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 University of Florida Sparse Matrix Collection is a large, widely available, and actively growing set of sparse matrices that arise in real applications. Its matrices cover a wide spectrum of problem domains, both those arising from problems with underlying 2D or 3D geometry (structural engineering, computational fluid dynamics, model reduction, electromagnetics, semiconductor devices, thermodynamics, materials, acoustics, computer graphics/vision, robotics/kinematics, and other discretizations) and those that typically do not have such geometry (optimization, circuit simulation, networks and graphs, economic and financial modeling, theoretical and quantum chemistry, chemical process simulation, mathematics and statistics, and power networks). The collection meets a vital need that artificially-generated matrices cannot meet, and is widely used by the sparse matrix algorithms community for the development and performance evaluation of sparse matrix algorithms. The collection includes software for accessing and managing the collection, from MATLAB, Fortran, and C.

We have extended the matrix computation language and environment Matlab to include sparse matrix storage and operations. The only change to the outward appearance of the Matlab language is a pair of commands to create full or sparse matrices. Nearly all the operations of Matlab now apply equally to full or sparse matrices, without any explicit action by the user. The sparse data structure represents a matrix in space proportional to the number of nonzero entries, and most of the operations compute sparse results in time proportionaltothenumber of arithmetic operations on nonzeros.

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Graph partitioning is a fundamental problem in several scientific and engineering applications. In this paper, we describe heuristics that improve the state-of-the-art practical algorithms used in graph-partitioning software in terms of both partitioning speed and quality. An important use of graph-partitioning is in ordering sparse matrices for obtaining direct solutions to sparse systems of linear equations arising in engineering and optimization applications. The experiments reported in this paper show that the use of these heuristics results in a considerable improvement in the quality of sparse-matrix orderings over conventional ordering methods, especially for sparse matrices arising in linear programming problems. In addition, our graph-partitioning-based ordering algorithm is more parallelizable than minimum-degree-based ordering algorithms, and it renders the ordered matrix more amenable to parallel factorization.

In this paper we provide a robust reordering scheme for sparse matrices. The scheme relies on the notion of multisection, a generalization of bisection. The reordering strategy is demonstrated to have consistently good performance in terms of fill reduction when compared with multiple minimum degree and generalized nested dissection. Experimental results show that by using multisection, we obtain an ordering which is consistently as good as or better than both for a wide spectrum of sparse problems. 1 Introduction It is well recognized that finding a fill-reducing ordering is crucial in the success of the numerical solution of sparse linear systems. For symmetric positive-definite systems, the minimum degree [38] and the nested dissection [11] orderings are perhaps the most popular ordering schemes. They represent two opposite approaches to the ordering problem. However, they share a common undesirable characteristic. Both schemes produce generally good orderings, but the ordering qua...

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We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using state-of-the-art graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.

This dissertation presents new techniques for solving large sparse unsymmetric linear systems on high performance computers, using Gaussian elimination with partial pivoting. The efficiencies of the new algorithms are demonstrated for matrices from various fields and for a variety of high performance machines. In the first part we discuss optimizations of a sequential algorithm to exploit the memory hierarchies that exist in most RISC-based superscalar computers. We begin with the left-looking supernode-column algorithm by Eisenstat, Gilbert and Liu, which includes Eisenstat and Liu's symmetric structural reduction for fast symbolic factorization. Our key contribution is to develop both numeric and symbolic schemes to perform supernodepanel updates to achieve better data reuse in cache and floating-point register...