Martin H. Schultz

GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

Topological properties of hypercubes

The n-dimensional hypercube is a highly concurrent loosely coupled multiprocessor based on the binary n-cube topology. Machines based on the hypercube topology have been advocated as ideal parallel architectures for their powerful interconnection features. The authors examine the hypercube from the graph-theory point of view and consider those features that make its connectivity so appealing. Among other things, they propose a theoretical characterization of the n-cube as a graph and and show how to map various other topologies into a hypercube. >

Variational Iterative Methods for Nonsymmetric Systems of Linear Equations

We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do not make use of any associated symmetric problems. Convergence results and error bounds are presented.

Conjugate gradient-like algorithms for solving nonsymmetric linear systems

This paper presents a unified formulation of a class of the conjugate gradient-like algorithms for solving nonsymmetric linear systems. The common framework is the Petrov- Galerkin method on Krylov subspaces. We discuss some practical points concerning the methods and point out some of the interrelations between them. 1. Introduction. In the recent few years, a large number of generalizations of the conjugate gradient and conjugate residual methods, which are very successful in solving symmetric positive-definite linear systems, have been proposed for solving nonsymmetric linear systems (3), (6), (5), (7), (12). In this paper we present an abstract framework which includes most of these methods and many new ones. Our goal is to understand the relationships among the methods and to synthesize. Consider the general linear system:

First- and Second-Order Diffusive Methods for Rapid, Coarse, Distributed Load Balancing

Abstract. We consider the following general problem modeling load balancing in a variety of distributed settings. Given an arbitrary undirected connected graph G=(V,E) and a weight distribution w0 on the nodes, determine a schedule to move weights across edges in each step so as to (approximately) balance the weights on the nodes. We focus on diffusive schedules for this problem. All previously studied diffusive schedules can be modeled as wt+1 = Mwt where wt is the weight distribution after t steps and M is a doubly stochastic matrix. We call these the first-order schedules. First-order schedules, although widely used in practice, are often slow. In this paper we introduce a new direction in diffusive schedules by considering schedules that are modeled as: w1=Mw0;wt+1=β Mwt + (1-β)wt-1 for some appropriate β; we call these the second-order schedules. In the idealized setting of weights being real numbers, we adopt known results to show that β can be chosen so that the second-order schedule involves significantly fewer steps than the first-order method for approximate load balancing. In the realistic setting when the weights are positive integers, we simulate the idealized schedules by maintaining I Owe You units on the edges. Extensive experiments with simulated data and real-life data from JOSTLE, a mesh-partitioning software, show that the resultant realistic schedule is close to the idealized schedule, and it again involves fewer steps than the first-order schedules for approximate load balancing. Our main result is therefore a fast algorithm for coarse load balancing that can be used in a variety of applications.

Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination

In this paper we present algorithms and data structures that may be used in the efficient implementation of symmetric Gaussian elimination for sparse systems of linear equations with positive definite coefficient matrices. The techniques described here serve as the basis for the symmetric codes in the Yale Sparse Matrix Package.

The tYNA platform for comparative interactomics: a web tool for managing, comparing and mining multiple networks

Summary: Biological processes involve complex networks of interactions between molecules. Various large-scale experiments and curation efforts have led to preliminary versions of complete cellular networks for a number of organisms. To grapple with these networks, we developed TopNet-like Yale Network Analyzer (tYNA), a Web system for managing, comparing and mining multiple networks, both directed and undirected. tYNA efficiently implements methods that have proven useful in network analysis, including identifying defective cliques, finding small network motifs (such as feed-forward loops), calculating global statistics (such as the clustering coefficient and eccentricity), and identifying hubs and bottlenecks. It also allows one to manage a large number of private and public networks using a flexible tagging system, to filter them based on a variety of criteria, and to visualize them through an interactive graphical interface. A number of commonly used biological datasets have been pre-loaded into tYNA, standardized and grouped into different categories. Availability: The tYNA system can be accessed at http://networks.gersteinlab.org/tyna. The source code, JavaDoc API and WSDL can also be downloaded from the website. tYNA can also be accessed from the Cytoscape software using a plugin. Contact: mark.gerstein@yale.edu Supplementary information: Additional figures and tables can be found at http://networks.gersteinlab.org/tyna/supp

Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations

We consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself-adjoint elliptic boundary value problems. We derive bounds on convergence rates that are independent of discretization mesh size. For two-dimensional problems on rectangular domains, discretized on an

Complexity of dense-linear-system solution on a multiprocessor ring☆

n \times n

L∞-Multivariate approximation theory

grid, these bounds lead to asymptotic operation counts of