Minimizing the Gross-Pitaevskii Energy Functional With the Sobolev Gradient — Analytical and Numerical Results

Kazemi, Parimah; Eckart, M.

doi:10.1142/s0219876210002301

Cited by 18 publications

(20 citation statements)

References 29 publications

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“…In this subsection we define the evolution equation we use in the Hilbert space setting and give our global existence and convergence result for the constrained minimization problem. Here we extend the results obtained in [23] for the case of the Gross-Pitaevskii energy without rotation. In this work, as well as in [23], we move away from the general theory of Sobolev gradients as presented in [27], since the criteria for asymptotic convergence of the evolution equation for constrained minimization problems is not available in [27].…”

Section: 2supporting

confidence: 80%

“…The idea below the following analysis is to show that the GP energy functional with rotation has the same properties as the GP energy without rotation if the norm · HA is used. We thus can adapt the results obtained in [23] to our case. We start by noting that, due to the mass conservation, one can add a multiple of D |u| 2 to the Gross-Pitaevskii energy and the resulting functional will have the same minimizers as the original functional.…”

Section: 2mentioning

confidence: 97%

“…In conclusion, by projecting the Sobolev gradient of E at z(t) into the null space of β ′ (z(t)) for each t, we get that z(t) satisfies the mass constraint for all t (see [27] for a more detailed development on this topic). For numerical implementation purposes, we give below an heuristic derivation of the explicit expression of the projection (see [23] for a more rigorous demonstration). If, for the sake of simplicity, G = ∇ X E(u) denotes the Sobolev gradient gradient of E at u, the projected gradient is determined from the following two conditions:…”

Section: Constrained Energy Minimizationmentioning

confidence: 99%

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A New Sobolev Gradient Method for Direct Minimization of the Gross–Pitaevskii Energy with Rotation

Danaila¹,

Kazemi²

2010

SIAM J. Sci. Comput.

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Abstract. In this paper we improve traditional steepest descent methods for the direct minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels. We first define a new inner product to equip the Sobolev space H 1 and derive the corresponding gradient. Secondly, for the treatment of the mass conservation constraint, we use a new projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods. The descent method with these two new ingredients is studied theoretically in a Hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the GP energy. The new method is implemented in both finite difference and finite element twodimensional settings and used to compute various complex configurations with vortices of rotating Bose-Einstein condensates. The new Sobolev gradient method shows better numerical performances compared to classical L 2 or H 1 gradient methods, especially when high rotation rates are considered.

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Section: 2supporting

confidence: 80%

Section: 2mentioning

confidence: 97%

Section: Constrained Energy Minimizationmentioning

confidence: 99%

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A New Sobolev Gradient Method for Direct Minimization of the Gross–Pitaevskii Energy with Rotation

Danaila¹,

Kazemi²

2010

SIAM J. Sci. Comput.

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“…tangent to the constraint manifold M at u n . As was shown in [30,41], the associated projection operator P un,X can be expressed in the following general form…”

Section: Steepest Descent Sobolev Gradientmentioning

confidence: 99%

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

Danaila

Protas

2017

SIAM J. Sci. Comput.

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Abstract. In this paper we combine concepts from Riemannian Optimization [P.-A. Absil, R. Mahony, R. Sepulchre, Optimization algorithms on matrix manifolds, Princeton University Press, 2008] and the theory of Sobolev gradients [J. W. Neuberger, Sobolev gradients, 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 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 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 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 non-linear interaction constant and the rotation rate as well as by strongly anisotropic trapping potentials.

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“…(1) One level minimization methods that use the variational structure of the problem to find a local minimum point of the variational energy functional J, e.g., [2,3,5,10,13]. These methods are usually stable, have less dependence on an initial guess and can treat degeneracy.…”

Section: λU(x) = −∆U(x) + V(x)u(x) + κF (X |U(x)|)u(x)mentioning

confidence: 99%

An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schrödinger equation with symmetry

Wang

Zhou

2013

Numerical Methods Partial

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An orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schrödinger equation with symmetry. Since such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive vs insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots.

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Minimizing the Gross-Pitaevskii Energy Functional With the Sobolev Gradient — Analytical and Numerical Results

Cited by 18 publications

References 29 publications

A New Sobolev Gradient Method for Direct Minimization of the Gross–Pitaevskii Energy with Rotation

A New Sobolev Gradient Method for Direct Minimization of the Gross–Pitaevskii Energy with Rotation

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

An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schrödinger equation with symmetry

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