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A REGULARIZED NEWTON METHOD FOR COMPUTING GROUND STATES OF BOSE-EINSTEIN CONDENSATES∗

Xinming Wu, Zaiwen Wen, Weizhu Bao

Published Jan 20, 2016• 56 citations
OpenAlex

Abstract

Abstract. 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 fi-nite 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 gra-dient 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 subprob-lem 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 di-mensions 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. Key words. Bose-Einstein condensation, Gross-Pitaevskii equation, ground state, energy func-tional, spherical constraint, gradient type method, regularized Newton method. 1. Introduction. Since

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Abstract. 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 fi-nite 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 gra-dient 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 subprob-lem 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 di-mensions 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. Key words. Bose-Einstein condensation, Gross-Pitaevskii equation, ground state, energy func-tional, spherical constraint, gradient type method, regularized Newton method. 1. Introduction. Since
The computation of the ground states of spin-FFF Bose--Einstein condensates (BECs) can be formulated as an energy minimization problem with two quadratic constraints. We discretize the energy functional and constraints using the Fourier pseudospectral schemes and view the discretized problem as an optimization problem on manifold. Three different types of retractions to the manifold are designed. They enable us to apply various optimization methods on manifold to solve the problem. Specifically, an adaptive regularized Newton method is used together with a cascadic multigrid technique to accelerate the convergence. Our method is the first applicable algorithm for BECs with an arbitrary integer spin, including the complicated spin-3 BECs. Extensive numerical results on ground states of spin-1, spin-2, and spin-3 BECs with diverse interaction and optical lattice potential in one/two/three dimensions are reported to show the efficiency of our method and to demonstrate interesting physical phenomena.
Ground states and their characterization of spin-FFF Bose-Einstein condensates
The computation of the ground states of spin-FFF Bose-Einstein condensates (BECs) can be formulated as an energy minimization problem with two quadratic constraints. We discretize the energy functional and constraints using the Fourier pseudospectral schemes and view the discretized problem as an optimization problem on manifold. Three different types of retractions to the manifold are designed. They enable us to apply various optimization methods on manifold to solve the problem. Specifically, an adaptive regularized Newton method is used together with a cascadic multigrid technique to accelerate the convergence. According to our limited knowledege, our method is the first applicable algorithm for BECs with an arbitrary integer spin, including the complicated spin-3 BECs. Extensive numerical results on ground states of spin-1, spin-2 and spin-3 BECs with diverse interaction and optical lattice potential in one/two/three dimensions are reported to show the efficiency of our method and to demonstrate some interesting physical phenomena.
We propose and analyze a new numerical method for computing the ground state of the modified Gross--Pitaevskii equation for modeling the Bose--Einstein condensate with a higher order interaction by adapting the density function formulation and the accelerated projected gradient method. By reformulating the energy functional E(ϕ)E(\phi)E(ϕ) with ϕ\phiϕ, the wave function, in terms of the density ρ=∣ϕ∣2\rho=|\phi|^2ρ=∣ϕ∣2, the original nonconvex minimization problem for defining the ground state is then reformulated to a convex minimization problem. In order to overcome the semismoothness of the function ρ\sqrt{\rho}ρ​ in the kinetic energy part, a regularization is introduced with a small parameter 0<ε≪10<\varepsilon\ll10<ε≪1. Convergence of the regularization is established when ε→0\varepsilon\to0ε→0. The regularized convex optimization problem is discretized by the second order finite difference method. The convergence rates in terms of the density and energy of the discretization are established. The accelerated projected gradient method is adapted for solving the discretized optimization problem. Numerical results are reported to demonstrate the efficiency and accuracy of the proposed numerical method. Our results show that the proposed method is much more efficient than the existing methods in the literature, especially in the strong interaction regime.
ABSTRACT In this paper, we present a highly accurate discretization model for directly computing the ground states of Bose–Einstein condensates (BECs). The energy functional and constraint are discretized by introducing the fourth‐order compact finite difference and the composite Simpson quadrature, and thus the original infinite‐dimensional constrained minimization problem is approximated by a finite‐dimensional minimization problem with a spherical constraint uniformly. We develop a projected Riemannian gradient (PRG) algorithm to solve the optimization problem and establish the corresponding convergence results. Extensive numerical tests of non‐rotating BECs as well as rotating BECs under different potentials and parameter regimes are provided to verify the feasibility and high accuracy of our method. In addition, we apply it to investigate the ground states of multi‐component BECs and the numerical results again validate the power of this method.
Abstract. The computation of the ground state of special multicomponent Bose–Einstein condensates (BECs) can be formulated as an energy functional minimization problem with spherical constraints. It leads to a nonconvex quartic-quadratic optimization problem after suitable discretizations. First, we generalize the Newton-based methods for single-component BECs to the alternating minimization scheme for multicomponent BECs. Second, the global convergent alternating Newton-Noda iteration (ANNI) is proposed. In particular, we prove the positivity preserving property of ANNI under mild conditions. Finally, our analysis is applied to a class of more general “multiblock” optimization problems with spherical constraints. Numerical experiments are performed to evaluate the performance of proposed methods for different multicomponent BECs, including pseudo spin-1/2, antiferromagnetic spin-1 and spin-2 BECs. These results support our theory and demonstrate the efficiency of our algorithms.
Abstract. In this paper, we propose a preconditioned Riemannian conjugate gradient method for computing ground states of spin-1 Bose-Einstein condensates, which can be reformulated as an optimization problem with the mass and magnetization constraints. The energy functional and constraints can be discretized using Fourier pseudospectral schemes, thereby transforming the problem into an optimization task on a manifold. We derive vector transports by differentiating three existing retractions and propose an initial step size selection strategy based on the second-order approximation of the energy function for the proposed Riemannian optimization algorithm. In addition, a preconditioner is derived and used. To further accelerate the convergence, we combine the proposed algorithm with a multigrid method. Numerical experiments demonstrate the efficiency and accuracy of the proposed method while confirming the usefulness of our preconditioning technique and the effectiveness of the proposed strategy for initial step size selection.