Riemannian optimization on low-rank tensor train manifolds for the Gross-Pitaevskii equation

Abstract

This work is concerned with the numerical computation of Gross-Pitaevskii ground states using the tensor train (TT) format. We regard the problem as the minimization of the Gross-Pitaevskii energy functional under a unit mass constraint, and propose a first-order Riemannian optimization scheme that preserves both the mass constraint and the low-rank representation throughout the optimization, by operating on the manifold of unit-norm functions of fixed TT-rank. On this manifold, we derive the energy-adaptive Riemannian gradient and show how to compute it efficiently in the TT format. Spatial discretization is performed using a spectral method with numerical integration, which allows the required computations to be carried out efficiently while preserving the low-rank format. The method is shown to substantially reduce computational time compared to full-rank computations, while maintaining accuracy for both single- and multicomponent Gross-Pitaevskii equations.

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BibTeX RIS

BibTeX

@misc{bioli2026riemannian,
  title = {{Riemannian optimization on low-rank tensor train manifolds for the Gross-Pitaevskii equation}},
  author = {Ivan Bioli and Carlo Marcati and Maxim Rakhuba},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37103},
  eprint = {2609.37103},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37103},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Riemannian optimization on low-rank tensor train manifolds for the Gross-Pitaevskii equation
AU  - Ivan Bioli
AU  - Carlo Marcati
AU  - Maxim Rakhuba
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37103
UR  - https://arxiv.org/abs/2609.37103
AB  - This work is concerned with the numerical computation of Gross-Pitaevskii ground states using the tensor train (TT) format. We regard the problem as the minimization of the Gross-Pitaevskii energy functional under a unit mass constraint, and propose a first-order Riemannian optimization scheme that preserves both the mass constraint and the low-rank representation throughout the optimization, by operating on the manifold of unit-norm functions of fixed TT-rank. On this manifold, we derive the energy-adaptive Riemannian gradient and show how to compute it efficiently in the TT format. Spatial discretization is performed using a spectral method with numerical integration, which allows the required computations to be carried out efficiently while preserving the low-rank format. The method is shown to substantially reduce computational time compared to full-rank computations, while maintaining accuracy for both single- and multicomponent Gross-Pitaevskii equations.
ER  -

Abstract

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