Results 11  20
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526
Robust approximate inverse preconditioning for the conjugate gradient method
 SIAM J. SCI. COMPUT
, 2000
"... We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdownfree and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly illcondit ..."
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Cited by 55 (11 self)
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We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdownfree and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly illconditioned linear systems. We also investigate an alternative approach to a stable approximate inverse algorithm, based on the idea of diagonally compensated reduction of matrix entries. The results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented.
ARMS: An Algebraic Recursive Multilevel Solver for general sparse linear systems
 Numer. Linear Alg. Appl
, 1999
"... This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The first subset which can be termed "coarse" is obtained by using "block" ..."
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Cited by 53 (24 self)
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This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The first subset which can be termed "coarse" is obtained by using "block" independent sets, or "aggregates". Two aggregates have no coupling between them, but nodes in the same aggregate may be coupled. The nodes not in the coarse set are part of what might be called the "Fringe" set. The idea of the methods is to form the Schur complement related to the fringe set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solver recursively using as preconditioner the factorization that could be obtained from the next level. Unlike other multilevel preconditioners available, iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other technique...
NLEVP: A Collection of Nonlinear Eigenvalue Problems
, 2010
"... We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems acco ..."
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Cited by 49 (12 self)
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We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear Eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.
Making Sparse Gaussian Elimination Scalable by Static Pivoting
 In Proceedings of Supercomputing
, 1998
"... We propose several techniques as alternatives to partial pivoting to stabilize sparse Gaussian elimination. From numerical experiments we demonstrate that for a wide range of problems the new method is as stable as partial pivoting. The main advantage of the new method over partial pivoting is th ..."
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Cited by 43 (6 self)
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We propose several techniques as alternatives to partial pivoting to stabilize sparse Gaussian elimination. From numerical experiments we demonstrate that for a wide range of problems the new method is as stable as partial pivoting. The main advantage of the new method over partial pivoting is that it permits a priori determination of data structures and communication pattern for Gaussian elimination, which makes it more scalable on distributed memory machines. Based on this a priori knowledge, we design highly parallel algorithms for both sparse Gaussian elimination and triangular solve and we show that they are suitable for largescale distributed memory machines. Keywords: sparse unsymmetric linear systems, static pivoting, iterative refinement, MPI, 2D matrix decomposition. 1 Introduction In our earlier work [8, 9, 22], we developed new algorithms to solve unsymmetric sparse linear systems using Gaussian elimination with partial pivoting (GEPP). The new algorithms are hi...
Approximating Betweenness Centrality
, 2007
"... Betweenness is a centrality measure based on shortest paths, widely used in complex network analysis. It is computationallyexpensive to exactly determine betweenness; currently the fastestknown algorithm by Brandes requires O(nm) time for unweighted graphs and O(nm + n 2 log n) time for weighted ..."
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Cited by 41 (4 self)
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Betweenness is a centrality measure based on shortest paths, widely used in complex network analysis. It is computationallyexpensive to exactly determine betweenness; currently the fastestknown algorithm by Brandes requires O(nm) time for unweighted graphs and O(nm + n 2 log n) time for weighted graphs, where n is the number of vertices and m is the number of edges in the network. These are also the worstcase time bounds for computing the betweenness score of a single vertex. In this paper, we present a novel approximation algorithm for computing betweenness centrality of a given vertex, for both weighted and unweighted graphs. Our approximation algorithm is based on an adaptive sampling technique that significantly reduces the number of singlesource shortest path computations for vertices with high centrality. We conduct an extensive experimental study on realworld graph instances, and observe that our random sampling algorithm gives very good betweenness approximations for biological networks, road networks and web crawls.
Strategies for scaling and pivoting for sparse symmetric indefinite problems
, 2004
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Towards a tighter coupling of bottomup and topdown sparse matrix ordering methods
 BIT
, 2001
"... Most stateoftheart ordering schemes for sparse matrices are a hybrid of a bottomup method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottomup and topdown methods. In our ..."
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Cited by 35 (0 self)
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Most stateoftheart ordering schemes for sparse matrices are a hybrid of a bottomup method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottomup and topdown methods. In our methodology vertex separators are interpreted as the boundaries of the remaining elements in an unfinished bottomup ordering. As a consequence, we are using bottomup techniques such as quotient graphs and special node selection strategies for the construction of vertex separators. Once all separators have been found, we are using them as a skeleton for the computation of several bottomup orderings. Experimental results show that the orderings obtained by our scheme are in general better than those obtained by other popular ordering codes.
On twodimensional sparse matrix partitioning: Models, methods, and a recipe
 SIAM J. SCI. COMPUT
, 2010
"... We consider twodimensional partitioning of general sparse matrices for parallel sparse matrixvector multiply operation. We present three hypergraphpartitioningbased methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces finegrain ..."
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Cited by 35 (18 self)
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We consider twodimensional partitioning of general sparse matrices for parallel sparse matrixvector multiply operation. We present three hypergraphpartitioningbased methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces finegrain partitions. The other two produce coarser partitions, where one of them imposes a limit on the number of messages sent and received by a single processor, and the other trades that limit for a lower communication volume. We also present a thorough experimental evaluation of the proposed twodimensional partitioning methods together with the hypergraphbased onedimensional partitioning methods, using an extensive set of public domain matrices. Furthermore, for the users of these partitioning methods, we present a partitioning recipe that chooses one of the partitioning methods according to some matrix characteristics.