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107
A fast and high quality multilevel scheme for partitioning irregular graphs
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 1998
"... Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. ..."
Abstract

Cited by 797 (12 self)
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Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc.
A column approximate minimum degree ordering algorithm
, 2000
"... Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A, where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q, based solely on the nonzero patt ..."
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Cited by 255 (52 self)
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Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A, where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q, based solely on the nonzero pattern of A such that the factorization remains as sparse as possible, regardless of the subsequent choice of P. The choice of Q can have a dramatic impact on the number of nonzeros in L and U. One scheme for determining a good column ordering for A is to compute a symmetric ordering that reduces fillin in the Cholesky factorization of ATA. This approach, which requires the sparsity structure of ATA to be computed, can be expensive both in
METIS  Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 2.0
, 1995
"... this paper is organized as follows: Section 2 briefly describes the various ideas and algorithms implemented in METIS. Section 3 describes the user interface to the METIS graph partitioning and sparse matrix ordering packages. Sections 4 and 5 describe the formats of the input and output files used ..."
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Cited by 122 (5 self)
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this paper is organized as follows: Section 2 briefly describes the various ideas and algorithms implemented in METIS. Section 3 describes the user interface to the METIS graph partitioning and sparse matrix ordering packages. Sections 4 and 5 describe the formats of the input and output files used by METIS. Section 6 describes the standalone library that implements the various algorithms implemented in METIS. Section 7 describes the system requirements for the METIS package. Appendix A describes and compares various graph partitioning algorithms that are extensively used.
An UnsymmetricPattern Multifrontal Method for Sparse LU Factorization
 SIAM J. MATRIX ANAL. APPL
, 1994
"... Sparse matrix factorization algorithms for general problems are typically characterized by irregular memory access patterns that limit their performance on parallelvector supercomputers. For symmetric problems, methods such as the multifrontal method avoid indirect addressing in the innermost loops ..."
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Cited by 118 (29 self)
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Sparse matrix factorization algorithms for general problems are typically characterized by irregular memory access patterns that limit their performance on parallelvector supercomputers. For symmetric problems, methods such as the multifrontal method avoid indirect addressing in the innermost loops by using dense matrix kernels. However, no efficient LU factorization algorithm based primarily on dense matrix kernels exists for matrices whose pattern is very unsymmetric. We address this deficiency and present a new unsymmetricpattern multifrontal method based on dense matrix kernels. As in the classical multifrontal method, advantage is taken of repetitive structure in the matrix by factorizing more than one pivot in each frontal matrix thus enabling the use of Level 2 and Level 3 BLAS. The performance is compared with the classical multifrontal method and other unsymmetric solvers on a CRAY YMP.
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 105 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
Solving unsymmetric sparse systems of linear equations with PARDISO
 Journal of Future Generation Computer Systems
, 2004
"... Supernode partitioning for unsymmetric matrices together with complete block diagonal supernode pivoting and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to ..."
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Cited by 91 (8 self)
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Supernode partitioning for unsymmetric matrices together with complete block diagonal supernode pivoting and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and unsymmetric prepermutation of rows is used to place large matrix entries on the diagonal. Complete block diagonal supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The level3 BLAS efficiency is retained and an advanced twolevel left–right looking scheduling scheme results in good speedup on SMP machines. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications. Key words: Computational sciences, numerical linear algebra, direct solver, unsymmetric linear systems
Analysis of multilevel graph partitioning
, 1995
"... Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity, and provide excellent graph partitions. However, there exists little theoretical analysis that could explain the ability of multileve ..."
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Cited by 90 (14 self)
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Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity, and provide excellent graph partitions. However, there exists little theoretical analysis that could explain the ability of multilevel algorithms to produce good partitions. In this paper we present such an analysis. We show under certain reasonable assumptions that even if no refinement is used in the uncoarsening phase, a good bisection of the coarser graph is worse than a good bisection of the finer graph by at most a small factor. We also show that the size of a good vertexseparator of the coarse graph projected to the finer graph (without performing refinement in the uncoarsening phase) is higher than the size of a good vertexseparator of the finer graph by at most a small factor.
SuperLU DIST: A scalable distributedmemory sparse direct solver for unsymmetric linear systems
 ACM Trans. Mathematical Software
, 2003
"... We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and sc ..."
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Cited by 87 (17 self)
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We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on largescale distributed machines.
Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate
 ACM Transactions on Mathematical Software
, 2008
"... CHOLMOD is a set of routines for factorizing sparse symmetric positive definite matrices of the form A or A A T, updating/downdating a sparse Cholesky factorization, solving linear systems, updating/downdating the solution to the triangular system Lx = b, and many other sparse matrix functions for b ..."
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Cited by 53 (7 self)
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CHOLMOD is a set of routines for factorizing sparse symmetric positive definite matrices of the form A or A A T, updating/downdating a sparse Cholesky factorization, solving linear systems, updating/downdating the solution to the triangular system Lx = b, and many other sparse matrix functions for both symmetric and unsymmetric matrices. Its supernodal Cholesky factorization relies on LAPACK and the Level3 BLAS, and obtains a substantial fraction of the peak performance of the BLAS. Both real and complex matrices are supported. CHOLMOD is written in ANSI/ISO C, with both C and MATLAB TM interfaces. It appears in MATLAB 7.2 as x=A\b when A is sparse symmetric positive definite, as well as in several other sparse matrix functions.