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June  2014, 1(2): 357-375. doi: 10.3934/jcd.2014.1.357

## Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

 1 United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108, United States, United States

Received  June 2013 Revised  June 2014 Published  December 2014

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.
Citation: José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357
##### References:
 [1] M. S. Allen and J. A. Camberos, Comparison of uncertainty propagation / response surface techniques for two aeroelastic systems, in 50th AIAA Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, May 4-7, 2009, 2009. doi: 10.2514/6.2009-2269.  Google Scholar [2] G. Blatman and B. Sudret, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, 25 (2010), 183-197. doi: 10.1016/j.probengmech.2009.10.003.  Google Scholar [3] G. Blatman and B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, Journal of Computational Physics, 230 (2011), 2345-2367. doi: 10.1016/j.jcp.2010.12.021.  Google Scholar [4] J. Bucklew, Introduction to Rare Event Simulation, Springer, 2004. doi: 10.1007/978-1-4757-4078-3.  Google Scholar [5] R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804.  Google Scholar [6] R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annals of Mathematics, 48 (1947), 385-392. doi: 10.2307/1969178.  Google Scholar [7] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, 1971.  Google Scholar [8] S. E. Geneser, R. M. Kirby and F. B. Sachse, Sensitivity analysis of cardiac electrophysiological models using polynomial chaos, in Engineering in Medicine and Biology Society, 2005. IEEE-EMBS 2005. 27th Annual International Conference of the, IEEE, (2006), 4042-4045. doi: 10.1109/IEMBS.2005.1615349.  Google Scholar [9] R. Ghanem, Probabilistic characterization of transport in heterogeneous media, Comput. Methods Appl. Mech. Engng., 158 (1998), 199-220. doi: 10.1016/S0045-7825(97)00250-8.  Google Scholar [10] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar [11] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31, Springer, 2006.  Google Scholar [12] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd edition, Springer, 2009. doi: 10.1007/978-0-387-84858-7.  Google Scholar [13] M. He, S. Murugesan and J. Zhang, Multiple timescale dispatch and scheduling for stochastic reliability in smart grids with wind generation integration, in Proceedings of the IEEE INFOCOM, April 10-15, 2011, Shanghai, China, 2011. doi: 10.1109/INFCOM.2011.5935204.  Google Scholar [14] P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, 1996. doi: 10.1017/CBO9780511622700.  Google Scholar [15] B. Huberman and J. P. Crutchfield, Chaotic states of anharmonic systems in periodic fields, Phys. Rev. Lett., 43 (1979), 1743-1747. doi: 10.1103/PhysRevLett.43.1743.  Google Scholar [16] S. Klus, T. Sahai, C. Liu and M. Dellnitz, An efficient algorithm for the parallel solution of high-dimensional differential equations, J. Comput. Appl. Math., 235 (2011), 3053-3062. doi: 10.1016/j.cam.2010.12.026.  Google Scholar [17] B. Kouchmeshky and N. Zabaras, The effect of multiple sources of uncertainty on the convex hull of material properties of polycrystals, Computational Materials Science, 47 (2009), 342-352. doi: 10.1016/j.commatsci.2009.08.010.  Google Scholar [18] J. Laskar, Large-scale chaos in the solar system, Astronomy and Astrophysics, 287 (1994), L9-L12. Google Scholar [19] R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 2nd edition, Wiley Interscience, 1998. Google Scholar [20] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the atmospheric sciences, 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar [21] X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, Journal of Computational Physics, 228 (2009), 3084-3113. doi: 10.1016/j.jcp.2009.01.006.  Google Scholar [22] X. Ma and N. Zabaras, Kernel principal component analysis for stochastic input model generation, Journal of Computational Physics, 230 (2011), 7311-7331. doi: 10.1016/j.jcp.2011.05.037.  Google Scholar [23] Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, Journal of Computational Physics, 228 (2009), 1862-1902. doi: 10.1016/j.jcp.2008.11.024.  Google Scholar [24] R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology, 3rd edition, Wiley, 2009.  Google Scholar [25] H. N. Najm, B. J. Debusschere, Y. M. Marzouk, S. Widmer and O. P. Le Maìtre, Uncertainty quantification in chemical systems, Int. J. Numer. Meth. Engng., 80 (2009), 789-814. doi: 10.1002/nme.2551.  Google Scholar [26] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bulletin of the American Mathematical Society, 84 (1978), 957-1041. doi: 10.1090/S0002-9904-1978-14532-7.  Google Scholar [27] F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with randon input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345. doi: 10.1137/060663660.  Google Scholar [28] H. Ogura, Orthogonal functions of the Poisson processes, IEEE Transactions on Information Theory, 18 (1972), 473-481. doi: 10.1109/TIT.1972.1054856.  Google Scholar [29] P. Parpas and M. Webster, A stochastic multiscale model for electricity generation capacity expansion, Eur. J. Oper. Res., 232 (2014), 359-374. doi: 10.1016/j.ejor.2013.07.022.  Google Scholar [30] R. H. Rand, Lecture Notes on Nonlinear Vibrations, Internet-First University Press, 2012, URL http://hdl.handle.net/1813/28989. Google Scholar [31] T. Sahai, V. Fonoberov and S. Loire, Uncertainty as a stabilizer of the head-tail ordered phase in carbon-monoxide monolayers on graphite, Physical Review B, 80 (2009), 115413. doi: 10.1103/PhysRevB.80.115413.  Google Scholar [32] T. Sahai, Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection, Nonlinear Dynamics, 62 (2010), 273-289. doi: 10.1007/s11071-010-9716-4.  Google Scholar [33] T. Sahai, R. B. Bhiladvala and A. T. Zehnder, Thermomechanical transitions in doubly-clamped micro-oscillators, International Journal of Non-Linear Mechanics, 42 (2007), 596-607. doi: 10.1016/j.ijnonlinmec.2006.12.009.  Google Scholar [34] T. Sahai and J. M. Pasini, Uncertainty quantification in hybrid dynamical systems, J. Comput. Phys., 237 (2013), 411-427. doi: 10.1016/j.jcp.2012.10.030.  Google Scholar [35] T. Sahai, A. Speranzon and A. Banaszuk, Hearing the clusters in a graph: A dristributed algorithm, Automatica, 48 (2012), 15-24. doi: 10.1016/j.automatica.2011.09.019.  Google Scholar [36] T. Sahai and A. T. Zehnder, Modeling of coupled dome-shaped microoscillators, Microelectromechanical Systems, Journal of, 17 (2008), 777-786. doi: 10.1109/JMEMS.2008.924844.  Google Scholar [37] S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Group, 2001. Google Scholar [38] A. Surana, T. Sahai and A. Banaszuk, Iterative methods for scalable uncertainty quantification in complex networks, International Journal for Uncertainty Quantification, 2 (2012), 413-439. doi: 10.1615/Int.J.UncertaintyQuantification.2012004138.  Google Scholar [39] Y. Susuki, I. Mezić and T. Hikihara, Coherent swing instability of power grids, J. Nonlinear Sci., 21 (2011), 403-439. doi: 10.1007/s00332-010-9087-5.  Google Scholar [40] G. Szegö, Orthogonal Polynomials, vol. 23, Amer Mathematical Society, 1967.  Google Scholar [41] Y. Ueda, Explosion of strange attractors exhibited by Duffing's equation, in Nonlinear Dynamics (ed. R. H. G. Hellerman), New York Academy of Sciences, New York, 357 (1980), 422-434. doi: 10.1111/j.1749-6632.1980.tb29708.x.  Google Scholar [42] X. Wan and G. E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, Journal of Scientific Computing, 27 (2006), 455-464. doi: 10.1007/s10915-005-9038-8.  Google Scholar [43] X. Wan and G. E. Karniadakis, Recent advances in polynomial chaos methods and extensions, in Computational Uncertainty in Military Vehicle Design Meeting Proceedings, NATO/OTAN, Paper Reference Number: RTO-MP-IST-999, 2008. Google Scholar [44] X. Wan and G. E. Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations, Computer methods in applied mechanics and engineering, 195 (2006), 5582-5596. doi: 10.1016/j.cma.2005.10.016.  Google Scholar [45] C. G. Webster, Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data, PhD thesis, Florida State University, 2007.  Google Scholar [46] N. Wiener, The homogeneous chaos, American Journal of Mathematics, 60 (1938), 897-936. doi: 10.2307/2371268.  Google Scholar [47] D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comp. Phys., 187 (2003), 137-167. doi: 10.1016/S0021-9991(03)00092-5.  Google Scholar [48] N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, J. Comput. Phys., 227 (2008), 4697-4735. doi: 10.1016/j.jcp.2008.01.019.  Google Scholar

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##### References:
 [1] M. S. Allen and J. A. Camberos, Comparison of uncertainty propagation / response surface techniques for two aeroelastic systems, in 50th AIAA Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, May 4-7, 2009, 2009. doi: 10.2514/6.2009-2269.  Google Scholar [2] G. Blatman and B. Sudret, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, 25 (2010), 183-197. doi: 10.1016/j.probengmech.2009.10.003.  Google Scholar [3] G. Blatman and B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, Journal of Computational Physics, 230 (2011), 2345-2367. doi: 10.1016/j.jcp.2010.12.021.  Google Scholar [4] J. Bucklew, Introduction to Rare Event Simulation, Springer, 2004. doi: 10.1007/978-1-4757-4078-3.  Google Scholar [5] R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804.  Google Scholar [6] R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annals of Mathematics, 48 (1947), 385-392. doi: 10.2307/1969178.  Google Scholar [7] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, 1971.  Google Scholar [8] S. E. Geneser, R. M. Kirby and F. B. Sachse, Sensitivity analysis of cardiac electrophysiological models using polynomial chaos, in Engineering in Medicine and Biology Society, 2005. IEEE-EMBS 2005. 27th Annual International Conference of the, IEEE, (2006), 4042-4045. doi: 10.1109/IEMBS.2005.1615349.  Google Scholar [9] R. Ghanem, Probabilistic characterization of transport in heterogeneous media, Comput. Methods Appl. Mech. Engng., 158 (1998), 199-220. doi: 10.1016/S0045-7825(97)00250-8.  Google Scholar [10] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar [11] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31, Springer, 2006.  Google Scholar [12] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd edition, Springer, 2009. doi: 10.1007/978-0-387-84858-7.  Google Scholar [13] M. He, S. Murugesan and J. Zhang, Multiple timescale dispatch and scheduling for stochastic reliability in smart grids with wind generation integration, in Proceedings of the IEEE INFOCOM, April 10-15, 2011, Shanghai, China, 2011. doi: 10.1109/INFCOM.2011.5935204.  Google Scholar [14] P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, 1996. doi: 10.1017/CBO9780511622700.  Google Scholar [15] B. Huberman and J. P. Crutchfield, Chaotic states of anharmonic systems in periodic fields, Phys. Rev. Lett., 43 (1979), 1743-1747. doi: 10.1103/PhysRevLett.43.1743.  Google Scholar [16] S. Klus, T. Sahai, C. Liu and M. Dellnitz, An efficient algorithm for the parallel solution of high-dimensional differential equations, J. Comput. Appl. Math., 235 (2011), 3053-3062. doi: 10.1016/j.cam.2010.12.026.  Google Scholar [17] B. Kouchmeshky and N. Zabaras, The effect of multiple sources of uncertainty on the convex hull of material properties of polycrystals, Computational Materials Science, 47 (2009), 342-352. doi: 10.1016/j.commatsci.2009.08.010.  Google Scholar [18] J. Laskar, Large-scale chaos in the solar system, Astronomy and Astrophysics, 287 (1994), L9-L12. Google Scholar [19] R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 2nd edition, Wiley Interscience, 1998. Google Scholar [20] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the atmospheric sciences, 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar [21] X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, Journal of Computational Physics, 228 (2009), 3084-3113. doi: 10.1016/j.jcp.2009.01.006.  Google Scholar [22] X. Ma and N. Zabaras, Kernel principal component analysis for stochastic input model generation, Journal of Computational Physics, 230 (2011), 7311-7331. doi: 10.1016/j.jcp.2011.05.037.  Google Scholar [23] Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, Journal of Computational Physics, 228 (2009), 1862-1902. doi: 10.1016/j.jcp.2008.11.024.  Google Scholar [24] R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology, 3rd edition, Wiley, 2009.  Google Scholar [25] H. N. Najm, B. J. Debusschere, Y. M. Marzouk, S. Widmer and O. P. Le Maìtre, Uncertainty quantification in chemical systems, Int. J. Numer. Meth. Engng., 80 (2009), 789-814. doi: 10.1002/nme.2551.  Google Scholar [26] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bulletin of the American Mathematical Society, 84 (1978), 957-1041. doi: 10.1090/S0002-9904-1978-14532-7.  Google Scholar [27] F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with randon input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345. doi: 10.1137/060663660.  Google Scholar [28] H. Ogura, Orthogonal functions of the Poisson processes, IEEE Transactions on Information Theory, 18 (1972), 473-481. doi: 10.1109/TIT.1972.1054856.  Google Scholar [29] P. Parpas and M. Webster, A stochastic multiscale model for electricity generation capacity expansion, Eur. J. Oper. Res., 232 (2014), 359-374. doi: 10.1016/j.ejor.2013.07.022.  Google Scholar [30] R. H. Rand, Lecture Notes on Nonlinear Vibrations, Internet-First University Press, 2012, URL http://hdl.handle.net/1813/28989. Google Scholar [31] T. Sahai, V. Fonoberov and S. Loire, Uncertainty as a stabilizer of the head-tail ordered phase in carbon-monoxide monolayers on graphite, Physical Review B, 80 (2009), 115413. doi: 10.1103/PhysRevB.80.115413.  Google Scholar [32] T. Sahai, Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection, Nonlinear Dynamics, 62 (2010), 273-289. doi: 10.1007/s11071-010-9716-4.  Google Scholar [33] T. Sahai, R. B. Bhiladvala and A. T. Zehnder, Thermomechanical transitions in doubly-clamped micro-oscillators, International Journal of Non-Linear Mechanics, 42 (2007), 596-607. doi: 10.1016/j.ijnonlinmec.2006.12.009.  Google Scholar [34] T. Sahai and J. M. Pasini, Uncertainty quantification in hybrid dynamical systems, J. Comput. Phys., 237 (2013), 411-427. doi: 10.1016/j.jcp.2012.10.030.  Google Scholar [35] T. Sahai, A. Speranzon and A. Banaszuk, Hearing the clusters in a graph: A dristributed algorithm, Automatica, 48 (2012), 15-24. doi: 10.1016/j.automatica.2011.09.019.  Google Scholar [36] T. Sahai and A. T. Zehnder, Modeling of coupled dome-shaped microoscillators, Microelectromechanical Systems, Journal of, 17 (2008), 777-786. doi: 10.1109/JMEMS.2008.924844.  Google Scholar [37] S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Group, 2001. Google Scholar [38] A. Surana, T. Sahai and A. Banaszuk, Iterative methods for scalable uncertainty quantification in complex networks, International Journal for Uncertainty Quantification, 2 (2012), 413-439. doi: 10.1615/Int.J.UncertaintyQuantification.2012004138.  Google Scholar [39] Y. Susuki, I. Mezić and T. Hikihara, Coherent swing instability of power grids, J. Nonlinear Sci., 21 (2011), 403-439. doi: 10.1007/s00332-010-9087-5.  Google Scholar [40] G. Szegö, Orthogonal Polynomials, vol. 23, Amer Mathematical Society, 1967.  Google Scholar [41] Y. Ueda, Explosion of strange attractors exhibited by Duffing's equation, in Nonlinear Dynamics (ed. R. H. G. Hellerman), New York Academy of Sciences, New York, 357 (1980), 422-434. doi: 10.1111/j.1749-6632.1980.tb29708.x.  Google Scholar [42] X. Wan and G. E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, Journal of Scientific Computing, 27 (2006), 455-464. doi: 10.1007/s10915-005-9038-8.  Google Scholar [43] X. Wan and G. E. Karniadakis, Recent advances in polynomial chaos methods and extensions, in Computational Uncertainty in Military Vehicle Design Meeting Proceedings, NATO/OTAN, Paper Reference Number: RTO-MP-IST-999, 2008. Google Scholar [44] X. Wan and G. E. Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations, Computer methods in applied mechanics and engineering, 195 (2006), 5582-5596. doi: 10.1016/j.cma.2005.10.016.  Google Scholar [45] C. G. Webster, Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data, PhD thesis, Florida State University, 2007.  Google Scholar [46] N. Wiener, The homogeneous chaos, American Journal of Mathematics, 60 (1938), 897-936. doi: 10.2307/2371268.  Google Scholar [47] D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comp. Phys., 187 (2003), 137-167. doi: 10.1016/S0021-9991(03)00092-5.  Google Scholar [48] N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, J. Comput. Phys., 227 (2008), 4697-4735. doi: 10.1016/j.jcp.2008.01.019.  Google Scholar
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