A Regularized Newton Method for Computing Ground States of Bose–Einstein Condensates
A Regularized Newton Method for Computing Ground States of Bose–Einstein Condensates
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DOI：
10.1007/s10915-017-0412-0
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发表时间：
2015-04
影响因子：
2.5
通讯作者：
Xinming Wu;Zaiwen Wen;W. Bao
中科院分区：
文献类型：
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作者：
Xinming Wu;Zaiwen Wen;W. Bao
在本文中，我们计算玻色-爱因斯坦凝聚体（BEC）的基态，它可以被表述为一个能量最小化问题与球约束。能量泛函和约束离散的有限差分，正弦或傅立叶伪谱离散计划，从而原来的无限维非凸极小化问题近似为有限维约束非凸极小化问题。然后，我们提出了一个可行的梯度型方法来解决这个最小化问题，这是一个显式的计划，并保持球面约束自动。为了加快梯度型方法的收敛速度，我们近似的能量泛函的二阶泰勒展开与正则化项在每个牛顿迭代，并采用级联多重网格技术选择初始数据。它导致了一个标准的信赖域子问题，我们再次解决它的可行梯度型方法。正则化牛顿法的收敛性是通过调整正则化参数作为标准的信赖域策略来建立的。大量的数值实验具有挑战性的例子，包括在三维的光学晶格势和旋转BEC在二维快速旋转和强烈排斥的相互作用，表明我们的方法是有效的，准确的和强大的。
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.