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

Bartosz Protas


Abstract: This presentation concerns a novel approach to the
computation of ground states in Bose-Einstein condensates, a topic
which receives significant attention in theoretical physic.  In the
proposed approach 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
in the system constraints the minimizers to lie on a Riemannian
manifold corresponding to the unit L2 norm. The idea developed in our
study is to transform the original constrained optimization problem to
an unconstrained problem on this (spherical) Riemannian manifold, so
that faster minimization algorithms can be applied. 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 new 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. 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. 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 non-linear interaction constant and
the rotation rate.

[Joint work with Ionut Danaila from Universite de Rouen]