We consider approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train (TT) format for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. We approximate several low-lying eigenvectors simultaneously in the block version of the TT format. The computation is done by the alternating minimization of the block Rayleigh quotient sequentially for all TT cores. The proposed method combines the advances of the density matrix renormalization group (DMRG) and the variational numerical renormalization group (vNRG) methods. We compare the performance of the proposed method with several versions of the DMRG codes, and show that it may be preferable for systems with large dimension and/or mode size, or when a large number of eigenstates is sought.
Bibliographical note© 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
- High-dimensional problems
- Tensor train format
- Low-lying eigenstates
Dolgov, S., Khoromskij, B., Oseledets, I., & Savostyanov, D. (2013). Computation of extreme eigenvalues in higher dimensions using block tensor train format. Computer Physics Communications, 185(4), 1207-1216. https://doi.org/10.1016/j.cpc.2013.12.017