TY - GEN

T1 - Efficient analysis of high dimensional data in tensor formats

AU - Espig, Mike

AU - Hackbusch, Wolfgang

AU - Litvinenko, Alexander

AU - Matthies, Hermann G.

AU - Zander, Elmar

PY - 2013/3/5

Y1 - 2013/3/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874431081&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-31703-3-2

DO - 10.1007/978-3-642-31703-3-2

M3 - Conference contribution

AN - SCOPUS:84874431081

SN - 9783642317026

T3 - Lecture Notes in Computational Science and Engineering

SP - 31

EP - 56

BT - Sparse Grids and Applications

A2 - Garcke, Jochen

A2 - Griebel, Michael

ER -