Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose–Einstein condensates

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.

The crystallization conjecture: a review

“…In order to formalize the crystallization problem at positive temperature, it is convenient to consider the empirical measures (also called k-point correlation functions [88,173]), which are similar to the measure µ N introduced in (15). To be more precise, we define the family of measures, obtained by integrating with respect to all variables except k of them:…”

Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

“…Nevertheless, in [12], some examples show that the method does not converge when the rotation speed Ω is too large. In [12], the introduction of Krylov subspace iterative solvers (GMRES, BiCGStab) accelerated by simple operator-based preconditioners provides robust and fast iterative methods that can be easily extended to the mutli-components case.…”

“…We use BESP and BEFD for computing a stationary state of (58) 12 ) the reference stationary state. We report in Table 1 the quadratic error, the infinity norm error and finally the energy norm error between the reference and computed stationary states for BESP and BEFD.…”

“…As in the one-component case, preconditioned Krylov subspace solvers can be used to iteratively solve the associated linear systems (see [12]). …”

“…The linear system appearing in (111) and depending on n is solved by a Krylov subspace iterative solver (CGS, BiCGStab, GMRES) [12]. The method is called Relaxation SPectral (ReSP) scheme.…”