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 $L^2$ 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 $H^1$ inner product which takes into account rotation. Then, the Riemannian gradient (RG) steepest desce...