Time Integration of Tensor Trains
Open accessAbstract
A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train/matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation, and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included.
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A natural order similar to a DMRG sweep optimizes computational overhead and gives robustness under overapproximation.
“There is a natural order similar to the DRMG sweep, which optimizes the computational overhead (and results in robustness in the case of overapproximation [27]).”
From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [27]
Each term in the TDVP equation corresponding to a single projector can be integrated exactly.
“The crucial insight leading to our new integration scheme is that, while the TDVP equation ( 3 N n=1 H(n) -N -1 n=1 K(n) FIG. 4. Right-hand side (up to the factor -i) of the TDVP equation [Eq.(3)]. [corresponding to a single term from Eq. ( 4)] can be integrated exactly [27].”
From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [27]
The tangent space projector can be decomposed as a sum of one-site projectors minus a sum of zero-site projectors.
“Equation ( 3 [27]), the tangent space projector can be decomposed as PT | (A) M MPS = N n=1 P [1:n-1] L ⊗ 1n ⊗ P [n+1:N] R - N-1 n=1 P [1:n] L ⊗ P [n+1:N] R , (4 )”
From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [27]
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Cited by
Sorted by citations
Unifying time evolution and optimization with matrix product states
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- Authors
- Lubich, C., Oseledets, I. V., & Vandereycken, B.
- Venue
- SIAM Journal on Numerical Analysis
- Published
- 2015-01
- Volume
- 53
- Issue
- 2
- Pages
- 917-941
- ISSN
- 0036-1429 (ISSN-L)
- 1095-7170
- OpenAlex
- W2051456448
- Cited by
- 183
- References
- 25
- Citations by year
- 2026: 24 · 2025: 24 · 2024: 25 · 2023: 19 · 2022: 19 · 2021: 24 · 2020: 13 · 2019: 12 · 2018: 13 · 2017: 2 · 2016: 6 · 2015: 2
- FWCI
- 8.64
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- SIAM Journal on Numerical Analysis (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutsche Forschungsgemeinschaft (GRK1838, SPP-1324)
- Russian Science Foundation (14-11-00659)
- Topics
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Physical Sciences › Mathematics › Computational Mathematics › Tensor decomposition and applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- tensor train, matrix product states, time integration, tangent space projection, quantum molecular dynamics, iterative processes
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