Efficient analysis of high dimensional data in tensor formats

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

27 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationSparse Grids and Applications
EditorsJochen Garcke, Michael Griebel
Pages31-56
Number of pages26
DOIs
StatePublished - Mar 5 2013

Publication series

NameLecture Notes in Computational Science and Engineering
Volume88
ISSN (Print)1439-7358

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering(all)
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Mathematics

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