Ivan V. Oseledets

Tensor-Train Decomposition

A simple nonrecursive form of the tensor decomposition in

Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions

d

Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time

dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.

Solution of Linear Systems and Matrix Inversion in the TT-Format

For

Approximation of

d

2^d\times2^d

-dimensional tensors with possibly large

Matrices Using Tensor Decomposition

d>3

Computation of extreme eigenvalues in higher dimensions using block tensor train format

, an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial “spatial” dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given.

Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs

We consider Tucker-like approximations with an

Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker--Planck Equation

r \times r \times r