A Regularized Newton Method for Computing Ground States of Bose–Einstein Condensates

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 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.describes the interaction between atoms in the condensate with the s-wave scattering length a s (positive for repulsive interaction and negative for attractive interaction) and

A Brief Introduction to Manifold Optimization

“…Adaptive regularized Newton method. From the perspective of Euclidean approximation, an adaptive regularized Newton algorithm (ARNT) is proposed for specific and general manifold optimization problems [92,97,41]. In the subproblem, the objective function is constructed by the second-order Taylor expansion in the Euclidean space and an extra regularization term, while the manifold constraint is kept.…”

Computing Ground States of Bose--Einstein Condensates with Higher Order Interaction via a Regularized Density Function Formulation

“…As a result, we can expect a great advantage of the rDF-APG method over methods based on the wave function formulation when the interaction energy part, especially the HOI part, is dominant. To better illustrate the advantage of the density function formulation we introduced, we compare the rDF-APG method with the regularized Newton method [41,44], which is one of the state-of-the-art numerical methods for computing ground states of BEC [44]. In fact, both methods share the similar strategy in designing the numerical methods, i.e.…”

“…In this paper, we aim to design a numerical method to compute the ground state defined in (1.7). When δ = 0, the MGPE (1.3) degenerates to the classical GPE and numerous methods have been proposed for this special case, such as the normalized gradient flow method [3,4,10,22,8,9], a Runge-Kutta spectral method with spectral discretization in space and Runge-Kutta type integration in time [35], Gauss-Seidel-type methods [20], a finite element method by directly minimizing the energy functional [11], a regularized Newton method [44], a method based on Riemannian optimization [24], a preconditioned nonlinear conjugate gradient method [2]. However, to our best knowledge, there are few numerical schemes proposed for the case δ > 0.…”

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

“…We next use the regularized Newton method in [28], denoted by RN, to solve the original BEC problem and the feasible gradient method with default parameters to solve the quadratic SDP (3.12). The deterministic and randomized versions are denoted by SDR1 and SDR2, respectively.…”