Acceleration of the imaginary time method for spectrally computing the stationary states of Gross-Pitaevskii equations

International audienceThe aim of this paper is to propose a simple accelerated spectral gradient flow formulation for solving the Gross-Pitaevskii Equation (GPE) when computing the stationary states of Bose-Einstein Condensates. The new algorithm, based on the recent iPiano minimization algorithm [35], converges three to four times faster than the standard implicit gradient scheme. To support the method, we provide a complete numerical study for 1d-2d-3d GPEs, including rotation and dipolar terms

Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization

In this paper we combine concepts from Riemannian Optimization and the theory of Sobolev gradients to derive a new conjugate gradient method for direct minimization of the Gross-Pitaevskii energy functional with rotation. The conservation of the number of particles constrains the minimizers to lie on a manifold corresponding to the unit $L^2$ norm. The idea developed here is to transform the original constrained optimization problem to an unconstrained problem on this (spherical) Riemannian manifold, so that fast minimization algorithms can be applied as alternatives to more standard constrained formulations. First, we obtain Sobolev gradients using an equivalent definition of an $H^1$ inner product which takes into account rotation. Then, the Riemannian gradient (RG) steepest descent method is derived based on projected gradients and retraction of an intermediate solution back to the constraint manifold. Finally, we use the concept of the Riemannian vector transport to propose a Riemannian conjugate gradient (RCG) method for this problem. It is derived at the continuous level based on the "optimize-then-discretize" paradigm instead of the usual "discretize-then-optimize" approach, as this ensures robustness of the method when adaptive mesh refinement is performed in computations. We evaluate various design choices inherent in the formulation of the method and conclude with recommendations concerning selection of the best options. Numerical tests demonstrate that the proposed RCG method outperforms the simple gradient descent (RG) method in terms of rate of convergence. While on simple problems a Newton-type method implemented in the {\tt Ipopt} library exhibits a faster convergence than the (RCG) approach, the two methods perform similarly on more complex problems requiring the use of mesh adaptation. At the same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure

A Preconditioned Conjugated Gradient Method for Computing Ground States of Rotating Dipolar Bose-Einstein Condensates via Kernel Truncation Method for Dipole-Dipole Interaction Evaluation

International audienceIn this paper, we propose an efficient and accurate method to compute the ground state of 2D/3D rotating dipolar BEC by incorporating the Kernel Truncation Method (KTM) for Dipole-Dipole Interaction (DDI) evaluation into the newly-developed Preconditioned Conjugate Gradient (PCG) method [9]. Adaptation details of KTM and PCG, including multidimensional discrete convolution acceleration for KTM, choice of the preconditioners in PCG, are provided. The performance of our method is confirmed with extensive numerical tests, with emphasis on spectral accuracy of KTM and efficiency of ground state computation with PCG. Application of our method shows some interesting vortex lattice patterns in 2D and 3D respectively