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On Spectral Approximations with Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

  • A. DitkowskiEmail author
  • R. Katz
Article

Abstract

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained, e.g, by Gram–Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm–Liouville problem, as is the case for known polynomials, such as the Jacobi polynomials. Therefore, the standard convergence theorems do not apply. Furthermore, since in general multidimensional cases the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a product of one-dimensional orthogonal polynomials, as well. This work provides a way of estimating the convergence rate using a comparison lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We also show that the use of non-standard weight functions can allow for efficient integration of singular functions. We demonstrate the use of this method to uncertainty quantification problem using Generalized Polynomial Chaos Expansions in the case of dependent random variables, as well.

Keywords

GPC Generalized chaos expansions Spectral methods Orthogonal polynomials Integration methods Collocation methods 

Notes

References

  1. 1.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in sobolev spaces. Math. Comput. 38(157), 67–86 (1982)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chen, X., Park, E.-J., Xiu, D.: A flexible numerical approach for quantification of epistemic uncertainty. J. Comput. Phys. 240, 211–224 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Conte, S.D., De Boor, C.W.: Elementary Numerical Analysis: an Algorithmic Approach. McGraw-Hill Higher Education, New York City, NY (1980)zbMATHGoogle Scholar
  4. 4.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation, vol. 303. Springer, Berlin, Germany (1993)zbMATHGoogle Scholar
  5. 5.
    Funaro, D.: Polynomial Approximation of Differential Equations, vol. 8. Springer, Berlin, Germany (2008)zbMATHGoogle Scholar
  6. 6.
    Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys 3(2), 505–518 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems, vol. 21. Cambridge University Press, Cambridge, UK (2007)zbMATHGoogle Scholar
  8. 8.
    Jakeman, J., Eldred, M., Xiu, D.: Numerical approach for quantification of epistemic uncertainty. J. Comput. Phys. 229(12), 4648–4663 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Li, J., Qi, X., Xiu, D.: On upper and lower bounds for quantity of interest in problems subject to epistemic uncertainty. SIAM J. Sci. Comput. 36(2), A364–A376 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lin, G., Su, C.-H., Karniadakis, G.E.: Random roughness enhances lift in supersonic flow. Phys. Rev. Lett. 99(10), 104501 (2007)Google Scholar
  11. 11.
    Mujumdar, A.S.: Handbook of Industrial Drying. CRC Press, Boca Raton, Florida (2014)Google Scholar
  12. 12.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, vol. 37. Springer, Berlin, Germany (2010)zbMATHGoogle Scholar
  14. 14.
    Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23(3), 470–472 (1952)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: algorithms, Analysis and Applications, vol. 41. Springer, Berlin, Germany (2011)zbMATHGoogle Scholar
  16. 16.
    Stein, E.M.: Harmonic Analysis (PMS-43), Volume 43: real-variable Methods, Orthogonality, and Oscillatory Integrals. (PMS-43), vol. 3. Princeton University Press, Princeton, NJ (2016)Google Scholar
  17. 17.
    Szabó, B.A., Yosibash, Z.: Numerical analysis of singularities in two dimensions. Part 2: computation of generalized flux/stress intensity factors. Int. J. Numer. Methods Eng. 39(3), 409–434 (1996)zbMATHGoogle Scholar
  18. 18.
    Szeg, G.: Orthogonal Polynomials, vol. 23. American Mathematical Soc., Providence, Rhode Island (1939)Google Scholar
  19. 19.
    Xiu, D.: Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys. 5(2–4), 242–272 (2009)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xiu, D.: Numerical Methods for Stochastic Computations: a Spectral Method Approach. Princeton University Press, Princeton, NJ (2010)zbMATHGoogle Scholar
  21. 21.
    Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Xiu, D., Karniadakis, G.E.: A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Transf. 46(24), 4681–4693 (2003)zbMATHGoogle Scholar
  24. 24.
    Xiu, D., Tartakovsky, D.M.: Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28(3), 1167–1185 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Yuan, X.: Lecture notes on orthogonal polynomials of several variables. Adv. Theory Spec. Funct. Orthogonal Polynomials Nova Sci. Publ. 135, 188 (2004)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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