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

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Abstract

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.

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Acknowledgements

Part of this work was done when the authors were visiting the Institute for Mathematical Sciences at the National University of Singapore in 2015. The work of Xinming Wu is supported in part by NSFC Grants 91330202 and 11301089 and Shanghai Science and Technology Commission Grant 17XD1400500. The work of Zaiwen Wen is supported in part by NSFC Grants 11322109 and 91330202. The work of Weizhu Bao is supported in part by the Ministry of Education of Singapore Grant R-146-000-196-112.

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Authors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Xinming Wu

  2. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Zaiwen Wen

  3. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Weizhu Bao

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  1. Xinming Wu
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  2. Zaiwen Wen
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Correspondence to Xinming Wu.

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Wu, X., Wen, Z. & Bao, W. A Regularized Newton Method for Computing Ground States of Bose–Einstein Condensates. J Sci Comput 73, 303–329 (2017). https://doi.org/10.1007/s10915-017-0412-0

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  • DOI: https://doi.org/10.1007/s10915-017-0412-0

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