During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

http://sma.epfl.ch/~anchpcommon/publications/tensorsurvey.pdf

A literature survey of low-rank tensor approximation techniques

What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Aspects of topology: On dimension theory

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the $n$ -dimensional setting, however, it requires the inversion of an $n \times n$ covariance matrix, $C$ , as well as the evaluation of its determinant, $\det (C)$ . In many cases, such as regression using Gaussian processes, the covariance matrix is of the form $C = \sigma ^2 I + K$ , where $K$ is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix $C$ is typically dense, causing standard direct methods for inversion and determinant evaluation to require $\mathcal {O}(n^3)$ work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix $C$ can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an $\mathcal {O} (n\,\log^2\, n)$ algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant $\det (C)$ , permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining $K$ . Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.

Fast Direct Methods for Gaussian Processes

In previous papers hierarchical matrices were introduced which are data-sparse and allow an approximate matrix arithmetic of nearly optimal complexity. In this paper we analyse the complexity (storage, addition, multiplication and inversion) of the hierarchical matrix arithmetics. Two criteria, the sparsity and idempotency, are sufficient to give the desired bounds. For standard finite element and boundary element applications we present a construction of the hierarchical matrix format for which we can give explicit bounds for the sparsity and idempotency.

/pdf/construction-and-arithmetics-of-h-matrices-3y9hejpxul.pdf

Construction and arithmetics of H -matrices

We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix – matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity. We give a review of specialised variants of H-matrices, especially of H 2 -matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory. q 2003 Elsevier Science Ltd. All rights reserved.

https://www.mis.mpg.de/preprints/2002/preprint2002_18.pdf

Introduction to Hierarchical Matrices with Applications

A class of matrices (H2-matrices) has recently been introduced for storing discretisations of elliptic problems and integral operators from the BEM. These matrices have the following properties: (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but after truncation to the H2-matrix format these operations are again of quasi-linear complexity.We introduce the basic ideas of H- and H2-matrices and present an algorithm that adaptively computes approximations of general matrices in the latter format.

/pdf/data-sparse-approximation-by-adaptive-h-2-matrices-478syv39v9.pdf

Data-sparse approximation by adaptive H 2 -matrices

Efficient Numerical Methods for Non-local Operators

The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the **-matrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by low-rank matrices. The low-rank matrices are assembled by a new hybrid algorithm (HCA) that has the same proven convergence as standard interpolation but also the same efficiency as the (heuristic) adaptive cross approximation (ACA).

/pdf/hybrid-cross-approximation-of-integral-operators-sjlt0mgd31.pdf

Hybrid cross approximation of integral operators

https://www.mis.mpg.de/preprints/2001/preprint2001_104.pdf

Steffen Börm

Papers

Introduction to Hierarchical Matrices with Applications

Data-sparse approximation by adaptive H 2 -matrices

Efficient Numerical Methods for Non-local Operators

Hybrid cross approximation of integral operators

H 2 -matrix approximation of integral operators by interpolation