An example of a tensor is a grid function in a d-dimensional cube with n grid points per direction. Its dimension is n^d. In cases of n=d=1000 it seems hopeless to apply numerical methods. Nevertheless, there are different possibilities to represent tensors by fewer data. Moreover, it is possible to perform operations with tensors. We introduce the n-term format, the subspace format, and the hierarchical format and discuss their properties. In particular, the HOSVD (higher order singular value decomposition) will be mentioned.
Finally we describe a technique called 'tensorisation' which treats vectors or matrices as
high-dimensional tensors in order to exploit the successful operations within the set of tensors.
W. Hackbusch: Tensor spaces and numerical tensor calculus. Springer, Berlin 2012
Max Planck Institute for Mathematics in the Sciences
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