An out-of-core sparse symmetric-indefinite factorization method

Meshar, Omer; Irony, Dror; Toledo, Sivan

doi:10.1145/1163641.1163645

Cited by 15 publications

(11 citation statements)

References 21 publications

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“…The factorized Q + λ S is used in the inner loop of the eigen solver (Line 4). To factorize Q + λ S , we used the sparse OOC (out-of-core) symmetric indefinite factorization [Meshar et al 2006] implemented in the future release of TAUCS, kindly provided by S. Toledo. We then recover the λ 's from the µ's (Line 6) and stream-write the new eigenpairs into a file (Line 7).…”

Section: Numerical Solution Mechanismmentioning

confidence: 99%

Spectral Geometry Processing with Manifold Harmonics

Vallet

Lévy

2008

Computer Graphics Forum

333

304

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We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices.In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.

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Section: Numerical Solution Mechanismmentioning

confidence: 99%

Spectral Geometry Processing with Manifold Harmonics

Vallet

Lévy

2008

Computer Graphics Forum

333

304

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“…The thesis surveys some out-of-core solvers including [50,97,147,169,186], provides NP-completeness results for the problem of minimising I/O volume in certain variations of factorisation methods, and also develops polynomial time algorithms for some other variations. A significant contribution of the thesis (also available in [2]) is the demonstration of the difference between the problem of minimising the I/O volume and that of minimising the active memory size (as is done by Liu [136]).…”

Section: Elimination Tree and The Multifrontal Methodsmentioning

confidence: 99%

Combinatorial Problems in Solving Linear Systems

Duff¹,

Uçar²

2012

Combinatorial Scientific Computing

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Abstract. Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.

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“…For our experimental evaluations, we used five applications: PSTT (parallel sparse FFT) [5], PaSTiX (a high performance parallel solver for very large sparse linear systems based on direct methods) [6], SSIF (sparse symmetric indefinite factorization) [26], PPS (a persistent, pervasive surveillance code), and REACT (Jacobian based combustion modeling code). The last two applications are written by us.…”

Section: Implementation Benchmarks and Setupmentioning

confidence: 99%

Trading cache hit rate for memory performance

Ding

Kandemir

Guttman

et al. 2014

Proceedings of the 23rd International Conference on Parallel Architectures and Compilation

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Most of the prior compiler based data locality optimization works target exclusively cache locality optimization, and row-buffer locality in DRAM banks received much less attention. In particular, to the best of our knowledge, there is no single compiler based approach that can improve row-buffer locality in executing irregular applications. This presents a critical problem considering the fact that executing irregular applications in a power and performance efficient manner will be a key requirement to extract maximum benefits from emerging multicore machines and exascale systems. Motivated by these observations, this paper makes the following contributions. First, it presents a compiler-runtime cooperative data layout optimization approach that takes as input an irregular program that has already been optimized for cache locality and generates an output code with the same cache performance but better row-buffer locality (lower number of row-buffer misses). Second, it discusses a more aggressive strategy that sacrifices some cache performance in order to further improve row-buffer performance (i.e., it trades cache performance for memory system performance). The ultimate goal of this strategy is to find the right tradeoff point between cache performance and rowbuffer performance so that the overall application performance is improved. Third, the paper performs a detailed evaluation of these two approaches using both an AMD Opteron based multicore system and a multicore simulator. The experimental results, collected using five real-world irregular applications, show that (i) conventional cache optimizations do not improve row-buffer locality significantly; (ii) our first approach achieves about 9.8% execution time improvement by keeping the number of cache misses the same as a cache-optimized code but reducing the number of row-buffer misses; and (iii) our second approach achieves even higher execution time improvements (13.8% on average) by sacrificing cache performance for additional memory performance.

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An out-of-core sparse symmetric-indefinite factorization method

Cited by 15 publications

References 21 publications

Spectral Geometry Processing with Manifold Harmonics

Spectral Geometry Processing with Manifold Harmonics

Combinatorial Problems in Solving Linear Systems

Trading cache hit rate for memory performance

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