| Abstract: |
| This presentation concerns a novel approach to the computation of ground states in Bose-Einstein condensates where 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. We first 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. Then, 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. Finally, we will also discuss our on-going work on the design and implementation of the Riemannian Newton method for the
minimization of the Gross-Pitaevskii energy functional. |
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