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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Spectral partitioning methods use the Fiedler vector---the eigenvector of the secondsmallest eigenvalue of the Laplacian matrix---to find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on bounded-degree planar graphs and finite element meshes--- the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for bounded-degree planar graphs and two-dimensional meshes and O i n 1=d j for well-shaped d-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...

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 stand-alone 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.

We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method’s novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain classes of “well-shaped” finite element meshes have good separators. The geometric method is quite simple to implement: we describe a Matlab code for it in some detail. The method is also quite efficient and effective: we compare it with some other methods, including spectral bisection.

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 vertex-separator 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.

Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innovative techniques have been implemented mostly in university research codes and only very few commercial codes use them. There are two reasons for this. Firstly, the multigrid solution and adaptive refinement for many engineering applications are still a topic of active research and cannot be considered to be mature enough for routine application. Secondly, the implementation of all these techniques in a code with sufficient generality requires a lot of time and know-how in different fields. UG (abbreviation for Unstructured Grids) has been designed to overcome these problems. It provides very general tools for the generation and manipulation of unstructured meshes in two and three space dime...

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 large-scale distributed machines.

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This document describes a collection of three related ANSI C subroutine libraries for solving sparse linear systems of equations AX = B. Here A is a square, nonsingular, n \Theta n sparse matrix, and X and B are dense n \Theta nrhs matrices, where nrhs is the number of right-hand sides and solution vectors. Matrix A need not be symmetric or definite; indeed, SuperLU is particularly appropriate for matrices with very unsymmetric structure. All three libraries use variations of Gaussian elimination optimized to take advantage both of sparsity and the computer architecture, in particular memory hierarchies (caches) and parallelism. In this introduction we refer to all three libraries collectively as SuperLU. The three libraries within SuperLU are as follows. Detailed references are also given (see also [19]).