×

Computing invariant sets of random differential equations using polynomial chaos. (English) Zbl 1441.37057

Random differential equations are those for which parameters are not fixed, but thought of as being randomly chosen from some distribution. For given values of the parameters one can solve the differential equation, and often one is interested in moments of the solution, where the expectation is taken over the distribution of the random parameters. This is typically done using a polynomial chaos expansion of the solution – writing it as a series in polynomials of the random parameters – together with forward time integration. This paper instead concentrates on the computation of invariant sets such as fixed points and periodic orbits, determining distributions of scalars associated with these, e.g., the period of a periodic orbit, or the curvature of a stable manifold at a fixed point. The main technique involves a finite polynomial chaos expansion of the invariant set of interest, and the numerical solution of the associated large set of defining equations.
The method is demonstrated on a Lotka-Volterra system and the Lorenz equations. Fixed points, eigenvalues and eigenvectors, local stable and unstable manifolds, periodic orbits, and heteroclinic orbits are calculated.

MSC:

37H10 Generation, random and stochastic difference and differential equations
37H05 General theory of random and stochastic dynamical systems
37M21 Computational methods for invariant manifolds of dynamical systems
37M22 Computational methods for attractors of dynamical systems
34F05 Ordinary differential equations and systems with randomness
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
65C30 Numerical solutions to stochastic differential and integral equations

Software:

Chebfun; DLMF
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Appl. Math. 13, SIAM, Philadelphia, 1987. · Zbl 0671.65063
[2] M. Breden and C. Kuehn, MATLAB Code for “Computing Invariant Sets of Random Differential Equations Using Polynomial Chaos,”https://sites.google.com/site/maximebreden/research. · Zbl 1427.60142
[3] M. Breden, J.-P. Lessard, and J. D. Mireles James, Computation of maximal local (un)stable manifold patches by the parameterization method, Indag. Math., 27 (2016), pp. 340-367. · Zbl 1336.65197
[4] X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), pp. 283-328. · Zbl 1034.37016
[5] X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), pp. 329-360. · Zbl 1034.37017
[6] X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations, 218 (2005), pp. 444-515. · Zbl 1101.37019
[7] B. J. Debusschere, H. N. Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem, and O. Le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput., 26 (2004), pp. 698-719. · Zbl 1072.60042
[8] M. Dellnitz, S. Klus, and A. Ziessler, A set-oriented numerical approach for dynamical systems with parameter uncertainty, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 120-138. · Zbl 1357.65308
[9] A. Desai, J. A. Witteveen, and S. Sarkar, Uncertainty quantification of a nonlinear aeroelastic system using polynomial chaos expansion with constant phase interpolation, J. Vibration Acoustics, 135 (2013), 051034.
[10] T. A. Driscoll, N. Hale, and L. N. Trefethen, Chebfun Guide, https://www.chebfun.org/, 2014.
[11] T. A. Driscoll and J. Weideman, Optimal domain splitting for interpolation by Chebyshev polynomials, SIAM J. Numer. Anal., 52 (2014), pp. 1913-1927. · Zbl 1302.65041
[12] O. G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann, On the convergence of generalized polynomial chaos expansions, Math. Model. Numer. Anal., 46 (2012), pp. 317-339. · Zbl 1273.65012
[13] J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1070-1088. · Zbl 1370.65046
[14] G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer, New York, 2013.
[15] R. Ghanem and D. Ghosh, Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition, Internat. J. Numer. Methods Engrg., 72 (2007), pp. 486-504. · Zbl 1194.74153
[16] R. G. Ghanem and P. D. Spanos, Stochastic finite element method: Response statistics, in Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991, pp. 101-119. · Zbl 0722.73080
[17] M. Grigoriu, Stochastic Systems: Uncertainty Quantification and Propagation, Springer, New York, 2012. · Zbl 1247.60002
[18] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. · Zbl 0515.34001
[19] A. Haro, M. Canadell, J.-L. Figueras, A. Luque, and J.-M. Mondelo, The parameterization method for invariant manifolds, Appl. Math. Sci, 195 (2016). · Zbl 1372.37002
[20] T. Hurth and C. Kuehn, Random switching near bifurcations, Stoch. Dyn., to appear. · Zbl 1441.60057
[21] B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bifur. Chaos, 15 (2005), pp. 763-791. · Zbl 1086.34002
[22] B. Krauskopf, H. M. Osinga, and J. Galán-Vique, eds., Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer, New York, 2007. · Zbl 1117.65005
[23] C. Kuehn, Deterministic continuation of stochastic metastable equilibria via Lyapunov equations and ellipsoids, SIAM J. Sci. Comput., 34 (2012), pp. A1635-A1658. · Zbl 1246.65244
[24] C. Kuehn, Quenched noise and nonlinear oscillations in bistable multiscale systems, Europhys. Lett., 120 (2017), 10001.
[25] C. Kuehn, Uncertainty transformation via Hopf bifurcation in fast-slow systems, Proc. A, 473 (2017), 20160346. · Zbl 1404.37063
[26] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci. 112, Springer, New York, 2013.
[27] O. Le Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer, New York, 2010. · Zbl 1193.76003
[28] O. Le Maître, L. Mathelin, O. M. Knio, and M. Y. Hussaini, Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics, Discrete Contin. Dyn. Syst., 28 (2010), pp. 199-226. · Zbl 1198.37072
[29] D. Lucor and G. E. Karniadakis, Adaptive generalized polynomial chaos for nonlinear random oscillators, SIAM J. Sci. Comput., 26 (2004), pp. 720-735. · Zbl 1075.65008
[30] D. R. Millman, P. I. King, R. C. Maple, P. S. Beran, and L. K. Chilton, Airfoil pitch-and-plunge bifurcation behavior with Fourier chaos expansions, J. Aircraft, 42 (2005), pp. 376-384.
[31] I. Molchanov, Theory of Random Sets, Springer, New York, 2017. · Zbl 1406.60006
[32] H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer Ser. Inform. Sci. 2, Springer, New York, 2012. · Zbl 0476.65097
[33] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, UK, 2010. · Zbl 1198.00002
[34] D. Potts, G. Steidl, and M. Tasche, Fast algorithms for discrete polynomial transforms, Math. Comp., 67 (1998), pp. 1577-1590. · Zbl 0904.65145
[35] K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, New York, 2014. · Zbl 0726.65162
[36] M. Schick, V. Heuveline, and O. Le Maître, A Newton-Galerkin method for fluid flow exhibiting uncertain periodic dynamics, SIAM/ASA J. Uncertain. Quantif., 2 (2014), pp. 153-173. · Zbl 1311.76071
[37] R. Szwarc, Orthogonal polynomials and banach algebras, in Inzell Lectures on Orthogonal Polynomials, Advances in the Theory of Special Functions and Orthogonal Polynomials, Vol. 2, Nova Science Publishers, Hauppauge, NY, 2005, pp. 103-139. · Zbl 1119.33011
[38] L. N. Trefethen, Approximation Theory and Approximation Practice, Appl. Math. 128, SIAM, Philadelphia, 2013. · Zbl 1264.41001
[39] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. · Zbl 1231.65077
[40] J. B. van den Berg, J. D. Mireles James, and C. Reinhardt, Computing (un)stable manifolds with validated error bounds: Non-resonant and resonant spectra, J. Nonlinear Sci., 26 (2016), pp. 1055-1095. · Zbl 1360.37176
[41] J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015). · Zbl 1338.68301
[42] J. B. van den Berg and R. Sheombarsing, Rigorous Numerics for ODEs Using Chebyshev Series and Domain Decomposition, preprint, 2015. · Zbl 1481.65115
[43] A. J. Veraart, E. J. Faassen, V. Dakos, E. H. van Nes, M. Lurling, and M. Scheffer, Stochastic bifurcation analysis of Rayleigh-Bénard convection, J. Fluid Mech., 650 (2010), pp. 391-413. · Zbl 1189.76213
[44] X. Wan and G. E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28 (2006), pp. 901-928. · Zbl 1128.65009
[45] N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897-936. · JFM 64.0887.02
[46] D. Xiu, Generalized (Wiener-Askey) Polynomial Chaos, Ph.D. thesis, Brown University, Providence, RI, 2004.
[47] D. Xiu, Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), pp. 293-309. · Zbl 1164.65302
[48] D. Xiu, Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys., 5 (2009), pp. 242-272. · Zbl 1364.65019
[49] D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619-644. · Zbl 1014.65004
[50] D. Xiu and G. E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 4927-4948. · Zbl 1016.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.