Efstratios Gallopoulos

SVD based initialization: A head start for nonnegative matrix factorization

We describe Nonnegative Double Singular Value Decomposition (NNDSVD), a new method designed to enhance the initialization stage of nonnegative matrix factorization (NMF). NNDSVD can readily be combined with existing NMF algorithms. The basic algorithm contains no randomization and is based on two SVD processes, one approximating the data matrix, the other approximating positive sections of the resulting partial SVD factors utilizing an algebraic property of unit rank matrices. Simple practical variants for NMF with dense factors are described. NNDSVD is also well suited to initialize NMF algorithms with sparse factors. Many numerical examples suggest that NNDSVD leads to rapid reduction of the approximation error of many NMF algorithms.

Efficient solution of parabolic equations by Krylov approximation methods

This paper takes a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus the resulting approximation consists of applying an evolution operator of very small dimension to a known vector, which is, in turn, computed accurately by exploiting high-order rational Chebyshev and Pade approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications and, as a result, the algorithm can easily be parallelized and vectorized. Further parallelism is introduced by expanding the rational approximations into partial fractions. Some relevant approximation and ...

An iterative method for nonsymmetric systems with multiple right-hand sides

We propose a method for the solution of linear systems

TMG: A MATLAB Toolbox for Generating Term-Document Matrices from Text Collections

AX = B

A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems

where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order

Convergence properties of block GMRES and matrix polynomials

n \times s

FALCON: A MATLAB Interactive Restructuring Compiler

with s of moderate size. The method uses a single Krylov subspace per step as a generator of approximations, a projection process, and a Richardson acceleration technique. It thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right-hand sides. Numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right-hand sides.

On the parallel solution of parabolic equations

A wide range of computational kernels in data mining and information retrieval from text collections involve techniques from linear algebra. These kernels typically operate on data that are presented in the form of large sparse term-document matrices (tdm). We present TMG, a research and teaching toolbox for the generation of sparse tdms from text collections and for the incremental modification of these tdms by means of additions or deletions. The toolbox is written entirely in MATLAB, a popular problem-solving environment that is powerful in computational linear algebra, in order to streamline document preprocessing and prototyping of algorithms for information retrieval. Several design issues that concern the use of MATLAB sparse infrastructure and data structures are addressed. We illustrate the use of the tool in numerical explorations of the effect of stemming and different term-weighting policies on the performance of querying and clustering tasks.

A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides

Motivated by a recent method of Freund [SIAM J. Sci. Comput., 14 (1993), pp. 470–482], who introduced a quasi-minimal residual (QMR) version of the conjugate gradients squared (CGS) algorithm, a QMR variant of the biconjugate gradient stabilized (Bi-CGSTAB) algorithm of van der Vorst that is called QMRCGSTAB, is proposed for solving nonsymmetric linear systems. The motivation for both QMR variants is to obtain smoother convergence behavior of the underlying method. The authors illustrate this by numerical experiments that also show that for problems on which Bi-CGSTAB performs better than CGS, the same advantage carries over to QMRCGSTAB.

Incomplete partial fractions for parallel evaluation of rational matrix functions

Abstract This paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right-hand sides. A convergence theory is developed based on a representation of the method using matrix-valued polynomials. Relations between the roots of the residual polynomial for block gmres and the matrix e-pseudospectrum are derived, and illustrated with numerical experiments. The role of invariant subspaces in the effectiveness of block methods is also discussed.