Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method

TL;DR

This paper introduces a preconditioned nonlinear conjugate gradient method combined with spectral discretization to efficiently compute ground states of rotating Bose-Einstein condensates, outperforming traditional methods especially in higher dimensions.

Contribution

The paper develops a novel preconditioned conjugate gradient algorithm tailored for spectral discretization to improve the computation of BEC ground states, with demonstrated efficiency in 1D, 2D, and 3D cases.

Findings

PCG method outperforms previous algorithms in 2D and 3D cases

Preconditioning significantly accelerates convergence

Spectral discretization enhances computational efficiency

Abstract

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial dis-cretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned gradient (PG) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.