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664
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 789 (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.
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 380 (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.
A Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems
 Experience
, 1994
"... Unstructured meshes are used in many largescale scientific and engineering problems, including finitevolume methods for computational fluid dynamics and finiteelement methods for structural analysis. If unstructured problems such as these are to be solved on distributedmemory parallel computers, ..."
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Cited by 277 (7 self)
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Unstructured meshes are used in many largescale scientific and engineering problems, including finitevolume methods for computational fluid dynamics and finiteelement methods for structural analysis. If unstructured problems such as these are to be solved on distributedmemory parallel computers, their data structures must be partitioned and distributed across processors; if they are to be solved efficiently, the partitioning must maximize load balance and minimize interprocessor communication. Recently the recursive spectral bisection method (RSB) has been shown to be very effective for such partitioning problems compared to alternative methods. Unfortunately, RSB in its simplest form is rather expensive. In this report we shall describe a multilevel implementation of RSB that can attain about an orderofmagnitude improvement in run time on typical examples. Keywords: graph partitioning, domain decomposition, MIMD machines, multilevel algorithm, spectral bisection, sp...
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 217 (34 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 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 188 (11 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.
Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations
 Concurrency
, 1991
"... If a finite element mesh has a sufficiently regular structure, it is easy to decide in advance how to distribute the mesh among the processors of a distributedmemory parallel processor, but if the mesh is unstructured, the problem becomes much more difficult. The distribution should be made so that ..."
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Cited by 155 (3 self)
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If a finite element mesh has a sufficiently regular structure, it is easy to decide in advance how to distribute the mesh among the processors of a distributedmemory parallel processor, but if the mesh is unstructured, the problem becomes much more difficult. The distribution should be made so that each processor has approximately equal work to do, and such that communication overhead is minimized. If the mesh is solutionadaptive, i.e. the mesh and hence the load balancing problem change discretely during execution of the code, then it is most efficient to decide the optimal mesh distribution in parallel. In this paper three parallel algorithms, Orthogonal Recursive Bisection (ORB), Eigenvector Recursive Bisection (ERB) and a simple parallelization of Simulated Annealing (SA) have been implemented for load balancing a dynamic unstructured triangular mesh on 16 processors of an NCUBE machine. The test problem is a solutionadaptive Laplace solver, with an initial mesh of 280 elements,...
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 object ..."
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Cited by 150 (12 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 di erential is better than that based on the Fiedler vector, providing another e ective partition method.
PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users
 Guide 6.0, Society for Industrial and Applied Mathematics
, 1990
"... Copyright (c) 2004, by the author. ..."
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 135 (14 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 119 (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.