to solve the Jacobian but is costly for large problems; an alternative is to use an iterative matrix method. This paper investigates effective ways of using a conjugate gradient type method with an incomplete LU factorization preconditioner for two-dimensional incompressible viscous flow problems. Special attention is paid to the ordering of unknowns, with emphasis on a minimum updating matrix (MUM) ordering. Numerical results are given for several test problems.

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A Graph-Theoretic Study of the Numerical Solution of Sparse Positive Definite Systems of Linear Equations

Cited by 327 publications

References 22 publications

Optimization of sparse matrix–vector multiplication on emerging multicore platforms

Optimization of sparse matrix–vector multiplication on emerging multicore platforms

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Preconditioned conjugate gradient methods for the incompressible Navier‐Stokes equations

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