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

In this paper we combine concepts from Riemannian optimization P. -A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008 and the theory of Sobolev gradients J. W. Neuberger, Sobolev Gradients and Differential Equation, 2nd ed. , Springer, 2010 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 in the system constrains the minimizers to lie on a manifold corresponding to the unit L² norm. The idea developed here is to transform the original constrained optimization problem into 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 H¹ 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 carried out in the finite-element setting based on Lagrangian piecewise quadratic space discretization 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 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. Finally, the RCG method is extensively tested by computing complicated vortex configurations in rotating Bose--Einstein condensates, a task made challenging by large values of the nonlinear interaction constant and the rotation rate as well as by strongly anisotropic trapping potentials.