On the convergence of generalized polynomial chaos expansions

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

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

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

Abstract

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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.

How to cite

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Ernst, Oliver G., et al. "On the convergence of generalized polynomial chaos expansions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 317-339. <http://eudml.org/doc/273193>.

@article{Ernst2012,
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 - Modélisation Mathématique et Analyse Numérique},
keywords = {equations with random data; polynomial chaos; generalized polynomial chaos; Wiener–Hermite expansion; Wiener integral; determinate measure; moment problem; stochastic Galerkin method; spectral elements; Wiener-Hermite expansion; numerical examples},
language = {eng},
number = {2},
pages = {317-339},
publisher = {EDP-Sciences},
title = {On the convergence of generalized polynomial chaos expansions},
url = {http://eudml.org/doc/273193},
volume = {46},
year = {2012},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2012
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; Wiener-Hermite expansion; numerical examples
UR - http://eudml.org/doc/273193
ER -

References

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  1. [1] M. Arnst, R. Ghanem and C. Soize, Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys.229 (2010) 3134–3154. Zbl1184.62034MR2601093
  2. [2] 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. Zbl1080.65003MR2084236
  3. [3] 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. Zbl1087.65004MR2121215
  4. [4] C. Berg, Moment problems and polynomial approximation. Ann. Fac. Sci. Toulouse Math. (Numéro spécial Stieltjes) 6 (1996) 9–32. Zbl0877.44003MR1462705
  5. [5] C. Berg and J.P.R. Christensen, Density questions in the classical theory of moments. Ann. Inst. Fourier31 (1981) 99–114. Zbl0437.42007MR638619
  6. [6] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005). Zbl1092.46001MR2176612
  7. [7] 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. Zbl0029.14302MR20230
  8. [8] T.S. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978). Zbl0389.33008MR481884
  9. [9] J.H. Curtiss, A note on the theory of moment generating functions. Ann. Stat.13 (1942) 430–433. Zbl0063.01024MR7577
  10. [10] 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. Zbl1072.60042MR2116369
  11. [11] R.V. Field Jr. and M. Grigoriu, On the accuracy of the polynomial chaos expansion. Probab. Engrg. Mech.19 (2004) 65–80. 
  12. [12] G. Freud, Orthogonal Polynomials. Akademiai, Budapest (1971). 
  13. [13] W. Gautschi, Orthogonal Polynomials: Computation and Approximation. Oxford University Press (2004). Zbl1130.42300MR2061539
  14. [14] R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). Zbl0722.73080MR1083354
  15. [15] A. Gut, On the moment problem. Bernoulli8 (2002) 407–421. Zbl1006.60016MR1913113
  16. [16] T. Hida, Brownian Motion. Springer, New York (1980). Zbl0432.60002MR562914
  17. [17] K. Itô, Multiple Wiener integral. J. Math. Soc. Jpn3 (1951) 157–169. Zbl0044.12202MR44064
  18. [18] S. Janson, Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997). Zbl1143.60005MR1474726
  19. [19] O. Kallenberg, Foundations of Modern Probability, 2nd edition. Springer-Verlag, New York (2002). Zbl0892.60001MR1876169
  20. [20] G. Kallianpur, Stochastic Filtering Theory. Springer, New York (1980). Zbl0458.60001MR583435
  21. [21] G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edition. Oxford University Press (2005). Zbl1256.76003MR2165335
  22. [22] 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). 
  23. [23] A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933). Zbl0007.21601MR494348JFM59.1152.03
  24. [24] G.D. Lin, On the moment problems. Stat. Probab. Lett. 35 (1997) 85–90. Correction: G.D. Lin, On the moment problems. Stat. Probab. Lett. 50 (2000) 205. Zbl0904.62021MR1467713
  25. [25] P. Masani, Wiener’s contributions to generalized harmonic analysis, prediction theory and filter theory. Bull. Amer. Math. Soc.72 (1966) 73–125. Zbl0144.19402MR187018
  26. [26] P.R. Masani, Norbert Wiener, 1894–1964. Number 5 in Vita mathematica, Birkhäuser (1990). Zbl0681.01016MR1032520
  27. [27] H.G. Matthies and C. Bucher, Finite elements for stochastic media problems. Comput. Methods Appl. Mech. Engrg.168 (1999) 3–17. Zbl0953.74065MR1666718
  28. [28] A. Mugler and H.-J. Starkloff, On elliptic partial differential equations with random coefficients, Stud. Univ. Babes-Bolyai Math.56 (2011) 473–487. MR2843705
  29. [29] A.T. Patera, A spectral element method for fluid dynamics – laminar flow in a channel expansion. J. Comput. Phys.54 (1984) 468–488. Zbl0535.76035
  30. [30] R.E.A.C. Payley and N. Wiener, Fourier Transforms in the Complex Domain. Number XIX in Colloquium Publications. Amer. Math. Soc. (1934). Zbl0011.01601
  31. [31] L.C. Petersen, On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand.51 (1982) 361–366. Zbl0514.44007MR690537
  32. [32] M. Reed and B. Simon, Methods of modern mathematical physics, Functional analysis 1. Academic press, New York (1972). Zbl0459.46001
  33. [33] M. Riesz, Sur le problème des moments et le théorème de Parseval correspondant. Acta Litt. Ac. Scient. Univ. Hung.1 (1923) 209–225. Zbl49.0708.02JFM49.0708.02
  34. [34] R.A. Roybal, A reproducing kernel condition for indeterminacy in the multidimensional moment problem. Proc. Amer. Math. Soc.135 (2007) 3967–3975. Zbl1133.47013MR2341947
  35. [35] I.E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956) 106–134. Zbl0070.34003MR76317
  36. [36] A.N. Shiryaev, Probability. Springer-Verlag, New York (1996). Zbl0835.60002MR1368405
  37. [37] I.C. Simpson, Numerical integration over a semi-infinite interval using the lognormal distribution. Numer. Math.31 (1978) 71–76. Zbl0421.65006MR508589
  38. [38] C. Soize and R. Ghanem, Physical systems with random uncertainties: Chaos representations with arbitrary probability measures. SIAM J. Sci. Comput.26 (2004) 395–410. Zbl1075.60084MR2116353
  39. [39] 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. Zbl1156.60303
  40. [40] J.M. Stoyanov, Counterexamples in Probability, 2nd edition. John Wiley & Sons Ltd., Chichester, UK (1997). Zbl0629.60001MR930671
  41. [41] G. Szegö, Orthogonal Polynomials. American Mathematical Society, Providence, Rhode Island (1939). MR106295
  42. [42] 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. Zbl1120.65004MR2317004
  43. [43] N. Wiener, Differential space. J. Math. Phys.2 (1923) 131–174. 
  44. [44] N. Wiener, Generalized harmonic analysis. Acta Math.55 (1930) 117–258. Zbl56.0954.02MR1555316JFM56.0954.02
  45. [45] N. Wiener, The homogeneous chaos. Amer. J. Math.60 (1938) 897–936. Zbl0019.35406MR1507356JFM64.0887.02
  46. [46] D. Xiu and J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput.27 (2005) 1118–1139. Zbl1091.65006MR2199923
  47. [47] 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. Zbl1016.65001MR1932024
  48. [48] D. Xiu and G.E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput.24 (2002) 619–644. Zbl1014.65004MR1951058
  49. [49] 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. Zbl1038.80003
  50. [50] D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phy.187 (2003) 137–167. Zbl1047.76111MR1977783
  51. [51] 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. 
  52. [52] 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). Zbl1188.60038MR2103735
  53. [53] 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 Communications 14 (1997) 247–270. Zbl0873.42016MR1448689

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