Journal Article10.1137/0915085
An efficient block-oriented approach to parallel sparse Cholesky factorization
Edward Rothberg,Anoop Gupta +1 more
91
TL;DR: An approach that is simple to implement, provides slightly higher performance than column (and panel) methods on small parallel machines, and has the potential to provide much higher performance on large parallel machines is described and evaluated.
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Abstract: This paper explores the use of a subblock decomposition strategy for parallel sparse Cholesky factorization in which the sparse matrix is decomposed into rectangular blocks. Such a strategy has enormous theoretical scalability advantages over more traditional column-oriented and panel-oriented decompositions. However, little progress has been made in producing a practical subblock method. This paper describes and evaluates an approach that is simple to implement, provides slightly higher performance than column (and panel) methods on small parallel machines, and has the potential to provide much higher performance on large parallel machines.
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Citations
A Supernodal Approach to Sparse Partial Pivoting
TL;DR: A sparse LU code is developed that is significantly faster than earlier partial pivoting codes and compared with UMFPACK, which uses a multifrontal approach; the code is very competitive in time and storage requirements, especially for large problems.
A survey of direct methods for sparse linear systems
TL;DR: 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.
254
PASTIX: a high-performance parallel direct solver for sparse symmetric positive definite systems
Pascal Hénon,Pierre Ramet,Jean Roman +2 more
- 01 Feb 2002
TL;DR: The block partitioning and scheduling problem for sparse parallel factorization without pivoting is considered, and the scalability of the parallel solver and the compromise between memory overhead and efficiency are considered.
246
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
TL;DR: An efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A = LDLT, where L is lower triangular and D is diagonal is described.
Developments and trends in the parallel solution of linear systems
Iain S. Duff,Henk A. van der Vorst +1 more
- 01 Dec 1999
TL;DR: This review paper considers some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism and considers preconditioning techniques for iterative solvers.
References
The Stanford Dash multiprocessor
Daniel E. Lenoski,James Laudon,Kourosh Gharachorloo,Wolf-Dietrich Weber,Abhinav Gupta,John L. Hennessy,Mark Horowitz,Monica S. Lam +7 more
TL;DR: The directory architecture for shared memory (Dash) as discussed by the authors allows shared data to be cached, significantly reducing the latency of memory accesses and yielding higher processor utilization and higher overall performance, and a distributed directory-based protocol that provides cache coherence without compromising scalability.
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The role of elimination trees in sparse factorization
TL;DR: The role of elimination trees in the direct solution of large sparse linear systems is examined and its relation to sparse Cholesky factorization is discussed.
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Modification of the minimum-degree algorithm by multiple elimination
TL;DR: Experimental results indicate that the modified version of the minimum-degree algorithm retains the fill-reducing property of (and is often better than) the original ordering algorithm and yet requires less computer time.
408
A New Implementation of Sparse Gaussian Elimination
TL;DR: An implementation of sparse ${LDL}^T$ and LU factorization and back-substitution, based on a new scheme for storing sparse matrices, is presented and appears to be as efficient in terms of work and storage as existing schemes.
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