Improving multifrontal methods by means of block low-rank representations

Submitted for publication to SIAMMatrices coming from elliptic Partial Differential Equations (PDEs) have been shown to have a low-rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives to the blocks a geometrical meaning, such approximations can be computed using an SVD or a rank-revealing QR factorization. The resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations. Several strategies have been proposed to exploit this property. We propose a low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method. We present experimental results that show how the BLR format delivers gains that are comparable to those obtained with hierarchical formats such as Hierarchical matrices (H matrices) and Hierarchically Semi-Separable (HSS matrices) but provides much greater flexibility and ease of use which are essential in the context of a general purpose, algebraic solver

Grouping variables in Frontal Matrices to improve Low-Rank Approximations in a Multifrontal Solver

Session 3International audienc

Applications of the Dulmage-Mendelsohn Decomposition and Network Flow to Graph Bisection Improvement

In this paper, we consider the use of the Dulmage-Mendelsohn decomposition and network flow on bipartite graphs to improve a graph bisection partition. Given a graph partition [S; B; W ] with a vertex separator S and two disconnected components B and W , different strategies are considered based on the Dulmage-Mendelsohn decomposition to reduce the separator size jSj and/or the imbalance between B and W . For the case when the vertices are weighted, we relate this with the bipartite network flow problem. A further enhancement is made on partition improvement by generalizing the bipartite network to solving a general network flow problem. We demonstrate the utility of these improvement techniques on a set of sparse test matrices, where we find top level separators and nested dissection and multisection orderings. Key words. Dulmage-Mendelsohn decomposition, network flow, graph bisection, ordering algorithms, nested dissection. multisection. AMS(MOS) subject classifications. 65F05, 65..

Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR^2 Format

International audienceWe investigate a special class of data sparse rank-structured matrices that combine a flat block low-rank (BLR) partitioning with the use of shared (called nested in the hierarchical case) bases. This format is to H 2 matrices what BLR is to H matrices: we therefore call it the BLR 2 matrix format. We present algorithms for the construction and LU factorization of BLR 2 matrices, and perform their cost analysis-both asymptotically and for a fixed problem size. With weak admissibility, BLR 2 matrices reduce to block separable matrices (the flat version of HBS/HSS). Our analysis and numerical experiments reveal some limitations of BLR 2 matrices with weak admissibility, which we propose to overcome with two approaches: strong admissibility, and the use of multiple shared bases per row and column

Robust Ordering of Sparse Matrices using Multisection

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..

Using Domain Decomposition to find Graph Bisectors

In this paper we introduce a three-step approach to find a vertex bisector of a graph. The first step finds a domain decomposition of the graph, a set of connected subgraphs, the domains, and a multisector, the remaining vertices that separate the domains from each other. The second step uses a block variant of the Kernighan-Lin scheme to find a bisector that is a subset of the multisector. The third step improves the bisector by bipartite graph matching. Experimental results show this domain decomposition method finds graph partitions that compare favorably with a state-of-the-art multilevel partitioning scheme in both quality and execution time. 1 Introduction Graph partitioning is a well-known practical problem that has many important applications, such as task allocation for parallel computations [13] and circuit partitioning for VLSI design [22]. Our driving interest is to find low-fill orderings for sparse matrix computation [4], [6], [15], [19]. An effective approach to find fi..

The Reference Manual for SPOOLES, Release 2.2: An Object Oriented Software Library for Solving Sparse Linear Systems of Equations

Solving sparse linear systems of equations is a common and important application of a multitude of scientific and engineering applications. The SPOOLES software package 1 provides this functionality with a collection of software objects. The first step to solving a sparse linear system is to find a good low-fill ordering of the rows and columns. The library contains several ways to perform this operation: minimum degree, generalized nested dissection, and multisection. The second step is to factor the matrix as a product of triangular and diagonal matrices. The library supports pivoting for numerical stability (when required), approximation techniques to reduce the storage for and work to compute the matrix factors, and the computations are based on BLAS3 numerical kernels to take advantage of high performance computing architectures. The third step is to solve the linear system using the computed factors. The library is written in ANSI C using object oriented design. Good design and..