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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Efficient Analysis of High Dimensional Data in Tensor Formats

Mike Espig, Wolfgang Hackbusch, Alexander Litvinenko, Hermann G. Matthies and Elmar Zander


In this article we introduce new methods for the analysis of high dimensional data in tensor formats, where the underling data come from the stochastic elliptic boundary value problem. After discretisation of the deterministic operator as well as the presented random fields via KLE and PCE, the obtained high dimensional operator can be approximated via sums of elementary tensors. This tensors representation can be effectively used for computing different values of interest, such as maximum norm, level sets and cumulative distribution function. The basic concept of the data analysis in high dimensions is discussed on tensors represented in the canonical format, however the approach can be easily used in other tensor formats. As an intermediate step we describe efficient iterative algorithms for computing the characteristic and sign functions as well as pointwise inverse in the canonical tensor format. Since during majority of algebraic operations as well as during iteration steps the representation rank grows up, we use lower-rank approximation and inexact recursive iteration schemes.


Related publications

2013 Repository Open Access
Mike Espig, Wolfgang Hackbusch, Alexander Litvinenko, Hermann G. Matthies and Elmar Zander

Efficient analysis of high dimensional data in tensor formats

In: Sparse grids and applications / Jochen Garcke... (eds.)
Berlin : Springer, 2013. - pp. 31-56
(Lecture notes in computational science and engineering ; 88)