Tensor Spaces and Numerical Tensor Calculus pp 249-280 | Cite as

*r*-Term Approximation

## Abstract

In general, one tries to approximate a tensor v by another tensor u requiring less data. The reason is twofold: the memory size should decrease and, hopefully, operations involving u should require less computational work. In fact, \({\rm u} \in \mathcal{R}_{{\rm r}}\) leads to decreasing cost for storage and operations as *r* decreases. However, the other side of the coin is an increasing approximation error. Correspondingly, in *Sect. 9.1* two approximation strategies are presented, where either the representation rank *r* of u or the accuracy is prescribed. Before we study the approximation problem in general, two particular situations are discussed. *Section 9.2* is devoted to *r* = 1, when \({\rm u} \in \mathcal{R}_{1}\) is an elementary tensor. The matrix case *d* = 2 is recalled in *Sect. 9.3*. The properties observed in the latter two sections contrast with the true tensor case studied in *Sect. 9.4*. Numerical algorithms solving the approximation problem will be discussed in *Sect. 9.5*. Modified approximation problems are addressed in *Sect. 9.6*.

## Keywords

Selfadjoint Operator Supremum Norm Sparse Grid Multivariate Function Elementary Tensor## Preview

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