Abstract
The hierarchical Tucker format is a way to decompose a high-dimensional tensor recursively into sums of products of lower-dimensional tensors. The number of degrees of freedom in such a representation is typically many orders of magnitude lower than the number of entries of the original tensor. This makes the hierarchical Tucker format a promising approach to solve ordinary differential equations for high-dimensional tensors. In order to propagate the approximation in time, differential equations for the parameters of the hierarchical Tucker format are derived from the Dirac-Frenkel variational principle. We prove an error bound for the dynamical approximation in the hierarchical Tucker format by extending previous results of Koch and Lubich for the non-hierarchical Tucker format.
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The authors thank the anonymous referees for their helpful remarks and suggestions.
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Communicated by Christian Lubich.
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Arnold, A., Jahnke, T. On the approximation of high-dimensional differential equations in the hierarchical Tucker format. Bit Numer Math 54, 305–341 (2014). https://doi.org/10.1007/s10543-013-0444-2
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DOI: https://doi.org/10.1007/s10543-013-0444-2
Keywords
- High-dimensional differential equations
- Tensor approximation
- Hierarchical Tucker format
- Dirac-Frenkel variational principle
- Error bounds