Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T14:19:53.438Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stochastic Collocation Method for Delay Differential Equations with Random Input

Published online by Cambridge University Press:  03 June 2015

Tao Zhou
Tao Zhou*
Affiliation:
Institute of Computational Mathematics, Academy of Mathematics and Systems Sciences, The Chinese Academy of Sciences, Beijing 100190, China
*
*Corresponding author. Email: [email protected]
Get access

Abstract

In this work, we concern with the numerical approach for delay differential equations with random coefficients. We first show that the exact solution of the problem considered admits good regularity in the random space, provided that the given data satisfy some reasonable assumptions. A stochastic collocation method is proposed to approximate the solution in the random space, and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations. Convergence property of the proposed method is analyzed. It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space. Numerical examples are given to illustrate the theoretical results.

Keywords

65C2065M70Delay differential equationsstochastic collocationsparse gridlegendre spectral method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 4 , August 2014 , pp. 403 - 418
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ali, I., Brunner, H. and Tang, T., A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput Math., 27 (2009), pp. 254–265.Google Scholar
[2] Back, Joakim, Nobile, Fabio, Tamellini, Lorenzo and Tempone, Raul, Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison, Hesthaven, J. S. and Ronquist, E. M. eds., Lecture Notes in Computational Science and Engineering, 76 (Springer, 2011) 43–62, Selected papers from the ICOSAHOM '09 conference.Google Scholar
[3] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004.CrossRefGoogle Scholar
[4] Campbell, S. A., Edwards, R. and Van Den Driessche, P., Delayed coupling between two neural network loops, SIAM J. Appl. Math., 65(1) (2004), pp. 316–335.Google Scholar
[5] Clenshaw, C. W. and Curtis, A. R., A method for numerical integration on an automatic computer, Numer. Math., 2 (1960), pp. 197–205.CrossRefGoogle Scholar
[6] Fishman, G., Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996.Google Scholar
[7] Gao, Z. and Zhou, T., On the choice of design points for least square polynomial approximations with application to uncertainty quantification, Commun. Comput. Phys., accepted, 2014.CrossRefGoogle Scholar
[8] Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, SpringerVerlag, New York, 1991.Google Scholar
[9] Hoang, V. H. and Schwab, C., Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs: I. Analytic regularity and gPC-approximation, Report 2010-11, Seminar for Applied Mathematics, ETH Zurich, 2010.Google Scholar
[10] Hoang, V. H. and Schwab, C., Analytic regularity and gPC approximation for parametric and random 2nd order hyperbolic PDEs, Report 2010-19, Seminar for Applied Mathematics, ETH Zurich, 2010.Google Scholar
[11] Lotka, A., Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.Google Scholar
[12] Migliorati, G., Nobile, F., Schwerin, E. and Tempone, R., Analysis of the discrete L2 projection on polynomial spaces with random evaluations, Found. Comput. Math., D0I:10.1007/s10208-013-9186-4, 2014.Google Scholar
[13] Nobile, F., Tempone, R. and Webster, C., A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46/5 (2008), pp. 2309–2345.Google Scholar
[14] Nobile, F., Tempone, R. and Webster, C., An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46/5 (2008), pp. 2411–2442.Google Scholar
[15] Novak, E. and Ritter, K., High dimensional integration of smooth functions over cubes, Numer. Math., 75 (1996), pp. 79–97.Google Scholar
[16] Novak, E. and Ritter, K., Simple cubature formulas with high polynomial exactness, Constructive Approx., 15 (1999), pp. 499–522.CrossRefGoogle Scholar
[17] Palanisamy, K. R., Balachandran, K. and Ramasamy, S.R., Optimal control of linear time-varying delay systems via single-term Walsh series, Proc. IEEE, 135 (1988), 332.Google Scholar
[18] Poleszczuk, Jan, Delayed differential equations in description of biochemical reactions channels, XXI Congress of Differential Equations and Applications XI Congress of Applied Mathematics Ciudad Real, 21-25 September 2009 (pp. 1–8).Google Scholar
[19] Ruiz-Herrera, A., Chaos in delay differential equations with applications in population dynamics, Discrete Contin. Dyn. Syst. 33(4) (2013), pp. 1633–1644.Google Scholar
[20] Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963), pp. 240–243.Google Scholar
[21] Tang, T. and Zhou, T., Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed, Commun. Comput. Phys., 8 (2010), pp. 226–248.Google Scholar
[22] Trefethen, L. N., Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 2006.Google Scholar
[23] Villasana, M. and Radunskaya, A., A delay differential equation model for tumor growth, J. Math. Biol., 47(3) (2003), pp. 270–294.Google Scholar
[24] Volterra, V., Varizioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. R. Acad. Naz. dei Lincei (ser. 6), 2 (1926), pp. 31–113.Google Scholar
[25] Wang, Z. Q. and Wang, L. L., A Legendre-Gauss collocation method for nonlinear delay differ-ential equations, Dis. Cont. Dyn. Sys. B, 13(3) (2010), pp. 685–708.Google Scholar
[26] Xiu, D., Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), pp. 293–309.Google Scholar
[27] Xiu, D. and Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139.CrossRefGoogle Scholar
[28] Xiu, D. and Karniadakis, G. E., Modeling uncertainty inflow simulations via generalized polynomial chaos, J. Comput. Phys., 187 (2003). pp. 137–167.Google Scholar
[29] Xiu, D. and Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24(2), pp. 619–644.Google Scholar
[30] Zhou, T., Narayan, A. and Xu, Z., Multivariate discrete least-squares approximations with a new type of collocation grid, available online: , January 2014.Google Scholar
[31] Zhou, T. and Tang, T., Galerkin methods for stochastic hyperbolic problems using bi-orthogonal polynomials, J. Sci. Comput., 51 (2012), pp. 274–292.CrossRefGoogle Scholar
[32] Zhou, T. and Tang, T., Convergence analysis for spectral approximation to a scalar transport equation with a random wave speed, J. Comput. Math., 30(6) (2012), pp. 643–656.Google Scholar
[33] Zhou, T., Stochastic Galerkin methods for elliptic interface problems with random input, J. Comput. App. Math., 236 (2011), pp. 782–792.Google Scholar