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Computation of extreme eigenvalues in higher dimensions using block tensor train format

Computation of extreme eigenvalues in higher dimensions using block tensor train format
Computation of extreme eigenvalues in higher dimensions using block tensor train format
We consider an approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train format (TT) for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. Applying a block version of the TT format to several vectors simultaneously, we compute the low-lying eigenstates of a system by minimization of a block Rayleigh quotient performed in an alternating fashion for all dimensions. For several numerical examples, we compare the proposed method with the deflation approach when the low-lying eigenstates are computed one-by-one, and also with the variational algorithms used in quantum physics.
high-dimensional problems, DMRG, MPS, tensor train format, low-lying eigenstates
0010-4655
1207-1216
Dolgov, Sergey
a6b5facb-530c-4324-bc62-73f81ea00e23
Khoromskij, Boris
6d972a5f-2ee0-4491-9fa3-d3547098154c
Oseledets, Ivan
bfd5503c-673b-47f2-8861-51d5fb5c77fc
Savostyanov, Dmitry
49d88c5f-b159-4dff-af88-5b9a5ff18322
Dolgov, Sergey
a6b5facb-530c-4324-bc62-73f81ea00e23
Khoromskij, Boris
6d972a5f-2ee0-4491-9fa3-d3547098154c
Oseledets, Ivan
bfd5503c-673b-47f2-8861-51d5fb5c77fc
Savostyanov, Dmitry
49d88c5f-b159-4dff-af88-5b9a5ff18322

Dolgov, Sergey, Khoromskij, Boris, Oseledets, Ivan and Savostyanov, Dmitry (2014) Computation of extreme eigenvalues in higher dimensions using block tensor train format. Computer Physics Communications, 185 (4), 1207-1216. (doi:10.1016/j.cpc.2013.12.017).

Record type: Article

Abstract

We consider an approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train format (TT) for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. Applying a block version of the TT format to several vectors simultaneously, we compute the low-lying eigenstates of a system by minimization of a block Rayleigh quotient performed in an alternating fashion for all dimensions. For several numerical examples, we compare the proposed method with the deflation approach when the low-lying eigenstates are computed one-by-one, and also with the variational algorithms used in quantum physics.

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Submitted date: 12 June 2013
e-pub ahead of print date: 27 December 2013
Published date: April 2014
Keywords: high-dimensional problems, DMRG, MPS, tensor train format, low-lying eigenstates
Organisations: Magnetic Resonance

Identifiers

Local EPrints ID: 353747
URI: http://eprints.soton.ac.uk/id/eprint/353747
ISSN: 0010-4655
PURE UUID: 32ca6ceb-0b0c-4ba9-a3e6-492e8ed9512e

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Date deposited: 17 Jun 2013 10:42
Last modified: 14 Mar 2024 14:09

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Contributors

Author: Sergey Dolgov
Author: Boris Khoromskij
Author: Ivan Oseledets
Author: Dmitry Savostyanov

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