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

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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

José Miguel Pasini ^1, and Tuhin Sahai ^1,

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

Abstract
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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.

Keywords: polynomial chaos, chaos., multiscale analysis, Hamiltonian systems, Uncertainty quantification.

Mathematics Subject Classification: Primary: 34F05, 65P10, 65P20, 34E13, 60H35; Secondary: 37H9.

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.
[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.
[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.
[4]	J. Bucklew, Introduction to Rare Event Simulation, Springer, 2004. doi: 10.1007/978-1-4757-4078-3.
[5]	R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804.
[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.
[7]	C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, 1971.
[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.
[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.
[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.
[11]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31, Springer, 2006.
[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.
[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.
[14]	P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, 1996. doi: 10.1017/CBO9780511622700.
[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.
[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.
[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.
[18]	J. Laskar, Large-scale chaos in the solar system, Astronomy and Astrophysics, 287 (1994), L9-L12.
[19]	R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 2nd edition, Wiley Interscience, 1998.
[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.
[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.
[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.
[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.
[24]	R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology, 3rd edition, Wiley, 2009.
[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.
[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.
[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.
[28]	H. Ogura, Orthogonal functions of the Poisson processes, IEEE Transactions on Information Theory, 18 (1972), 473-481. doi: 10.1109/TIT.1972.1054856.
[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.
[30]	R. H. Rand, Lecture Notes on Nonlinear Vibrations, Internet-First University Press, 2012, URL http://hdl.handle.net/1813/28989.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Group, 2001.
[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.
[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.
[40]	G. Szegö, Orthogonal Polynomials, vol. 23, Amer Mathematical Society, 1967.
[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.
[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.
[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.
[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.
[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.
[46]	N. Wiener, The homogeneous chaos, American Journal of Mathematics, 60 (1938), 897-936. doi: 10.2307/2371268.
[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.
[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.

show all references

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.
[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.
[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.
[4]	J. Bucklew, Introduction to Rare Event Simulation, Springer, 2004. doi: 10.1007/978-1-4757-4078-3.
[5]	R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804.
[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.
[7]	C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, 1971.
[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.
[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.
[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.
[11]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31, Springer, 2006.
[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.
[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.
[14]	P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, 1996. doi: 10.1017/CBO9780511622700.
[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.
[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.
[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.
[18]	J. Laskar, Large-scale chaos in the solar system, Astronomy and Astrophysics, 287 (1994), L9-L12.
[19]	R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 2nd edition, Wiley Interscience, 1998.
[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.
[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.
[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.
[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.
[24]	R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology, 3rd edition, Wiley, 2009.
[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.
[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.
[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.
[28]	H. Ogura, Orthogonal functions of the Poisson processes, IEEE Transactions on Information Theory, 18 (1972), 473-481. doi: 10.1109/TIT.1972.1054856.
[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.
[30]	R. H. Rand, Lecture Notes on Nonlinear Vibrations, Internet-First University Press, 2012, URL http://hdl.handle.net/1813/28989.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Group, 2001.
[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.
[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.
[40]	G. Szegö, Orthogonal Polynomials, vol. 23, Amer Mathematical Society, 1967.
[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.
[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.
[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.
[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.
[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.
[46]	N. Wiener, The homogeneous chaos, American Journal of Mathematics, 60 (1938), 897-936. doi: 10.2307/2371268.
[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.
[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.

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