×

zbMATH — the first resource for mathematics

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.

MSC:
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
Software:
GPELab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008) · Zbl 1147.65043
[2] Adhikari, SK, Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms, Phys. Lett. A, 265, 91-96, (2000)
[3] Aftalion, A; Du, Q, Vortices in a rotating Bose-Einstein condensate: critical angular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A, 64, 063603, (2001)
[4] Aftalion, A; Danaila, I, Three-dimensional vortex configurations in a rotating Bose-Einstein condensate, Phys. Rev. A, 68, 023603, (2003)
[5] Anderson, MH; Ensher, JR; Mattews, MR; Wieman, CE; Cornell, EA, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 198-201, (1995)
[6] Anglin, JR; Ketterle, W, Bose-Einstein condensation of atomic gases, Nature, 416, 211-218, (2002)
[7] Antoine, X; Duboscq, R, Gpelab, a Matlab toolbox to solve Gross-Pitaevskii equations I: computation of stationary solutions, Comput. Phys. Commun., 185, 2969-2991, (2014) · Zbl 1348.35003
[8] Antoine, X; Duboscq, R, Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates, J. Comput. Phys., 258, 509-523, (2014) · Zbl 1349.82027
[9] Bao, W, Ground states and dynamics of multi-component Bose-Einstein condensates, Multiscale Model. Simul., 2, 210-236, (2004) · Zbl 1062.82034
[10] Bao, W; Cai, Y, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135, (2013) · Zbl 1266.82009
[11] Bao, W; Cai, Y, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asia J. Appl. Math., 1, 49-81, (2011) · Zbl 1290.35236
[12] Bao, W; Cai, Y, Ground states and dynamics of spin-orbit-coupled Bose-Einstein condensates, SIAM J. Appl. Math., 75, 492-517, (2015) · Zbl 1335.35227
[13] Bao, W; Cai, Y; Wang, H, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229, 7874-7892, (2010) · Zbl 1198.82036
[14] Bao, W; Chern, IL; Lim, FY, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates, J. Comput. Phys., 219, 836-854, (2006) · Zbl 1330.82031
[15] Bao, W; Chern, IL; Zhang, Y, Efficient numerical methods for computing ground states of spin-1 Bose-Einstein condensates based on their characterizations, J. Comput. Phys., 253, 189-208, (2013) · Zbl 1349.82069
[16] Bao, W; Du, Q, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25, 1674-1697, (2004) · Zbl 1061.82025
[17] Bao, W; Tang, W, Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187, 230-254, (2003) · Zbl 1028.82500
[18] Bao, W; Wang, H, A mass and magnetization conservative and energy-diminishing numerical method for computing ground state of spin-1 Bose-Einstein condensates, SIAM J. Numer. Anal., 45, 2177-2200, (2007) · Zbl 1149.82018
[19] Bao, W; Wang, H; Markowich, PA, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci., 3, 57-88, (2005) · Zbl 1073.82004
[20] Barzilai, J; Borwein, JM, Two-point step size gradient methods, IMA J. Numer. Anal., 8, 141-148, (1988) · Zbl 0638.65055
[21] Bornemann, FA; Deuflhard, P, The cascadic multigrid method for elliptic problems, Numer. Math., 75, 135-152, (1996) · Zbl 0873.65107
[22] Bradley, CC; Sackett, CA; Tollett, JJ; Hulet, RG, Evidence of Bose-Einstein condensation in an atomic gas with attractive interations, Phys. Rev. Lett., 75, 1687-1690, (1995)
[23] Cancès, E; Chakir, R; Maday, Y, Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput., 45, 90-117, (2010) · Zbl 1203.65237
[24] Cerimele, MM; Chiofalo, ML; Pistella, F; Succi, S; Tosi, MP, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates, Phys. Rev. E, 62, 1382-1389, (2009)
[25] Chang, S-L; Chien, C-S; Jeng, B-W, Computing wave functions of nonlinear Schrödinger equations: a time-independent approach, J. Comput. Phys., 226, 104-130, (2007) · Zbl 1129.65077
[26] Chang, SM; Lin, WW; Shieh, SF, Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. Comput. Phys., 202, 367-390, (2005) · Zbl 1056.81088
[27] Chiofalo, ML; Succi, S; Tosi, MP, Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62, 7438-7444, (2000)
[28] Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods, MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2000) · Zbl 0958.65071
[29] Dalfovo, F; Giorgini, S; Pitaevskii, LP; Stringari, S, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 463-512, (1999)
[30] Danaila, I; Kazemi, P, A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32, 2447-2467, (2010) · Zbl 1216.35006
[31] Davis, KB; Mewes, MO; Andrews, MR; Druten, NJ; Durfee, DS; Kurn, DM; Ketterle, W, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75, 3969-3973, (1995)
[32] Dodd, RJ, Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates, J. Res. Natl. Inst. Stand. Technol., 101, 545-552, (1996)
[33] Edwards, M; Burnett, K, Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms, Phys. Rev. A, 51, 1382-1386, (1995)
[34] Fetter, AL, Rotating trapped Bose-Einstein condensates, Rev. Mod. Phys., 81, 647-691, (2009)
[35] Garcia-Ripoll, JJ; Perez-Garcia, VM, Optimizing Schrödinger functional using Sobolev gradients: applications to quantum mechanics and nonlinear optics, SIAM J. Sci. Comput., 23, 1315-1333, (2001) · Zbl 0999.65058
[36] Gross, EP, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20, 454-477, (1961) · Zbl 0100.42403
[37] Jiang, B; Dai, Y, A framework of constraint preserving update schemes for optimization on Stiefel manifold, Math. Program. A, 153, 535-575, (2015) · Zbl 1325.49037
[38] Leggett, AJ, Bose-Einstein condensation in the alkali gases: some fundamental concepts, Rev. Mod. Phys., 73, 307-356, (2001)
[39] Lieb, EH; Seiringer, R, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., 264, 505-537, (2006) · Zbl 1233.82004
[40] Lieb, EH; Seiringer, R; Yngvason, J, Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61, 043602, (2000)
[41] Matthews, MR; Anderson, BP; Haljan, PC; Hall, DS; Wieman, CE; Cornell, EA, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83, 2498-2501, (1999)
[42] Nocedal, J., Wright, S.J.: Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)
[43] Pethick, C.J., Smith, H.: Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2002)
[44] Pitaevskii, LP, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13, 451-454, (1961)
[45] Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Calrendon Press, Oxford (2003) · Zbl 1110.82002
[46] Raman, C; Abo-Shaeer, JR; Vogels, JM; Xu, K; Ketterle, W, Vortex nucleation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 87, 210402, (2001)
[47] Ruprecht, PA; Holland, MJ; Burrett, K; Edwards, M, Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms, Phys. Rev. A, 51, 4704-4711, (1995)
[48] Schneider, BI; Feder, DL, Numerical approach to the ground and excited states of a Bose-Einstein condensated gas confined in a completely anisotropic trap, Phys. Rev. A, 59, 2232, (1999)
[49] Sun, W., Yuan, Y.-X.: Optimization Theory and Methods, vol. 1 of Springer Optimization and Its Applications. Springer, New York (2006)
[50] Wen, Z; Milzarek, A; Ulbrich, M; Zhang, H, Adaptive regularized self-consistent field iteration with exact Hessian for electronic structure calculation, SIAM J. Sci. Comput., 35, a1299-a1324, (2013) · Zbl 1273.82004
[51] Wen, Z; Yin, W, A feasible method for optimization with orthogonality constraints, Math. Program. Ser. A., 142, 397-434, (2013) · Zbl 1281.49030
[52] Zhou, AH, An analysis of finite-dimensional approximations for the ground state solution of Bose-Einstein condensates, Nonlinearity, 17, 541-550, (2004) · Zbl 1051.35094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.