# On the convergence of generalized polynomial chaos expansions

Oliver G. Ernst; Antje Mugler; Hans-Jörg Starkloff; Elisabeth Ullmann

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 46, Issue: 2, page 317-339
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topErnst, Oliver G., et al. "On the convergence of generalized polynomial chaos expansions." ESAIM: Mathematical Modelling and Numerical Analysis 46.2 (2011): 317-339. <http://eudml.org/doc/222104>.

@article{Ernst2011,

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.},

author = {Ernst, Oliver G., Mugler, Antje, Starkloff, Hans-Jörg, Ullmann, Elisabeth},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Equations with random data; polynomial chaos; generalized polynomial chaos; Wiener–Hermite expansion; Wiener integral; determinate measure; moment problem; stochastic Galerkin method; spectral elements; equations with random data; Wiener-Hermite expansion; numerical examples},

language = {eng},

month = {10},

number = {2},

pages = {317-339},

publisher = {EDP Sciences},

title = {On the convergence of generalized polynomial chaos expansions},

url = {http://eudml.org/doc/222104},

volume = {46},

year = {2011},

}

TY - JOUR

AU - Ernst, Oliver G.

AU - Mugler, Antje

AU - Starkloff, Hans-Jörg

AU - Ullmann, Elisabeth

TI - On the convergence of generalized polynomial chaos expansions

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/10//

PB - EDP Sciences

VL - 46

IS - 2

SP - 317

EP - 339

AB - 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.

LA - eng

KW - Equations with random data; polynomial chaos; generalized polynomial chaos; Wiener–Hermite expansion; Wiener integral; determinate measure; moment problem; stochastic Galerkin method; spectral elements; equations with random data; Wiener-Hermite expansion; numerical examples

UR - http://eudml.org/doc/222104

ER -

## References

top- M. Arnst, R. Ghanem and C. Soize, Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys.229 (2010) 3134–3154.
- I. Babuška, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal.42 (2004) 800–825.
- I. Babuška, R. Tempone and G.E. Zouraris, 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.
- C. Berg, Moment problems and polynomial approximation. Ann. Fac. Sci. Toulouse Math. (Numéro spécial Stieltjes)6 (1996) 9–32.
- C. Berg and J.P.R. Christensen, Density questions in the classical theory of moments. Ann. Inst. Fourier31 (1981) 99–114.
- A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005).
- R.H. Cameron and W.T. Martin, The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals. Ann. Math.48 (1947) 385–392.
- T.S. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978).
- J.H. Curtiss, A note on the theory of moment generating functions. Ann. Stat.13 (1942) 430–433.
- B.J. Debusschere, H.N. Najm, Ph.P. Pébay, O.M. Knio, R.G. Ghanem and O.P. le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput.26 (2004) 698–719.
- R.V. FieldJr. and M. Grigoriu, On the accuracy of the polynomial chaos expansion. Probab. Engrg. Mech.19 (2004) 65–80.
- 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).
- A. Gut, On the moment problem. Bernoulli8 (2002) 407–421.
- T. Hida, Brownian Motion. Springer, New York (1980).
- K. Itô, Multiple Wiener integral. J. Math. Soc. Jpn3 (1951) 157–169.
- 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).
- G.D. Lin, On the moment problems. Stat. Probab. Lett.35 (1997) 85–90
- P. Masani, Wiener’s contributions to generalized harmonic analysis, prediction theory and filter theory. Bull. Amer. Math. Soc.72 (1966) 73–125.
- P.R. Masani, Norbert Wiener, 1894–1964. Number 5 in Vita mathematica, Birkhäuser (1990).
- H.G. Matthies and C. Bucher, Finite elements for stochastic media problems. Comput. Methods Appl. Mech. Engrg.168 (1999) 3–17.
- A. Mugler and H.-J. Starkloff, On elliptic partial differential equations with random coefficients, Stud. Univ. Babes-Bolyai Math.56 (2011) 473–487.
- A.T. Patera, A spectral element method for fluid dynamics – laminar flow in a channel expansion. J. Comput. Phys.54 (1984) 468–488.
- R.E.A.C. Payley and N. Wiener, Fourier Transforms in the Complex Domain. Number XIX in Colloquium Publications. Amer. Math. Soc. (1934).
- L.C. Petersen, On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand.51 (1982) 361–366.
- M. Reed and B. Simon, Methods of modern mathematical physics, Functional analysis1. Academic press, New York (1972).
- M. Riesz, Sur le problème des moments et le théorème de Parseval correspondant. Acta Litt. Ac. Scient. Univ. Hung.1 (1923) 209–225.
- R.A. Roybal, A reproducing kernel condition for indeterminacy in the multidimensional moment problem. Proc. Amer. Math. Soc.135 (2007) 3967–3975.
- I.E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc.81 (1956) 106–134.
- A.N. Shiryaev, Probability. Springer-Verlag, New York (1996).
- I.C. Simpson, Numerical integration over a semi-infinite interval using the lognormal distribution. Numer. Math.31 (1978) 71–76.
- C. Soize and R. Ghanem, Physical systems with random uncertainties: Chaos representations with arbitrary probability measures. SIAM J. Sci. Comput.26 (2004) 395–410.
- 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).
- R.-A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal.27 (2007) 232–261.
- N. Wiener, Differential space. J. Math. Phys.2 (1923) 131–174.
- N. Wiener, Generalized harmonic analysis. Acta Math.55 (1930) 117–258.
- N. Wiener, The homogeneous chaos. Amer. J. Math.60 (1938) 897–936.
- D. Xiu and J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput.27 (2005) 1118–1139.
- D. Xiu and G.E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg.191 (2002) 4927–4948.
- D. Xiu and G.E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput.24 (2002) 619–644.
- D. Xiu and G.E. Karniadakis, A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Trans.46 (2003) 4681–4693.
- D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phy.187 (2003) 137–167.
- D. Xiu, D. Lucor, C.-H. Su and G.E. Karniadakis, Stochastic modeling of flow-structure interactions using generalized polynomial chaos. J. Fluids Eng.124 (2002) 51–59.
- 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 Science2660, edited by P.M.A. Sloot, D. Abramson, A.V. Bogdanov, J.J. Dongarra, A.Y. Zomaya and Y.E. Gorbachev. Springer-Verlag (2003).
- Y. Xu, 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 Communications14 (1997) 247–270.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.