On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations
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Authors
J. Calatayud
- Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain.
J.-C. Cortés
- Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain.
M. Jornet
- Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain.
Abstract
In this paper, the application of adaptive generalized polynomial chaos (gPC) to quantify the uncertainty for non-linear random difference equations is analyzed. It is proved in detail that, under certain assumptions, the stochastic Galerkin projection technique converges algebraically in mean square to the solution process of the random recursive equation. The effect of the numerical errors on the convergence is also studied. A full numerical experiment illustrates our theoretical findings and gives useful insights to reduce the accumulation of numerical errors in practice.
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ISRP Style
J. Calatayud, J.-C. Cortés, M. Jornet, On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 9, 1077--1084
AMA Style
Calatayud J., Cortés J.-C., Jornet M., On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations. J. Nonlinear Sci. Appl. (2018); 11(9):1077--1084
Chicago/Turabian Style
Calatayud, J., Cortés, J.-C., Jornet, M.. "On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations." Journal of Nonlinear Sciences and Applications, 11, no. 9 (2018): 1077--1084
Keywords
- Adaptive gPC
- stochastic Galerkin projection technique
- non-linear random difference equations
- uncertainty quantification
- numerical analysis
MSC
References
-
[1]
C. Canuto, A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 67–86.
-
[2]
B.-M. Chen-Charpentier, J.-C. Cortés, J.-A. Licea, J.-V. Romero, M.-D. Roselló, F.-J. Santonja, R.-J. Villanueva, Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach, Math. Comput. Simulation, 109 (2015), 113–129.
-
[3]
J.-C. Cortés, J.-V. Romero, M.-D. Roselló, R.-J. Villanueva, Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique, Commun. Nonlinear Sci. Numer. Simul., 50 (2017), 1–15.
-
[4]
A. Ditkowski, R. Kats, On Spectral approximations with nonstandard weight functions and their implementations to generalized chaos expansions, preprint arXiv, (2017), 36 pages.
-
[5]
O. G. Ernst, A. Mugler, H.-J. Starkloff, E. Ullmann, On the convergence of generalized polynomial chaos expansions, ESAIM Math. Model. Numer. Anal., 46 (2012), 317–339.
-
[6]
L. Giraud, J. Langou, M. Rozloznik, The loss of orthogonality in the Gram-Schmidt orthogonalization process, Comput. Math. Appl., 50 (2005), 1069–1075.
-
[7]
G. González-Parra, B. M. Chen-Charpentier, A. J. Arenas, Polynomial Chaos for random fractional order differential equations, Appl. Math. Comput., 226 (2014), 123–130.
-
[8]
D. Lucor, C.-H. Su, G. E. Karniadakis, Generalized polynomial chaos and random oscillators, Internat. J. Numer. Methods Engrg, 60 (2004), 571–596.
-
[9]
F. Santonja, B. Chen-Charpentier, Uncertainty quantification in simulations of epidemics using polynomial chaos, Comput. Math. Methods Med., 2012 (2012), 8 pages.
-
[10]
W. J. Shi, C. J. Zhang, Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations, Appl. Numer. Math., 62 (2012), 1954–1964.
-
[11]
D. B. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, Princeton (2010)
-
[12]
D. B. Xiu, G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619–644.
-
[13]
G. Zou, A. Atangana, Y. Zhou, Error estimates of a semidiscrete finite element method for fractional stochastic diffusionwave equations, Numer Methods Partial Differ. Equ., 2018 (2018), 1–15.
