Foundations of Computational Mathematics

, Volume 10, Issue 6, pp 615–646

Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs

Article

Abstract

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(yi(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable xD and the in general countably many parameters y.

We establish new regularity theorems describing the smoothness properties of the solution u as a map from yU=(−1,1) to \(V=H^{1}_{0}(D)\). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u.

Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable yU are established. These estimates are of the form Nr, where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients.

A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family \(\{V_{l}\}_{l=0}^{\infty}\subset V\) of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from yU=(−1,1) to a smoothness space WV are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with \(H^{2}(D)\cap H^{1}_{0}(D)\) in the case where D is a smooth or convex domain.

Our analysis shows that in realistic settings a convergence rate \(N_{\mathrm{dof}}^{-s}\) in terms of the total number of degrees of freedom Ndof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.

Keywords

Stochastic and parametric elliptic equations Wiener polynomial chaos Approximation rates Nonlinear approximation Sparsity 

Mathematics Subject Classification (2000)

41A 65N 65C30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Babuška, R. Tempone, G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. 42, 800–825 (2004). MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    I. Babuška, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45, 1005–1034 (2007). MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    P.G. Ciarlet, The Finite Element Methods for Elliptic Problems (Elsevier, Amsterdam, 1978). Google Scholar
  4. 4.
    A. Cohen, Numerical Analysis of Wavelet Methods (Elsevier, Amsterdam, 2003). MATHGoogle Scholar
  5. 5.
    A. Cohen, W. Dahmen, R. DeVore, Adaptive wavelet methods for elliptic operator equations—convergence rates, Math. Comput. 70, 27–75 (2001). MATHMathSciNetGoogle Scholar
  6. 6.
    A. Cohen, R. DeVore, C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s, Anal. Appl. (2010, to appear). Google Scholar
  7. 7.
    R. DeVore, Nonlinear approximation, Acta Numer. 7, 51–150 (1998). CrossRefMathSciNetGoogle Scholar
  8. 8.
    T. Gantumur, H. Harbecht, R. Stevenson, An adaptive wavelet method without coarsening of the iterands, Math. Comput. 76, 615–629 (2007). MATHCrossRefGoogle Scholar
  9. 9.
    R. Ghanem, P. Spanos, Spectral techniques for stochastic finite elements, Arch. Comput. Methods Eng. 4, 63–100 (1997). CrossRefMathSciNetGoogle Scholar
  10. 10.
    G.E. Karniadakis, D.B. Xiu, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24, 619–644 (2002). MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Kleiber, T.D. Hien, The Stochastic Finite Element Methods (Wiley, New York, 1992). Google Scholar
  12. 12.
    M. Ledoux, M. Talagrand, Probability in Banachspaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 23 (Verlag, Berlin, 1991). Google Scholar
  13. 13.
    F. Nobile, R. Tempone, C.G. Webster, A sparse grid stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 46, 2309–2345 (2008). MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    F. Nobile, R. Tempone, C.G. Webster, An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 46, 2411–2442 (2008). MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations and application to transport and continuum mechanics, Arch. Comput. Methods Eng. 15, 229–275 (2008). MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    W. Schoutens, Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, vol. 146 (Springer, New York, 2000). MATHGoogle Scholar
  17. 17.
    C. Schwab, R. Todor, Karhúnen–Loève approximation of random fields by generalized fast multipole methods, J. Comput. Phys. 217, 100–122 (2006). MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 4, 240–243 (1963). Google Scholar
  19. 19.
    R. Todor, Robust eigenvalue computation for smoothing operators, SIAM J. Numer. Anal. 44, 865–878 (2006). MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    R. Todor, C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal. 45, 232–261 (2007). MathSciNetGoogle Scholar
  21. 21.
    T. von Petersdorff, C. Schwab, Sparse finite element methods for operator equations with stochastic data, Appl. Math. 51, 145–180 (2006). MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    N. Wiener, The homogeneous chaos, Am. J. Math. 60, 897–936 (1938). CrossRefMathSciNetGoogle Scholar

Copyright information

© SFoCM 2010

Authors and Affiliations

  • Albert Cohen
    • 1
    • 2
  • Ronald DeVore
    • 3
  • Christoph Schwab
    • 4
  1. 1.Laboratoire Jacques-Louis Lions, UMR 7598UPMC Univ. Paris 06ParisFrance
  2. 2.Laboratoire Jacques-Louis Lions, UMR 7598CNRSParisFrance
  3. 3.Department of MathematicsTexas A& M UniversityCollege StationUSA
  4. 4.Seminar for Applied MathematicsETH ZürichZürichSwitzerland

Personalised recommendations