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MiS Preprint
59/2013

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

Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets and Dmitry Savostyanov

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.

Received:
Jun 9, 2013
Published:
Jun 14, 2013
MSC Codes:
15A69, 65F10, 65F15, 82B28, 82B20
PACS:
02.10.Ud, 02.10.Xm, 02.10.Yn, 02.60.Dc, 02.60.Pn, 75.10.Pq, 05.10.Cc
Keywords:
high--dimensional problems, DMRG, MPS, tensor train format, low--lying eigenstates, Spin models

Related publications

inJournal
2014 Repository Open Access
Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets and Dmitry V. Savostyanov

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

In: Computer physics communications, 185 (2014) 4, pp. 1207-1216