Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

The discrete-dipole approximation (DDA) for scattering calculations, including the relationship between the DDA and other methods, is reviewed. Computational considerations, i.e., the use of complex-conjugate gradient algorithms and fast-Fourier-transform methods, are discussed. We test the accuracy of the DDA by using the DDA to compute scattering and absorption by isolated, homogeneous spheres as well as by targets consisting of two contiguous spheres. It is shown that, for dielectric materials (|m| ≲ 2), the DDA permits calculations of scattering and absorption that are accurate to within a few percent.

Discrete-Dipole Approximation For Scattering Calculations

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...

/pdf/nineteen-dubious-ways-to-compute-the-exponential-of-a-matrix-49qnou6xyl.pdf

Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later ⁄

This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF) This includes NMFs various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD) NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models Key features: Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors own recently developed techniques in the subject area Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms Provides a comparative analysis of the different methods in order to identify approximation error and complexity Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia

Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation

Matrix Factorization Techniques for Recommender Systems

http://www.cb.uu.se/~milan/histo/before2011august/Boutsidis.pdf

SVD based initialization: A head start for nonnegative matrix factorization

During the early 1960s, scientists began to envision problem-solving computing environments not only powerful enough to solve complex problems but also able to interact with users on human terms. While many tried to create PSEs over the next few years, by the early 1970s they had abandoned almost all of these attempts. Technology could not yet support PSEs in computational science. But the dream of the 1960s can be the reality of the 1990s: high-performance computers combined with better understanding of computing and computational science have put PSEs well within our reach. The term 'problem-solving environment' means different things to different people. A PSE is a computer system that provides all the computational facilities necessary to solve a target class of problems. These features include advanced solution methods, automatic or semi-automatic selection of solution methods, and ways to easily incorporate novel solution methods. Simple PSEs appeared early in computing without being recognized as such. Some of the capabilities of future problem-solving environments seem like science fiction, but whatever form they eventually take, their scientific and economic impact will be enormous. >

Computer as thinker/doer: problem-solving environments for computational science

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 ...

Efficient solution of parabolic equations by Krylov approximation methods

We propose a method for the solution of linear systems $AX = B$ where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order $n \times s$ 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.

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

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.

Efstratios Gallopoulos

Papers

SVD based initialization: A head start for nonnegative matrix factorization

Computer as thinker/doer: problem-solving environments for computational science

Efficient solution of parabolic equations by Krylov approximation methods

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

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