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Energy preserving model order reduction of the nonlinear Schrödinger equation

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Abstract

An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.

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References

  1. Afkham, B.M., Hesthaven, J.S.: Structure preserving model reduction of parametric Hamiltonian systems. SIAM J. Sci. Comput. 39(6), A2616–A2644 (2017). https://doi.org/10.1137/17M1111991

    Article  MathSciNet  MATH  Google Scholar 

  2. Alla, A., Kutz, J.: Randomized model order reduction ArXiv e-prints (2016)

  3. Alla, A., Kutz, J.N.: Nonlinear model order reduction via dynamic mode decomposition. SIAM J. Sci. Comput. 39(5), B778–B796 (2017). https://doi.org/10.1137/16M1059308

    Article  MathSciNet  MATH  Google Scholar 

  4. Antil, H., Heinkenschloss, M., Sorensen Danny, C.: Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems. In: Quarteroni, A., Rozza, G. (eds.) Reduced Order Methods for Modeling and Computational Reduction, MS & A - Modeling, Simulation and Applications, vol. 9, pp 101–136. Springer International Publishing (2014), https://doi.org/10.1007/978-3-319-02090-7_4

    MATH  Google Scholar 

  5. Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621–2633 (2013). https://doi.org/10.1016/j.cpc.2013.07.012

    Article  MathSciNet  MATH  Google Scholar 

  6. Antoine, X., Duboscq, R.: GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations i: Computation of stationary solutions. Comput. Phys. Commun. 185(11), 2969–2991 (2014). https://doi.org/10.1016/j.cpc.2014.06.026

    Article  MATH  Google Scholar 

  7. Antoine, X., Duboscq, R.: GPELab, a matlab toolbox to solve Gross-Pitaevskii equations ii: Dynamics and stochastic simulations. Comput. Phys. Commun. 193, 95–117 (2015). https://doi.org/10.1016/j.cpc.2015.03.012

    Article  MATH  Google Scholar 

  8. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002). https://doi.org/10.1137/S0036142901384162

    Article  MathSciNet  MATH  Google Scholar 

  9. Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control 53(10), 2237–2251 (2008). https://doi.org/10.1109/TAC.2008.2006102

    Article  MathSciNet  MATH  Google Scholar 

  10. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinetic Relat. Models 6(1), 1–135 (2013). https://doi.org/10.3934/krm.2013.6.1

    Article  MathSciNet  MATH  Google Scholar 

  11. Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9), 667–672 (2004). https://doi.org/10.1016/j.crma.2004.08.006

    Article  MathSciNet  MATH  Google Scholar 

  12. Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear port-Hamiltonian systems. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 6564–6569. https://doi.org/10.1109/CDC.2011.6161504 (2011)

  13. Bistrian, D.A., Navon, I.M.: Randomized dynamic mode decomposition for nonintrusive reduced order modelling. Int. J. Numer. Methods Eng. https://doi.org/10.1002/nme.5499 (2017)

    Article  MathSciNet  Google Scholar 

  14. Bridges, T.J., Reich, S.: Numerical methods for Hamiltonian PDEs. J. Phys. A Math. Gen. 39(19), 5287–5320 (2006). https://doi.org/10.1088/0305-4470/39/19/S02

    Article  MathSciNet  MATH  Google Scholar 

  15. Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The {GNAT} method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013). https://doi.org/10.1016/j.jcp.2013.02.028

    Article  MathSciNet  MATH  Google Scholar 

  16. Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015). https://doi.org/10.1137/140959602

    Article  MathSciNet  MATH  Google Scholar 

  17. Celledoni, E., Owren, B., Sun, Y.: The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the averaged vector field method. Math. Comp. 83(288), 1689–1700 (2014). https://doi.org/10.1090/S0025-5718-2014-02805-6

    Article  MathSciNet  MATH  Google Scholar 

  18. Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D.J., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. J. Comput. Phys. 231, 6770–6789 (2012). https://doi.org/10.1016/j.jcp.2012.06.022

    Article  MathSciNet  MATH  Google Scholar 

  19. Charnyi, S., Heister, T., Olshanskii, M. A., Rebholz, L.G.: On conservation laws of Navier-Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017). https://doi.org/10.1016/j.jcp.2017.02.039

    Article  MathSciNet  Google Scholar 

  20. Chaturantabut, S., Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear Port-Hamiltonian systems. SIAM J. Sci. Comput. 38(5), B837–B865 (2016). https://doi.org/10.1137/15M1055085

    Article  MathSciNet  MATH  Google Scholar 

  21. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). https://doi.org/10.1137/090766498

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, J.B., Qin, M.Z., Tang, Y.F.: Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 43(8), 1095–1106 (2002). https://doi.org/10.1016/S0898-1221(02)80015-3

    Article  MathSciNet  MATH  Google Scholar 

  23. Cohen, D., Hairer, E.: Linear energy-preserving integrators for Poisson systems. BIT Numer. Math. 51(1), 91–101 (2011). https://doi.org/10.1007/s10543-011-0310-z

    Article  MathSciNet  MATH  Google Scholar 

  24. Debussche, A., Faou, E.: Modified energy for split-step methods applied to the linear Schrödinger equation. SIAM J. Numer. Anal. 47(5), 3705–3719 (2009). https://doi.org/10.1137/080744578

    Article  MathSciNet  MATH  Google Scholar 

  25. Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012). https://doi.org/10.1137/10081157X

    Article  MathSciNet  MATH  Google Scholar 

  26. Erichson, N.B., Donovan, C.: Randomized low-rank dynamic mode decomposition for motion detection. Comput. Vis. Image Underst. 146, 40–50 (2016). https://doi.org/10.1016/j.cviu.2016.02.005

    Article  Google Scholar 

  27. Everson, R., Sirovich, L.: Karhunen–Loève procedure for gappy data. J. Opt. Soc. Am. A 12(8), 1657–1664 (1995). https://doi.org/10.1364/JOSAA.12.001657

    Article  Google Scholar 

  28. Galati, L., Zheng, S.: Nonlinear Schrödinger equations for Bose-Einstein condensates. AIP Conf. Proc. 1562(1), 50–64 (2013). https://doi.org/10.1063/1.4828682

    Article  Google Scholar 

  29. Gao, Y., Mei, L.: Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations. Appl. Numer. Math. 106, 41–60 (2016). https://doi.org/10.1016/j.apnum.2016.06.003

    Article  MATH  Google Scholar 

  30. Gong, Y., Cai, J., Wang, Y.: Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian {PDEs}. J. Comput. Phys. 279, 80–102 (2014). https://doi.org/10.1016/j.jcp.2014.09.001

    Article  MathSciNet  MATH  Google Scholar 

  31. Gong, Y., Wang, Q., Wang, Z.: Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems. Comput. Methods Appl. Mech. Eng. 315, 780–798 (2017). https://doi.org/10.1016/j.cma.2016.11.016

    Article  MathSciNet  Google Scholar 

  32. Gong, Y., Wang, Y.: An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs. Commun. Comput. Phys. 20 (5), 1313–1339 (2016). https://doi.org/10.4208/cicp.231014.110416a

    Article  MathSciNet  MATH  Google Scholar 

  33. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics. Springer, Heidelberg (2010)

    MATH  Google Scholar 

  34. Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011). https://doi.org/10.1137/090771806

    Article  MathSciNet  MATH  Google Scholar 

  35. Islas, A., Karpeev, D., Schober, C.: Geometric integrators for the nonlinear Schrödinger equation. J. Comput. Phys. 173(1), 116–148 (2001). https://doi.org/10.1006/jcph.2001.6854

    Article  MathSciNet  MATH  Google Scholar 

  36. Karasözen, B., Şimşek, G.: Energy preserving integration of bi-Hamiltonian partial differential equations. Appl. Math. Lett. 26(12), 1125–1133 (2013). https://doi.org/10.1016/j.aml.2013.06.005

    Article  MathSciNet  MATH  Google Scholar 

  37. Karasözen, B., Akkoyunlu, C., Uzunca, M.: Model order reduction for nonlinear Schrödinger equation. Appl. Math. Comput. 258, 509–519 (2015). https://doi.org/10.1016/j.amc.2015.02.001

    Article  MathSciNet  MATH  Google Scholar 

  38. Karasözen, B., Küċükseyhan, T., Uzunca, M.: Structure preserving integration and model order reduction of skew-gradient reaction–diffusion systems. Ann. Oper. Res. 258(1), 79–106 (2017). https://doi.org/10.1007/s10479-015-2063-6

    Article  MathSciNet  MATH  Google Scholar 

  39. Karasözen, B., Uzunca, M., Sarıaydın-Filibelioğlu, A., Yücel, H.: Energy stable discontinuous Galerkin finite element method for the Allen-Cahn equation. Int. J. Comput. Methods 0(0), 1850,013 (0) (2017). https://doi.org/10.1142/S0219876218500135

    Article  MATH  Google Scholar 

  40. Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–318 (1931)

    Article  Google Scholar 

  41. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90(1), 117–148 (2001). https://doi.org/10.1007/s002110100282

    Article  MathSciNet  MATH  Google Scholar 

  42. Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic Mode Decomposition: Data-driven Modeling of Complex Systems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2016)

    Book  Google Scholar 

  43. Lall, S., Krysl, P., Marsden, J. E.: Structure-preserving model reduction for mechanical systems. Phys. D 184(1-4), 304–318 (2003). https://doi.org/10.1016/S0167-2789(03)00227-6

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, Y. W., Wu, X.: General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs. J. Comput. Phys. 301, 141–166 (2015). https://doi.org/10.1016/j.jcp.2015.08.023

    Article  MathSciNet  MATH  Google Scholar 

  45. Mahoney, M.W.: Randomized algorithms for matrices and data. Found. Trends Mach. Learn. 3(2), 123–224 (2011). https://doi.org/10.1561/2200000035

    Article  MATH  Google Scholar 

  46. Martinsson, P.G.: Randomized methods for matrix computations and analysis of high dimensional data ArXiv e-prints (2016)

  47. Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45(1), 357–378 (2013). https://doi.org/10.1146/annurev-fluid-011212-140652

    Article  MathSciNet  MATH  Google Scholar 

  48. Mohebujjaman, M., Rebholz, L.G., Xie, X., Iliescu, T.: Energy balance and mass conservation in reduced order models of fluid flows. J. Comput. Phys. 346 (Supplement C), 262–277 (2017). https://doi.org/10.1016/j.jcp.2017.06.019

    Article  MathSciNet  MATH  Google Scholar 

  49. Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), A1–A27 (2016). https://doi.org/10.1137/140978922

    Article  MathSciNet  MATH  Google Scholar 

  50. Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Clarendon Press, Oxford (2003)

    MATH  Google Scholar 

  51. Quispel, G., McLaren, D.: A new class of energy-preserving numerical integration methods. J. Phys. Math. Theor. 41(4), 045206 (7pp) (2008). https://doi.org/10.1088/1751-8113/41/4/045206

    Article  MathSciNet  MATH  Google Scholar 

  52. Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM. https://doi.org/10.1137/1.9780898717440 (2008)

  53. Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D. S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009). https://doi.org/10.1017/S0022112009992059

    Article  MathSciNet  MATH  Google Scholar 

  54. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010). https://doi.org/10.1017/S0022112010001217

    Article  MathSciNet  MATH  Google Scholar 

  55. Sulem, C., Sulem, P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences. Springer, New York (1999)

    MATH  Google Scholar 

  56. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: Theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014). https://doi.org/10.3934/jcd.2014.1.391

    Article  MathSciNet  MATH  Google Scholar 

  57. Uzunca, M., Karasözen, B.: Energy stable model order reduction for the Allen-Cahn equation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, pp 403–419. Springer International Publishing, Cham (2017), https://doi.org/10.1007/978-3-319-58786-8_25

    Chapter  Google Scholar 

  58. Vemaganti, K.: Discontinuous Galerkin methods for periodic boundary value problems. Numer. Methods Partial Differ. Equ. 23(3), 587–596 (2007). https://doi.org/10.1002/num.20191

    Article  MathSciNet  MATH  Google Scholar 

  59. Wang, T., Guo, B., Xu, Q.: Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. J. Comput. Phys. 243, 382–399 (2013). https://doi.org/10.1016/j.jcp.2013.03.007

    Article  MathSciNet  MATH  Google Scholar 

  60. Williams, M.O., Schmid, P.J., Kutz, J.N.: Hybrid reduced-order integration with proper orthogonal decomposition and dynamic mode decomposition. Multiscale Model. Simul. 11(2), 522–544 (2013). https://doi.org/10.1137/120874539

    Article  MathSciNet  MATH  Google Scholar 

  61. Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205(1), 72–97 (2005). https://doi.org/10.1016/j.jcp.2004.11.001

    Article  MathSciNet  MATH  Google Scholar 

  62. Xu, Y., Zhang, L.: Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation. Comput. Phys. Commun. 183(5), 1082–1093 (2012). https://doi.org/10.1016/j.cpc.2012.01.006

    Article  MathSciNet  MATH  Google Scholar 

  63. Zimmermann, R., Willcox, K.: An accelerated greedy missing point estimation procedure. SIAM J. Sci. Comput. 38(5), A2827–A285 (2016). https://doi.org/10.1137/15M1042899

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institute of Applied Mathematics & Department of Mathematics, Middle East Technical University, Ankara, Turkey

    Bülent Karasözen

  2. Department of Mathematics, Sinop University, Sinop, Turkey

    Murat Uzunca

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  2. Murat Uzunca
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Karasözen, B., Uzunca, M. Energy preserving model order reduction of the nonlinear Schrödinger equation. Adv Comput Math 44, 1769–1796 (2018). https://doi.org/10.1007/s10444-018-9593-9

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