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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
21/2010

Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances

Boris N. Khoromskij

Abstract

In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing.

The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent $N$-$d$ tensors with log-volume complexity, $O(d \log N)$.

We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modeling---the tool apparently works and gives the promise for future use in challenging high-dimensional applications.

Received:
May 2, 2010
Published:
May 4, 2010
MSC Codes:
65F30, 65F50, 65N35
Keywords:
high-dimensional problems, rank structured tensor approximation, quantics folding of vectors, FEM/BEM, computational quantum chemistry, stochastic PDEs

Related publications

inJournal
2012 Repository Open Access
Boris N. Khoromskij

Tensors-structured numerical methods in scientific computing : survey on recent advances

In: Chemometrics and intelligent laboratory systems, 110 (2012) 1, pp. 1-19