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Low-rank tensor trains for Gross–Pitaevskii ground states

Computing ground states of Bose–Einstein condensates with Riemannian optimization on manifolds of fixed-rank tensor trains.

Role: First author
With: C. Marcati (Lyon), M. Rakhuba
Tools: Python, JAX
Links: arXiv

The problem

The ground state of a Bose–Einstein condensate is the minimiser of the Gross–Pitaevskii energy under a unit-mass constraint, over $u \in H^1_0(\Omega)$ on a product domain $\Omega \subset \mathbb{R}^d$ in two or three dimensions:

\min_{\|u\|_{L^2(\Omega)} = 1} \; E(u) = \frac12 \int_\Omega |\nabla u|^2 + \frac12 \int_\Omega V u^2 + \frac{\beta}{2} \int_\Omega u^4 .

On a fine grid — and even more so for multicomponent condensates — the number of unknowns grows quickly and full-rank computations become expensive.

Approach

We represent the discrete wavefunction in the tensor train (TT) format and minimise the energy directly on the manifold of unit-norm functions of fixed TT-rank, so that both the mass constraint and the low-rank structure are preserved at every step. On this manifold we

derive the energy-adaptive Riemannian gradient and show how to compute it efficiently in TT format, and
discretise in space with a spectral method with numerical integration, which keeps all required operations within the low-rank format.

Outcome

The method substantially reduces computational time compared to full-rank computations while maintaining accuracy, for both single- and multicomponent Gross–Pitaevskii equations.