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Numerical tensor calculus*

Published online by Cambridge University Press:  12 May 2014

Wolfgang Hackbusch
Wolfgang Hackbusch*
Affiliation:
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, D-04103 Leipzig, Germany, E-mail: [email protected]
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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 with n grid points per direction, a spatial dimension d yields nd grid points. A grid function defined on such a grid is an example of a tensor of order d. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly in d. In this way, data of size nd = 10001000 can 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.

Type
Research Article
Information
Acta Numerica , Volume 23 , May 2014 , pp. 651 - 742
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

References

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