Numerical tensor calculus

Abstract

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.