References
- X. Antoine and R. Duboscq, GPELab, a matlab toolbox to solve Gross–Pitaevskii equations i: Computation of stationary solutions, Comput. Phys. Commun. 185(11) (2014), pp. 2969–2991.
- X. Antoine and R. Duboscq, 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 (2014), pp. 509–523.
- X. Antoine and R. Duboscq, Modeling and computation of Bose-Einstein condensates: Stationary states, nucleation, dynamics, stochasticity. In Nonlinear Optical and Atomic Systems: At the Interface of Physics and Mathematics, C. Besse, and J.C. Garreau, eds., Lecture Notes in Mathematics Vol. 2146, 2015, pp. 49–145.
- X. Antoine, A. Levitt, and Q. Tang, Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods, J. Comput. Phys. 343 (2017), pp. 92–109.
- X. Antoine, Q. Tang, and J. Zhang, On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations, Int. J. Comput. Math. 95(6–7) (2018), pp. 1423–1443.
- X. Antoine, Q. Tang, and Y. Zhang, On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys. 325 (2016), pp. 74–97.
- X. Antoine, Q. Tang, and Y. Zhang, A preconditioned conjugated gradient method for computing ground states of rotating dipolar Bose–Einstein condensates via kernel truncation method for dipole-dipole interaction evaluation, Commun. Comput. Phys. 24(4) (2018), pp. 966–988.
- W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose–Einstein condensation, Kinet. Relat. Models 6(1) (2013), pp. 1–135.
- W. Bao, I. Chern, and F. Lim, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates, J. Comput. Phys. 219(2) (2006), pp. 836–854.
- W. Bao and X. Dong, Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime, Numer. Math. 120 (2012), pp. 189–229.
- W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25(5) (2004), pp. 1674–1697.
- W. Bao, S. Jiang, Q. Tang, and Y. Zhang, Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys. 296 (2015), pp. 72–89.
- W. Bao and W. Tang, Ground-state solution of Bose–Einstein condensate by directly minimizing the energy functional, J. Comput. Phys. 187(1) (2003), pp. 230–254.
- W. Bao and X. Zhao, Comparison of numerical methods for the nonlinear Klein–Gordon equation in the nonrelativistic limit regime, J. Comput. Phys. 398 (2019), pp. 108886.
- D. Baye and J.M. Sparenberg, Resolution of the Gross–Pitaevskii equation with the imaginary-time method on a Lagrange mesh, Phys. Rev. E 82(5) (2010).
- M. Caliari, A. Ostermann, S. Rainer, and M. Thalhammer, A minimisation approach for computing the ground state of Gross–Pitaevskii systems, J. Comput. Phys. 228(2) (2009), pp. 349–360.
- M. Caliari and S. Rainer, GSGPEs: A MATLAB code for computing the ground state of systems of Gross–Pitaevskii equations, Comput. Phys. Commun. 184(3) (2013), pp. 812–823.
- J.L. Chiff, The Laplace Transform: Theory and Applications, Undergraduate Texts in Mathematics, Springer, 1999.
- I. Danaila and P. Kazemi, A new Sobolev gradient method for direct minimization of the Gross–Pitaevskii energy with rotation, SIAM J. Sci. Comput. 32(5) (2010), pp. 2447–2467.
- I. Danaila and B. Protas, Computation of ground states of the Gross–Pitaevskii functional via Riemannian optimization, SIAM J. Sci. Comput. 39(6) (2017), pp. B1102–B1129.
- C.M. Dion and E. Cances, Ground state of the time-independent Gross–Pitaevskii equation, Comput. Phys. Commun. 177(10) (2007), pp. 787–798.
- W. Liu and Y. Cai, Normalized gradient flow with Lagrange multiplier for computing ground states of Bose–Einstein condensates, SIAM J. Sci. Comput. 43(1) (2021), pp. 219–242.
- Y. Nesterov, Gradient methods for minimizing composite objective function. Core discussion paper, (2007). Available at http://www.ecore.be/DPs/dp_1191313936.pdf.
- Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k2). In Doklady Akademii Nauk, Vol. 269, Russian Academy of Sciences, 1983, pp. 543–547.
- Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, Massachusetts, 2004.
- B. O'Donoghue and E. Candès, Adaptive restart for accelerated gradient schemes, Found. Comput. Math. 15 (2013), pp. 715–732.
- J. Royer, Energy decay for the Klein–Gordon equation with highly oscillating damping, Ann. Henri Lebesgues 120 (2018), pp. 297–312.
- W. Su, S. Boyd, and E.J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, J. Mach. Learn. Res. 17 (2016), pp. 1–43.
- B. Wang, T.M. Nguyen, A.L. Bertozzi, R.G. Baraniuk, and S.J. Osher, Scheduled restart momentum for accelerated stochastic gradient descent. preprint, 2020. arXiv:2002.10583.
- X. Wu, Z. Wen, and W. Bao, A regularized Newton method for computing ground states of Bose–Einstein condensates, J. Sci. Comput. 73 (2017), pp. 303–329.