On explicit QTT representation of Laplace operator and its inverse

Vladimir A. Kazeev and Boris N. Khoromskij

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Submission date: 14. Dec. 2010
Pages: 30
published in: SIAM journal on matrix analysis and applications, 33 (2012) 3, p. 742-758 
DOI number (of the published article): 10.1137/100820479
Bibtex
with the following different title: Low-Rank explicit QTT representation of the Laplace operator and inverse
MSC-Numbers: 15A69, 65F99
Keywords and phrases: tensor decompositions, low-rank approximation, Quantics Tensor Train, QTT, inverse Laplace operator
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Abstract:
Ranks and explicit structure of some matrices in the Quantics Tensor Train format, which allows representation with logarithmic complexity in many cases, are investigated. The matrices under consideration are Laplace operator with various boundary conditions in D dimensions and inverse Laplace operator with Dirichlet and Dirichlet-Neumann boundary conditions in one dimension. The minimal-rank explicit QTT representations of these matrices presented are suitable for any high mode sizes and, in the multi-dimensional case, for any high dimensions.