解一类非线性特征值问题的数值算例
|
摘要
刻画玻色-爱因斯坦凝聚态(BEC)的Gross-Pitaevskii方程通过差分方法离散,转化成一类非线性特征值问题(BEC问题).在这篇文章中,讨论了对BEC问题的求解方法,并给出数值算例.通过半定松弛的方法(SDP松弛方法)和交替方向乘子法(ADMM),计算BEC问题的最小非线性特征值的一个界;通过Lasserre半定松弛,可以依次地计算BEC问题的所有实非线性特征值.在数值算例中,从求解问题的规模和求解速度两方面比较了SDP松弛方法和ADMM,同时用matlab自带的fmincon方法来求解,初步比较了它们的数值计算结果.
The Gross-Pitaevskii equation, which depicts the Bose-Einstein condensate state(BEC),is discretized by a differential method and transformed into a class of nonlinear eigenvalue problems(BEC problem). This article discusses the solution to the BEC problem and give numerical examples. A bound of the minimum nonlinear eigenvalue of the BEC problem is calculated by the semidefinite programming relaxation method(SDP relaxation method)and the alternating direction multiplier method(ADMM); all real nonlinear eigenvalues of the BEC problem can be calculated sequentially by Lasserre semidefinite programming relaxation. In the numerical example, comparing the SDP relaxation method and ADMM from the scale of the problem and the speed of solving the problem. At the same time, using the fmincon method that comes with matlab to solve the problem and compare their results preliminarily.
The Gross-Pitaevskii equation, which depicts the Bose-Einstein condensate state(BEC),is discretized by a differential method and transformed into a class of nonlinear eigenvalue problems(BEC problem). This article discusses the solution to the BEC problem and give numerical examples. A bound of the minimum nonlinear eigenvalue of the BEC problem is calculated by the semidefinite programming relaxation method(SDP relaxation method)and the alternating direction multiplier method(ADMM); all real nonlinear eigenvalues of the BEC problem can be calculated sequentially by Lasserre semidefinite programming relaxation. In the numerical example, comparing the SDP relaxation method and ADMM from the scale of the problem and the speed of solving the problem. At the same time, using the fmincon method that comes with matlab to solve the problem and compare their results preliminarily.
引文
[1] Bao W, Cai Y. Mathematical theory and numerical methods for bose-einstein condensation[J].Kinetic and Related Models(KRM), 2013, 6(1):1-135.
[2]谢和虎.非线性特征值问题的多重网格算法[J].中国科学:数学,2015, 45(8):1193-1204.
[3] Hu J, Jiang B, Liu X, et al. A note on semidefinite programming relaxations for polynomial optimization over a single sphere[J]. Science China Mathematics, 2016, 59(8):1543-1562.
[4] Yang Y, Yang Q. On solving biquadratic optimization via semidefinite relaxation[J]. Computational Optimization and Applications, 2012, 53(3):845-867.
[5] Wang Y, Yin W, Zeng J. Global convergence of admm in nonconvex nonsmooth optimization[J].Journal of Scientific Computing, 2019, 78(1):29-63.
[6] Cui C, Dai Y, Nie J. All real eigenvalues of symmetric tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35(4):1582-1601.
[7] Nie J. The hierarchy of local minimums in polynomial optimization[J]. Mathematical Programming, 2015, 151(2):555-583.
[8] Jia S, Xie H, Xie M, et al. A full multigrid method for nonlinear eigenvalue problems[J]. Science China, 2016, 59(10):1-12.
[9] Henrion D, Lasserre J B, Lofberg J. Gloptipoly 3:moments, optimization and semidefinite programming[J]. Optimization Methods and Software, 2009, 24(4-5):761-779.
[10] Sturm J F. Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones[J]. Optimization Methods and Software, 1999, 11(1-4):625-653.
[11] Kolda T G, Mayo J R, et al. An adaptive shifted power method for computing generalized tensor eigenpairs[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35(4):1095-1124.
[2]谢和虎.非线性特征值问题的多重网格算法[J].中国科学:数学,2015, 45(8):1193-1204.
[3] Hu J, Jiang B, Liu X, et al. A note on semidefinite programming relaxations for polynomial optimization over a single sphere[J]. Science China Mathematics, 2016, 59(8):1543-1562.
[4] Yang Y, Yang Q. On solving biquadratic optimization via semidefinite relaxation[J]. Computational Optimization and Applications, 2012, 53(3):845-867.
[5] Wang Y, Yin W, Zeng J. Global convergence of admm in nonconvex nonsmooth optimization[J].Journal of Scientific Computing, 2019, 78(1):29-63.
[6] Cui C, Dai Y, Nie J. All real eigenvalues of symmetric tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35(4):1582-1601.
[7] Nie J. The hierarchy of local minimums in polynomial optimization[J]. Mathematical Programming, 2015, 151(2):555-583.
[8] Jia S, Xie H, Xie M, et al. A full multigrid method for nonlinear eigenvalue problems[J]. Science China, 2016, 59(10):1-12.
[9] Henrion D, Lasserre J B, Lofberg J. Gloptipoly 3:moments, optimization and semidefinite programming[J]. Optimization Methods and Software, 2009, 24(4-5):761-779.
[10] Sturm J F. Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones[J]. Optimization Methods and Software, 1999, 11(1-4):625-653.
[11] Kolda T G, Mayo J R, et al. An adaptive shifted power method for computing generalized tensor eigenpairs[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35(4):1095-1124.