We present an accelerated, stabilizer-based normalized gradient flow (ASNGF) framework with a nonlocal inertial (or momentum) mechanism for computing stationary states of Bose-Einstein condensates. Two schemes, ASNGF-I and ASNGF-II, are developed, and their discrete Lyapunov energy dissipation properties are established under the stated conditions. Linearized stability analyses and numerical parameter studies provide practical guidance for selecting the tunable parameters and demonstrate the important role of the nonlocal Laplacian term in the inertial dynamics. For suitable parameter scalings, the observed iteration counts are essentially insensitive to the time step over the tested range, including τ=10±60. Numerical experiments on several Gross-Pitaevskii models show that ASNGF can substantially reduce the iteration count compared with conventional normalized gradient-flow schemes for the tested problems and parameter settings. The results also reveal that its convergence behavior is problem- and parameter-dependent. The inertial mechanism can enhance exploration of non-convex energy landscapes but does not guarantee convergence to a global minimizer. These results demonstrate the potential of ASNGF as an efficient framework for stationary-state computations in BECs.
We present an accelerated, stabilizer-based normalized gradient flow (ASNGF) framework with a nonlocal inertial (or momentum) mechanism for computing stationary states of Bose-Einstein condensates. Two schemes, ASNGF-I and ASNGF-II, are developed, and their discrete Lyapunov energy dissipation properties are established under the stated conditions. Linearized stability analyses and numerical parameter studies provide practical guidance for selecting the tunable parameters and demonstrate the important role of the nonlocal Laplacian term in the inertial dynamics. For suitable parameter scalings, the observed iteration counts are essentially insensitive to the time step over the tested range, including τ=10±60. Numerical experiments on several Gross-Pitaevskii models show that ASNGF can substantially reduce the iteration count compared with conventional normalized gradient-flow schemes for the tested problems and parameter settings. The results also reveal that its convergence behavior is problem- and parameter-dependent. The inertial mechanism can enhance exploration of non-convex energy landscapes but does not guarantee convergence to a global minimizer. These results demonstrate the potential of ASNGF as an efficient framework for stationary-state computations in BECs.
The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. We apply GFSI to the two-component scenario with Josephson junction and rotating term, which is one of the most important and topical models in multi-component Bose-Einstein condensates (MBECs), and rigorously establish the following fundamental results for the first time. By introducing a Lagrange multiplier to reformulate GFSI into an equivalent form, we prove its energy dissipation property and global convergence to stationary states. More significantly, we uncover an intrinsic connection between this classical numerical PDE discretization rooted in imaginary-time evolution and Riemannian optimization, a state-of-the-art mathematical framework for manifold-constrained optimization. This connection enables us to fully characterize the local convergence behavior of GFSI within the Riemannian optimization framework. Together with the aforementioned global convergence result, this yields a complete global--local convergence theory for GFSI. Finally, numerical experiments comprehensively validate the theoretically predicted energy dissipation and convergence properties.
The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evolution method is one of the most popular techniques for computing the ground states of Bose--Einstein condensates (BECs). In this paper, we revisit the time discretizations for the GFDN and its generalization to the multicomponent BECs. Several widely used time discretizations are demonstrated to be not accurate for computing the ground state solution in the general case, especially for the multicomponent BECs with two or more constraints even for the most accepted linearized backward Euler schemes. More precisely, these schemes usually converge to a solution with an error depending on the time step size. To accurately and efficiently compute the ground state solution of BECs, we propose the gradient flow with Lagrange multiplier (GFLM) method which can be viewed as the modified GFDN by introducing the explicit Lagrange multiplier terms or an approximation of the continuous normalized gradient flow. Through analysis and numerical computation, we clarify that, in order to accurately compute the ground state solution, the GFDN method must be discretized in very special ways in time such as the linearized backward Euler scheme, while the GFLM can be discretized by various schemes and works for multicomponent BEC with multiple constraints. We propose a simple backward-forward Euler scheme to discretize the GFLM, where only a linear system with constant coefficients needs to be solved at each time step. Taking the spin-1 BEC as an example, we demonstrate how the traditional GFDN approach fails in computing the ground state solution of general multicomponent BECs accurately. The proposed GFLM with backward-forward Euler discretization is shown to be accurate and efficient. Extensive numerical results are provided to verify our claims.
Abstract. We propose and analyze an efficient and accurate numerical method for computing ground states of spin-2 Bose–Einstein condensates (BECs) by using the normalized gradient flow (NGF). In order to successfully extend the NGF to spin-2 BECs which have five components in the vector wave function but with only two physical constraints on total mass conservation and magnetization conservation, two important techniques are introduced for designing the proposed numerical method. The first one is to systematically investigate the ground state structure and property of spin-2 BECs within a spatially uniform system, which can be used for how to properly choose initial data in the NGF for computing ground states of spin-2 BECs. The second one is to introduce three additional projection conditions based on the relations between the chemical potentials, together with the two existing physical constraints, such that the five projection parameters used in the projection step of the NGF can be uniquely determined. Then a backward-forward Euler finite difference method is adapted to discretize the NGF. We prove rigorously that there exists a unique solution of the nonlinear system for determining the five projection parameters in the full discretization of the NGF under a mild condition on the time step size. Extensive numerical results on ground states of spin-2 BECs with different types of phases and under different potentials are reported to show the efficiency and accuracy of the proposed numerical method and to demonstrate several interesting physical phenomena on ground states of spin-2 BECs.
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) with ϕ, the wave function, in terms of the density ρ=∣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 in the kinetic energy part, a regularization is introduced with a small parameter 0<ε≪1. Convergence of the regularization is established when ε→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 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
In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose–Einstein condensates (BECs) by using the normalized gradient flow or imaginary time method. The key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization. This allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of a single-component BEC to compute the ground state of spin-1 BECs. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method, is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BECs with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method.