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Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times
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 |
References:
[1] |
M. S. Allen and J. A. Camberos, Comparison of uncertainty propagation / response surface techniques for two aeroelastic systems,, in 50th AIAA Structures, (2009), 4.
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.
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.
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.
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.
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, (2006), 4042.
doi: 10.1109/IEMBS.2005.1615349. |
[9] |
R. Ghanem, Probabilistic characterization of transport in heterogeneous media,, Comput. Methods Appl. Mech. Engng., 158 (1998), 199.
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, (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, (2011), 10.
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.
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.
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.
doi: 10.1016/j.commatsci.2009.08.010. |
[18] |
J. Laskar, Large-scale chaos in the solar system,, Astronomy and Astrophysics, 287 (1994). Google Scholar |
[19] |
R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions,, 2nd edition, (1998). Google Scholar |
[20] |
E. N. Lorenz, Deterministic nonperiodic flow,, Journal of the atmospheric sciences, 20 (1963), 130.
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.
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.
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.
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, (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.
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.
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.
doi: 10.1137/060663660. |
[28] |
H. Ogura, Orthogonal functions of the Poisson processes,, IEEE Transactions on Information Theory, 18 (1972), 473.
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.
doi: 10.1016/j.ejor.2013.07.022. |
[30] |
R. H. Rand, Lecture Notes on Nonlinear Vibrations,, Internet-First University Press, (2012). 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).
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.
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.
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.
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.
doi: 10.1016/j.automatica.2011.09.019. |
[36] |
T. Sahai and A. T. Zehnder, Modeling of coupled dome-shaped microoscillators,, Microelectromechanical Systems, 17 (2008), 777.
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). 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.
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.
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), 357 (1980), 422.
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.
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, (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.
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, (2007).
|
[46] |
N. Wiener, The homogeneous chaos,, American Journal of Mathematics, 60 (1938), 897.
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.
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.
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, (2009), 4.
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.
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.
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.
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.
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, (2006), 4042.
doi: 10.1109/IEMBS.2005.1615349. |
[9] |
R. Ghanem, Probabilistic characterization of transport in heterogeneous media,, Comput. Methods Appl. Mech. Engng., 158 (1998), 199.
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, (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, (2011), 10.
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.
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.
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.
doi: 10.1016/j.commatsci.2009.08.010. |
[18] |
J. Laskar, Large-scale chaos in the solar system,, Astronomy and Astrophysics, 287 (1994). Google Scholar |
[19] |
R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions,, 2nd edition, (1998). Google Scholar |
[20] |
E. N. Lorenz, Deterministic nonperiodic flow,, Journal of the atmospheric sciences, 20 (1963), 130.
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.
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.
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.
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, (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.
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.
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.
doi: 10.1137/060663660. |
[28] |
H. Ogura, Orthogonal functions of the Poisson processes,, IEEE Transactions on Information Theory, 18 (1972), 473.
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.
doi: 10.1016/j.ejor.2013.07.022. |
[30] |
R. H. Rand, Lecture Notes on Nonlinear Vibrations,, Internet-First University Press, (2012). 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).
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.
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.
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.
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.
doi: 10.1016/j.automatica.2011.09.019. |
[36] |
T. Sahai and A. T. Zehnder, Modeling of coupled dome-shaped microoscillators,, Microelectromechanical Systems, 17 (2008), 777.
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). 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.
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.
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), 357 (1980), 422.
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.
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, (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.
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, (2007).
|
[46] |
N. Wiener, The homogeneous chaos,, American Journal of Mathematics, 60 (1938), 897.
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.
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.
doi: 10.1016/j.jcp.2008.01.019. |
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