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

Abstract

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.

28 pages, 13 figures

References in corpus (2)

• Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method
• Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation

Cited by in corpus (11)

• Mathematical models and numerical methods for spinor Bose-Einstein condensates
• Second-order flows for computing the ground states of rotating Bose-Einstein condensates
• Vortex patterns in the almost-bosonic anyon gas
• Computing the least action ground state of the nonlinear Schrödinger equation by a normalized gradient flow
• Computing the action ground state for the rotating nonlinear Schrödinger equation
• A normalized gradient flow method with attractive-repulsive splitting for computing ground states of Bose-Einstein condensates with higher-order interaction
• On the characterization of vortex configurations in the steady rotating Bose--Einstein condensates
• Convergence analysis of direct minimization and self-consistent iterations
• Systematic search for singularities in 3D Euler flows
• Computing ground states of Bose-Einstein Condensates with higher order interaction via a regularized density function formulation
• Nehari manifold optimization and its application for finding unstable solutions of semilinear elliptic PDEs