Tensor-product Decomposition of Operators and Functions

There is a wide range of traditional as well as an increasing number of modern applications involving multi-dimensional data/operators described by higher-order tensors, which are, in fact, the higher-order analogues of vectors and matrices. Naive numerical tensor calculus suffers from the so-called "curse of dimensionality" (Bellman).

The modern concept to treat numerically higher-order tensors is based on their structured low-rank decomposition via the canonical (rank-$1$) tensors.

There are numerous successful applications of such methods in chemometrics, higher-order statistics, data mining, mathematical biology, physics, etc.. We focus on the recent applications of tensor-product approximation to multi-dimensional operators arising in many-particle modeling and in standard FEM/BEM (say, discretisation of the electron- and molecular density functions, the classical potentials in $\mathbb{R}^d$, convolution and other integral transforms, elliptic Green's functions).

In particular, we discuss the following issues:
-- Why the orthogonal Tucker/canonical models are efficient (approximation theory);
-- Methods of numerical multi-linear algebra;
-- How to represent nonlocal operators arising from the Hartree-Fock, Kohn-Sham, Ornstein-Zernike and the Boltzmann equations in the data-sparse tensor-product form;
-- Numerical examples in the electron- and molecular structure calculations;
-- Future perspectives.