Computation of extreme eigenvalues in higher dimensions using block tensor train format
Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets,and Dmitry Savostyanov
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Submission date: 09. Jun. 2013
published in: Computer physics communications, 185 (2014) 4, p. 1207-1216
DOI number (of the published article): 10.1016/j.cpc.2013.12.017
MSC-Numbers: 15A69, 65F10, 65F15, 82B28, 82B20
PACS-Numbers: 02.10.Ud, 02.10.Xm, 02.10.Yn, 02.60.Dc, 02.60.Pn, 75.10.Pq, 05.10.Cc
Keywords and phrases: high--dimensional problems, DMRG, MPS, tensor train format, low--lying eigenstates, Spin models
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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.