17 research outputs found
A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates
In this paper, we propose a regularized Newton method for computing ground
states of Bose-Einstein condensates (BECs), which can be formulated as an
energy minimization problem with a spherical constraint. The energy functional
and constraint are discretized by either the finite difference, or sine or
Fourier pseudospectral discretization schemes and thus the original infinite
dimensional nonconvex minimization problem is approximated by a finite
dimensional constrained nonconvex minimization problem. Then an initial
solution is first constructed by using a feasible gradient type method, which
is an explicit scheme and maintains the spherical constraint automatically. To
accelerate the convergence of the gradient type method, we approximate the
energy functional by its second-order Taylor expansion with a regularized term
at each Newton iteration and adopt a cascadic multigrid technique for selecting
initial data. It leads to a standard trust-region subproblem and we solve it
again by the feasible gradient type method. The convergence of the regularized
Newton method is established by adjusting the regularization parameter as the
standard trust-region strategy. Extensive numerical experiments on challenging
examples, including a BEC in three dimensions with an optical lattice potential
and rotating BECs in two dimensions with rapid rotation and strongly repulsive
interaction, show that our method is efficient, accurate and robust.Comment: 25 pages, 6 figure
Acceleration of the imaginary time method for spectrally computing the stationary states of Gross-Pitaevskii equations
International audienceThe aim of this paper is to propose a simple accelerated spectral gradient flow formulation for solving the Gross-Pitaevskii Equation (GPE) when computing the stationary states of Bose-Einstein Condensates. The new algorithm, based on the recent iPiano minimization algorithm [35], converges three to four times faster than the standard implicit gradient scheme. To support the method, we provide a complete numerical study for 1d-2d-3d GPEs, including rotation and dipolar terms
Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold
We propose a class of multipliers correction methods to minimize a
differentiable function over the Stiefel manifold. The proposed methods combine
a function value reduction step with a proximal correction step. The former one
searches along an arbitrary descent direction in the Euclidean space instead of
a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter
one minimizes a first-order proximal approximation of the objective function in
the range space of the current iterate to make Lagrangian multipliers
associated with orthogonality constraints symmetric at any accumulation point.
The global convergence has been established for the proposed methods.
Preliminary numerical experiments demonstrate that the new methods
significantly outperform other state-of-the-art first-order approaches in
solving various kinds of testing problems.Comment: 21 pages, 9 figures, 1 tabl
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
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 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 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.Comment: 28 pages, 13 figure
Constrained high-index saddle dynamics for the solution landscape with equality constraints
We propose a constrained high-index saddle dynamics (CHiSD) to search
index- saddle points of an energy functional subject to equality
constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax
framework, and its linear stability at the index- saddle point is proved. To
ensure the manifold properties, the CHiSD is numerically implemented using
retractions and vector transport. Then we present a numerical approach by
combining CHiSD with downward and upward search algorithms to construct the
solution landscape with equality constraints. We apply the Thomson problem and
the Bose--Einstein condensation as the numerical examples to demonstrate the
efficiency of the proposed method