Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part I. Separable Approximation of Multi-variate Functions

Authors:
W. Hackbusch;N. Khoromskij
Affiliations:
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22–26, 04103, Leipzig, Germany;Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22–26, 04103, Leipzig, Germany
Venue:
Computing
Year:
2006

Citing 0
Cited 10

A structured low-rank wavelet solver for the Ornstein-Zernike integral equation

Computing
Tensor-product approximation to operators and functions in high dimensions

Journal of Complexity
On the robustness of elliptic resolvents computed by means of the technique of hierarchical matrices

Applied Numerical Mathematics
Tensor decomposition in electronic structure calculations on 3D Cartesian grids

Journal of Computational Physics
Kronecker product approximation of demagnetizing tensors for micromagnetics

Journal of Computational Physics
Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

Journal of Computational and Applied Mathematics
Tensor ranks for the inversion of tensor-product binomials

Journal of Computational and Applied Mathematics
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition

SIAM Journal on Matrix Analysis and Applications
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs

SIAM Journal on Scientific Computing
Tensor-Train Decomposition

SIAM Journal on Scientific Computing

Quantified Score

Hi-index	0.01

Visualization

Abstract

The Kronecker tensor-product approximation combined with the **-matrix techniques provides an efficient tool to represent integral operators as well as certain functions F(A) of a discrete elliptic operator A in ℝd with a high spatial dimension d. In particular, we approximate the functions A−1 and sign(A) of a finite difference discretisation A∈ℝN×N with a rather general location of the spectrum. The asymptotic complexity of our data-sparse representations can be estimated by ** (np log qn), p = 1, 2, with q independent of d, where n=N1/d is the dimension of the discrete problem in one space direction. In this paper (Part I), we discuss several methods of a separable approximation of multi-variate functions. Such approximations provide the base for a tensor-product representation of operators. We discuss the asymptotically optimal sinc quadratures and sinc interpolation methods as well as the best approximations by exponential sums. These tools will be applied in Part II continuing this paper to the problems mentioned above.