Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
75/2010

On explicit QTT representation of Laplace operator and its inverse

Vladimir A. Kazeev and Boris N. Khoromskij

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.

Received:
Dec 14, 2010
Published:
Dec 15, 2010
MSC Codes:
15A69, 65F99
Keywords:
tensor decompositions, low-rank approximation, Quantics Tensor Train, QTT, inverse Laplace operator

Related publications

inJournal
2012 Repository Open Access
Vladimir A. Kazeev and Boris N. Khoromskij

Low-Rank explicit QTT representation of the Laplace operator and inverse

In: SIAM journal on matrix analysis and applications, 33 (2012) 3, pp. 742-758