Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T12:09:09.342Z Has data issue: false hasContentIssue false
  • English
  • Français

On the convergence of generalized polynomial chaos expansions

Published online by Cambridge University Press:  12 October 2011

Oliver G. Ernst ,
Antje Mugler ,
Hans-Jörg Starkloff  and
Elisabeth Ullmann
Oliver G. Ernst
Affiliation:
Institut für Numerische Mathematik und Optimierung, TU Bergakademie Freiberg, 09596 Freiberg, Germany. ernst@math.tu-freiberg.de; ullmann@math.tu-freiberg.de
Antje Mugler
Affiliation:
Fachgruppe Mathematik, University of Applied Sciences Zwickau, 08012 Zwickau, Germany; Antje.Mugler@fh-zwickau.de; hans.joerg.starkloff@fh-zwickau.de
Hans-Jörg Starkloff
Affiliation:
Fachgruppe Mathematik, University of Applied Sciences Zwickau, 08012 Zwickau, Germany; Antje.Mugler@fh-zwickau.de; hans.joerg.starkloff@fh-zwickau.de
Elisabeth Ullmann
Affiliation:
Institut für Numerische Mathematik und Optimierung, TU Bergakademie Freiberg, 09596 Freiberg, Germany. ernst@math.tu-freiberg.de; ullmann@math.tu-freiberg.de
Get access

Abstract

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

Keywords

Equations with random datapolynomial chaosgeneralized polynomial chaosWiener–Hermite expansionWiener integraldeterminate measuremoment problemstochastic Galerkin methodspectral elements
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 317 - 339
Copyright
© EDP Sciences, SMAI, 2011

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

Arnst, M., Ghanem, R. and Soize, C., Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys. 229 (2010) 3134–3154. Google Scholar
Babuška, I., Tempone, R. and Zouraris, G.E., Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800–825. Google Scholar
Babuška, I., Tempone, R. and Zouraris, G.E., Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1251–1294. Google Scholar
Berg, C., Moment problems and polynomial approximation. Ann. Fac. Sci. Toulouse Math. (Numéro spécial Stieltjes) 6 (1996) 9–32. Google Scholar
Berg, C. and Christensen, J.P.R., Density questions in the classical theory of moments. Ann. Inst. Fourier 31 (1981) 99–114. Google Scholar
A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005).
Cameron, R.H. and Martin, W.T., The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals. Ann. Math. 48 (1947) 385–392. Google Scholar
T.S. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978).
Curtiss, J.H., A note on the theory of moment generating functions. Ann. Stat. 13 (1942) 430–433. Google Scholar
Debusschere, B.J., Najm, H.N., Pébay, Ph.P., Knio, O.M., Ghanem, R.G. and le Maître, O.P., Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698–719. Google Scholar
Field, R.V. Jr. and Grigoriu, M., On the accuracy of the polynomial chaos expansion. Probab. Engrg. Mech. 19 (2004) 65–80. Google Scholar
G. Freud, Orthogonal Polynomials. Akademiai, Budapest (1971).
W. Gautschi, Orthogonal Polynomials: Computation and Approximation. Oxford University Press (2004).
R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991).
Gut, A., On the moment problem. Bernoulli 8 (2002) 407–421. Google Scholar
T. Hida, Brownian Motion. Springer, New York (1980).
Itô, K., Multiple Wiener integral. J. Math. Soc. Jpn 3 (1951) 157–169. Google Scholar
S. Janson, Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997).
O. Kallenberg, Foundations of Modern Probability, 2nd edition. Springer-Verlag, New York (2002).
G. Kallianpur, Stochastic Filtering Theory. Springer, New York (1980).
G.E. Karniadakis and S. Sherwin, Spectral/ hp Element Methods for Computational Fluid Dynamics, 2nd edition. Oxford University Press (2005).
G.E. Karniadakis, C.-H. Shu, D. Xiu, D. Lucor, C. Schwab and R.-A. Todor, Generalized polynomial chaos solution for differential equations with random inputs. Technical Report 2005-1, Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland (2005).
A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933).
Lin, G.D., On the moment problems. Stat. Probab. Lett. 35 (1997) 85–90
Masani, P., Wiener’s contributions to generalized harmonic analysis, prediction theory and filter theory. Bull. Amer. Math. Soc. 72 (1966) 73–125. Google Scholar
P.R. Masani, Norbert Wiener, 1894–1964. Number 5 in Vita mathematica, Birkhäuser (1990).
Matthies, H.G. and Bucher, C., Finite elements for stochastic media problems. Comput. Methods Appl. Mech. Engrg. 168 (1999) 3–17. Google Scholar
Mugler, A. and Starkloff, H.-J., On elliptic partial differential equations with random coefficients, Stud. Univ. Babes-Bolyai Math. 56 (2011) 473–487. Google Scholar
Patera, A.T., A spectral element method for fluid dynamics – laminar flow in a channel expansion. J. Comput. Phys. 54 (1984) 468–488. Google Scholar
R.E.A.C. Payley and N. Wiener, Fourier Transforms in the Complex Domain. Number XIX in Colloquium Publications. Amer. Math. Soc. (1934).
Petersen, L.C., On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand. 51 (1982) 361–366. Google Scholar
M. Reed and B. Simon, Methods of modern mathematical physics, Functional analysis 1. Academic press, New York (1972).
Riesz, M., Sur le problème des moments et le théorème de Parseval correspondant. Acta Litt. Ac. Scient. Univ. Hung. 1 (1923) 209–225. Google Scholar
Roybal, R.A., A reproducing kernel condition for indeterminacy in the multidimensional moment problem. Proc. Amer. Math. Soc. 135 (2007) 3967–3975. Google Scholar
Segal, I.E., Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956) 106–134. Google Scholar
A.N. Shiryaev, Probability. Springer-Verlag, New York (1996).
Simpson, I.C., Numerical integration over a semi-infinite interval using the lognormal distribution. Numer. Math. 31 (1978) 71–76. Google Scholar
Soize, C. and Ghanem, R., Physical systems with random uncertainties: Chaos representations with arbitrary probability measures. SIAM J. Sci. Comput. 26 (2004) 395–410. Google Scholar
H.-J. Starkloff, On the number of independent basic random variables for the approximate solution of random equations, in Celebration of Prof. Dr. Wilfried Grecksch’s 60th Birthday, edited by C. Tammer and F. Heyde. Shaker Verlag, Aachen (2008) 195–211.
J.M. Stoyanov, Counterexamples in Probability, 2nd edition. John Wiley & Sons Ltd., Chichester, UK (1997).
G. Szegö, Orthogonal Polynomials. American Mathematical Society, Providence, Rhode Island (1939).
Todor, R.-A. and Schwab, C., Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232–261. Google Scholar
Wiener, N., Differential space. J. Math. Phys. 2 (1923) 131–174. Google Scholar
Wiener, N., Generalized harmonic analysis. Acta Math. 55 (1930) 117–258. Google Scholar
Wiener, N., The homogeneous chaos. Amer. J. Math. 60 (1938) 897–936. Google Scholar
Xiu, D. and Hesthaven, J.S., High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118–1139. Google Scholar
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927–4948. Google Scholar
Xiu, D. and Karniadakis, G.E., The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002) 619–644. Google Scholar
Xiu, D. and Karniadakis, G.E., A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Trans. 46 (2003) 4681–4693. Google Scholar
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phy. 187 (2003) 137–167. Google Scholar
Xiu, D., Lucor, D., Su, C.-H. and Karniadakis, G.E., Stochastic modeling of flow-structure interactions using generalized polynomial chaos. J. Fluids Eng. 124 (2002) 51–59. Google Scholar
D. Xiu, D. Lucor, C.-H. Su and G.E. Karniadakis, Performance evaluation of generalized polynomial chaos, in Computational Science – ICCS 2003, Lecture Notes in Computer Science 2660, edited by P.M.A. Sloot, D. Abramson, A.V. Bogdanov, J.J. Dongarra, A.Y. Zomaya and Y.E. Gorbachev. Springer-Verlag (2003).
Xu, Y., On orthogonal polynomials in several variables, in Special functions, q-series, and related topics, edited by M. Ismail, D.R. Masson and M. Rahman. Fields Institute Communications 14 (1997) 247–270. Google Scholar