### Abstract

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT)/matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.

Original language | English |
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Title of host publication | Numerical Mathematics and Advanced Applications - ENUMATH 2013 |

Editors | A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso |

Place of Publication | Cham, Switzerland |

Publisher | Springer International Publishing |

Pages | 335-343 |

Number of pages | 9 |

ISBN (Print) | 9783319107042 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 103 |

### ASJC Scopus subject areas

- Modelling and Simulation
- Engineering(all)
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Mathematics

### Cite this

*Numerical Mathematics and Advanced Applications - ENUMATH 2013*(pp. 335-343). (Lecture Notes in Computational Science and Engineering; Vol. 103). Cham, Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-10705-9_33

**Corrected one-site density matrix renormalization group and alternating minimal energy algorithm.** / Dolgov, Sergey V.; Savostyanov, Dmitry V.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Numerical Mathematics and Advanced Applications - ENUMATH 2013.*Lecture Notes in Computational Science and Engineering, vol. 103, Springer International Publishing, Cham, Switzerland, pp. 335-343. https://doi.org/10.1007/978-3-319-10705-9_33

}

TY - CHAP

T1 - Corrected one-site density matrix renormalization group and alternating minimal energy algorithm

AU - Dolgov, Sergey V.

AU - Savostyanov, Dmitry V.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT)/matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.

AB - Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT)/matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.

UR - http://www.scopus.com/inward/record.url?scp=84919797964&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-10705-9_33

DO - 10.1007/978-3-319-10705-9_33

M3 - Chapter

SN - 9783319107042

T3 - Lecture Notes in Computational Science and Engineering

SP - 335

EP - 343

BT - Numerical Mathematics and Advanced Applications - ENUMATH 2013

A2 - Abdulle, A.

A2 - Deparis, S.

A2 - Kressner, D.

A2 - Nobile, F.

A2 - Picasso, M.

PB - Springer International Publishing

CY - Cham, Switzerland

ER -