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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. ..."
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Cited by 1173 (16 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.
Parallel Numerical Linear Algebra
, 1993
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illust ..."
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Cited by 766 (23 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
The geometry of algorithms with orthogonality constraints
 SIAM J. MATRIX ANAL. APPL
, 1998
"... In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal proces ..."
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Cited by 642 (1 self)
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In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.
Fast Linear Iterations for Distributed Averaging
 Systems and Control Letters
, 2003
"... We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging linear ..."
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Cited by 429 (13 self)
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We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging linear iteration can be cast as a semidefinite program, and therefore efficiently and globally solved. These optimal linear iterations are often substantially faster than several common heuristics that are based on the Laplacian of the associated graph.
Informationbased complexity
, 1988
"... Computational complexity studies the intrinsic difficulty of mathematically posed problems and seeks optimal means for their solutions. This is a rich and diverse field; for the purpose of this paper we present a greatly simplified picture. Computational complexity may be divided into two branches, ..."
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Cited by 308 (37 self)
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Computational complexity studies the intrinsic difficulty of mathematically posed problems and seeks optimal means for their solutions. This is a rich and diverse field; for the purpose of this paper we present a greatly simplified picture. Computational complexity may be divided into two branches, discrete and continuous.
Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems, Concurrency: Practice and Experience 6
, 1994
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ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods.
, 1997
"... this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the ..."
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Cited by 215 (18 self)
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this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the software. 1.7 Dependence on LAPACK and BLAS
Krylov Projection Methods For Model Reduction
, 1997
"... This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov p ..."
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Cited by 213 (3 self)
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This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczosbased methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete modelreduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multipleinput multipleoutput systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.
A Minmax Cut Algorithm for Graph Partitioning and Data Clustering
, 2001
"... An important application of graph partitioning is data clustering using a graph model  the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. Here we propose a new algorithm for graph partition with an objec ..."
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Cited by 211 (15 self)
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An important application of graph partitioning is data clustering using a graph model  the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. Here we propose a new algorithm for graph partition with an objective function that follows the minmax clustering principle. The relaxed version of the optimization of the minmax cut objective function leads to the Fiedler vector in spectral graph partition. Theoretical analyses of minmax cut indicate that it leads to balanced partitions, and lower bonds are derived. The minmax cut algorithm is tested on newsgroup datasets and is found to outperform other current popular partitioning/clustering methods. The linkagebased re nements in the algorithm further improve the quality of clustering substantially. We also demonstrate that the linearized search order based on linkage differential is better than that based on the Fiedler vector, providing another effective partition method.