Global convergence of an efficient splitting method for the defocusing Gross--Pitaevskii ground state problem
Abstract
For computing the ground state of the defocusing Gross--Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis--Yin three-operator splitting, which treats the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs one inversion of I−γΔ for the first scheme, and of I−γΔ+γV1 for the second, with V1 denoting the separable part of the potential, and on structured meshes both operators can be inverted by simple fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped linear finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the iteration variable. In three-dimensional tests with up to 9993 unknowns on one GPU, a simple variable step size rule makes the proposed splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.
Related
- Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system
- Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem
- Riemannian Hessian analysis of Sobolev gradient flows for the Gross--Pitaevskii ground state problem
- Weak and strong convergence of a general splitting approach for monotone multioperator inclusions
- Accelerated Convergence of a Second-Order Dynamical System and its Application to Splitting Algorithms for Comonotone Inclusions
BibTeX
@misc{li2026global,
title = {{Global convergence of an efficient splitting method for the defocusing Gross--Pitaevskii ground state problem}},
author = {Jiaxing Li and Shixin Zheng and Xiangxiong Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13509},
eprint = {2609.13509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13509},
}
RIS
TY - UNPB
TI - Global convergence of an efficient splitting method for the defocusing Gross--Pitaevskii ground state problem
AU - Jiaxing Li
AU - Shixin Zheng
AU - Xiangxiong Zhang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13509
UR - https://arxiv.org/abs/2609.13509
AB - For computing the ground state of the defocusing Gross--Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis--Yin three-operator splitting, which treats the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs one inversion of I−γΔ for the first scheme, and of I−γΔ+γV_1 for the second, with V_1 denoting the separable part of the potential, and on structured meshes both operators can be inverted by simple fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped linear finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the iteration variable. In three-dimensional tests with up to 999^3 unknowns on one GPU, a simple variable step size rule makes the proposed splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.
ER -