Generalised Cross Approximation

Abstract

An important feature is the computation of a tensor from comparably few tensor entries. The input tensor ${\rm v} \in {\rm V}$ is assumed to be given in a full functional representation so that, on request, any entry can be determined. This partial information can be used to determine an approximation $\tilde{{\rm v}} \in {\rm V}$. In the matrix case (d = 2) an algorithm for this purpose is well-known under the name ‘cross approximation’ or ‘adaptive cross approximation’ (ACA). The generalisation to the multi-dimensional case is not straightforward. We present a generalised cross approximation, which fits to the hierarchical format. If ${\rm v} \in \mathcal{H}_{{\rm r}}$ holds with a known rank vector r, this tensor can be reproduced exactly, i.e., $\tilde{{\rm v}} = {\rm v}$. In the general case, the approximation is heuristic. Exact error estimates require either inspection of all tensor entries (which is practically impossible) or strong theoretical a priori knowledge.

There are many different applications, where tensors are given in a full functional representation. Section 15.1 gives examples of multivariate functions constructed via integrals and describes multiparametric solutions of partial differential equations, which may originate from stochastic coefficients. Section 15.2 introduces the definitions of fibres and crosses. The matrix case is recalled in Sect. 15.3., while the true tensor case (d ≥ 3) is considered in Sect. 15.4.