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A regularized Newton method for computing ground states of Bose-Einstein condensates. (English) Zbl 1433.82025
Summary: In this paper, we compute 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 we present a feasible gradient type method to solve this minimization problem, 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.

82M20 Finite difference methods applied to problems in statistical mechanics
82M22 Spectral, collocation and related (meshless) methods applied to problems in statistical mechanics
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
90C53 Methods of quasi-Newton type
65K10 Numerical optimization and variational techniques
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