CONVERGENCE RATE ESTIMATION ON SS-HOPM FOR NONLINEAR EIGENVALUE PROBLEMS

Tang Yaozong, Yang Qingzhi

Mathematica Numerica Sinica ›› 2021, Vol. 43 ›› Issue (4) : 529-538.

PDF(363 KB)
PDF(363 KB)
Mathematica Numerica Sinica ›› 2021, Vol. 43 ›› Issue (4) : 529-538. DOI: 10.12286/jssx.j2020-0713
Articles

CONVERGENCE RATE ESTIMATION ON SS-HOPM FOR NONLINEAR EIGENVALUE PROBLEMS

  • Tang Yaozong1,2, Yang Qingzhi1,2
Author information +
History +

Abstract

In solving the nonlinear eigenvalue problems originated from Bose-Einstein Condensation, the shifted symmetric higher-order power method (SS-HOPM for short) not only has high computational efficiency, but also has point-wise convergence. However, the convergence rate of SS-HOPM has not been given. We apply the bound of the Kurdyka-Lojasiewicz (K-L) exponent of polynomial to the Lagrange function of the optimization problem involved in this paper, then we obtain sublinear convergence rate of the SS-HOPM, which can explain the calculation efficiency of the algorithm theoretically.

Key words

nonlinear eigenvalues / Bose-Einstein Condensation / SS-HOPM / convergence rate estimation

Cite this article

Download Citations
Tang Yaozong, Yang Qingzhi. CONVERGENCE RATE ESTIMATION ON SS-HOPM FOR NONLINEAR EIGENVALUE PROBLEMS. Mathematica Numerica Sinica, 2021, 43(4): 529-538 https://doi.org/10.12286/jssx.j2020-0713

References

[1] Bao W Z, Cai Y Y. Mathematical theory and numerical methods for Bose-Einstein condensation[J]. Kinetic and Related Models, 2013, 6(1):1-135.
[2] Bao W Z, Chern I L, Zhang Y Z. Efficient numerical methods for computing ground states of spin-1 Bose-Einstein condensates based on their characterizations[J]. Journal of Computational Physics, 2013, 253:189-208.
[3] Antoine X, Duboscq R. Modeling and computation of Bose-Einstein condensates:stationary states, nucleation, dynamics, stochasticity[C]. Lecture Notes in Mathematics, 2015, 49-145.
[4] Gross E P. Structure of a quantized vortex in Boson systems[J]. Nuovo Cimento, 1961, 20(3):454.
[5] Pitaevskii L P. Vortex lines in an imperfect Bose gas[J]. Soviet Physics JETP, 1961, 13(2):451-454.
[6] Zhang N, Xu F, Xie H H. An efficient multigrid for ground state solution of Bose-Einstein condensates[J]. International Journal of Numerical Analysis and Modeling, 2019, 16(5):789-803.
[7] Jia S H, et al. A full multigrid method for nonlinear eigenvalue problems[J]. Science China Mathematics, 2016, 59(10):2037-2048.
[8] 谢和虎. 非线性特征值的多重网格算法[J]. 中国科学:数学, 2015, 45(8):1193——1204.
[9] Wu X M, Wen Z W, Bao W Z. A regularized Newton method for computing ground states of Bose-Einstein condensates[J]. Journal of Scientific Computing, 2017, 73(1):303-329.
[10] Cancès E, Chakir R, Maday Y. Numerical analysis of nonlinear eigenvalue problems, Journal of Scientific Computing[J]. 2010, 45(1-3):90-117.
[11] Tang Y Z, Yang Q Z, Huang P F. SS-HOPM for BEC-like nonlinear eigenvalue problems[J]. 高等学校计算数学学报, 2020, 42(2):163——192.
[12] Lieb E H, Seiringer R, Yngvason J. Bosons in a trap:A rigorous derivation of the Gross-Pitaevskii energy functional, Physical Review[J]. 2001, A61:043602.
[13] Hu J, et al. A note on semi-definite programming relaxations for polynomial optimization over a single sphere[J]. Science China Mathematics, 2016, 59(8):1543-1560.
[14] 谢和虎. 子空间扩展算法及应用[J]. 数值计算与计算机应用, 2020, 41(3):169——191.
[15] Adhikari S K. Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms[J]. Physics Letters A, 2000, 265(1):91-96.
[16] Edwards M, Burnett K. Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms[J]. Physical Review A, 1995, 51(2):1382-1386.
[17] Dodd R J. Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates[J]. Journal of Research of the National Institute of Standards and Technology, 1996, 101(2):545-552.
[18] Schneider B I, Fede D L. Numerical approach to the ground and excited states of a Bose-Einstein condensed gas confined in a completely anisotropic trap[J]. Physical Review, 1999, 59(3):2232-2242.
[19] Bao W Z, Tang W J. Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional[J]. Journal of Computational Physics, 2003, 187(1):230-254.
[20] Chang S L, Chien C S, Jeng B W. Computing wave functions of nonlinear Schrödinger equations:A time-independent approach[J]. Journal of Computational Physics, 2007, 226(1):104-130.
[21] Chang S M, Lin W W, Shieh S F. Gauss-Seidel type methods for energy states of a multicomponent Bose-Einstein condensate[J]. Journal of Computational Physics, 2005, 202(1):367-390.
[22] Aftalion A, Danaila I. Three-dimensional vortex configurations in a rotating Bose Einstein condensate[J]. Physical Review, 2003, 68(2):92-94.
[23] Cerimele M M, et al. Numerical solution of the Gross-Pitaevskii equation using an explicit finitedifference scheme:An application to trapped Bose-Einstein condensates[J]. Physical Review, 2009, 62(1):1382-1389.
[24] Chiofalo M, Succi S, Tosi M. Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm[J]. Physical Review E, 2000, 62(5):7438-7444.
[25] Ruprecht P A, et al. Time-dependent solution of the nonlinear Schrödinger equation for Bosecondensed trapped neutral atoms[J]. Physical Review A, 1995, 51(6):4704-4711.
[26] Garcia-Ripoll J J, Perez-Garcia V M. Optimizing Schrödinger functionals using sobolev gradients:Applications to quantum mechanics and nonlinear optics[J]. SIAM Journal on Scientific Computing, 2001, 23(4):1316-1334.
[27] Bao W Z, Wang H Q. A mass and magnetization conservative and energy-diminishing numerical method for computing ground state of spin-1 Bose-Einstein condensates[J]. SIAM Journal on Numerical Analysis, 2007, 45(5):2177-2200.
[28] Danaila I, Kazemi P. A new Sobolev gradient method for direct minimization of the GrossPitaevskii energy with rotation[J]. SIAM Journal on Scientific Computing, 2010, 32(5):2447-2467.
[29] 杨庆之, 黄鹏斐, 刘亚君. 解一类非线性特征值的数值算例[J]. 数值计算与计算机应用, 2019, 40(2):130——142.
[30] Tang Y Z, Yang Q Z, Luo G. Convergence analysis on SS-HOPM for BEC-like nonlinear eigenvalue problems[J]. Journal of Computational Mathematics, 2021, 39(4):621-632.
[31] Didier D, Krzysztof K. Explicit bounds for the Lojasiewicz exponent in the gradient inequality for polynomials[J]. Annal Polonici Mathematici, 2005, 87(1):51-61.
[32] Kolda T G, Bader B W, Kenny J P. Higher-order web link analysis using multilinear algebra[C]. Proceedings of the 5th IEEE International Conference on Data Mining, Houston, TX., 2005, 242-249.
[33] Pierre Comon, et al. Symmetric tensors and symmetric tensor rank, SCCM Technical Report 06-02, Stanford University, 2006.
[34] Kolda T G, Mayo J R. Shifted power method for computing tensor eigenpairs[J]. SIAM Journal on Matrix Analysis and Applications, 2011, 32(4):1095-1124.
[35] Absil P A, Manhony R, Andrews B. Convergence of the iterates of descent methods for analytic cost functions[J]. SIAM Journal on Optimization, 2005, 16(2):531-547.
[36] Uschmajew A. A new convergence proof for the higher-order power method and generalizations[J]. Pacific Journal of Optimization, 2015, 11(2):309-321.
[37] Hu S L, Li G Y. Convergence rate analysis for the higher order power method in best rank one approximations of tensors[J]. Numerische Mathematik, 2018, 140(4):993-1031.
PDF(363 KB)

720

Accesses

0

Citation

Detail

Sections
Recommended

/