Applications of Mathematics

, Volume 51, Issue 2, pp 145–180 | Cite as

Sparse finite element methods for operator equations with stochastic data

  • Tobias von Petersdorff
  • Christoph Schwab
Article

Abstract

Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation Au = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments \(\mathcal{M}^1 u \in V,\mathcal{M}^2 u \in V \otimes V,...,\mathcal{M}^k u \in V \otimes ... \otimes V\) of the solution.

We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment \(\mathcal{M}^k u\) for k⩾1.

The key tool in both algorithms is a “sparse tensor product” space for the approximation of \(\mathcal{M}^k u\) with O(N(log N)k−1) degrees of freedom, instead of Nk degrees of freedom for the full tensor product FEM space.

A sparse Monte-Carlo FEM with M samples (i.e., deterministic solver) is proved to yield approximations to \(\mathcal{M}^k u\) with a work of O(M N(log N)k−1) operations. The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples.

The deterministic FEM is based on deterministic equations for \(\mathcal{M}^k u\) in Dk ⊂ ℝkd. Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of \(\mathcal{M}^k u\) with O(N(log N)k−1) degrees of freedom converging at an optimal rate (up to logs).

For nonlocal operators wavelet compression of the operators is used. The linear systems are solved iteratively with multilevel preconditioning. This yields an approximation for \(\mathcal{M}^k u\) with at most O(N (log N)k+1) operations.

Keywords

wavelet compression of operators random data Monte-Carlo method wavelet finite element method 

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References

  1. [1]
    I. Babuška: On randomized solution of Laplace’s equation. Čas. Pěst. Mat. 86 (1961), 269–276.Google Scholar
  2. [2]
    I. Babuška: Error-bounds for finite element method. Numer. Math. 16 (1971), 322–333.MathSciNetCrossRefGoogle Scholar
  3. [3]
    I. Babuška, R. Tempone, and G. Zouraris: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004), 800–825.MathSciNetCrossRefGoogle Scholar
  4. [4]
    V. I. Bogachev: Gaussian Measures. AMS Mathematical Surveys and Monographs Vol. 62. AMS, Providence, 1998.Google Scholar
  5. [5]
    L. Breiman: Probability. Addison-Wesley, Reading, 1968.Google Scholar
  6. [6]
    W. A. Light, E.W. Cheney: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics Vol. 1169. Springer-Verlag, Berlin, 1985.Google Scholar
  7. [7]
    P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Elsevier Publ. North Holland, Amsterdam, 1978.Google Scholar
  8. [8]
    M. Dahmen, H. Harbrecht, and R. Schneider: Compression techniques for boundary integral equations— optimal complexity estimates. SIAM J. Numer. Anal. 43 (2006), 2251–2271.MathSciNetCrossRefGoogle Scholar
  9. [9]
    W. Dahmen, R. Stevenson: Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal. 37 (1999), 319–352.MathSciNetCrossRefGoogle Scholar
  10. [10]
    S. C. Eisenstat, H. C. Elman, M. H. Schultz: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20 (1983), 345–357.MathSciNetCrossRefGoogle Scholar
  11. [11]
    M. Griebel, P. Oswald, T. Schiekofer: Sparse grids for boundary integral equations. Numer. Math. 83 (1999), 279–312.MathSciNetCrossRefGoogle Scholar
  12. [12]
    S. Hildebrandt, N. Wienholtz: Constructive proofs of representation theorems in separable Hilbert space. Commun. Pure Appl. Math. 17 (1964), 369–373.MathSciNetGoogle Scholar
  13. [13]
    G. C. Hsiao, W. L. Wendland: A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58 (1977), 449–481.MathSciNetCrossRefGoogle Scholar
  14. [14]
    S. Janson: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge, 1997.Google Scholar
  15. [15]
    S. Larsen: Numerical analysis of elliptic partial differential equations with stochastic input data. Doctoral Dissertation. Univ. of Maryland, 1985.Google Scholar
  16. [16]
    M. Ledoux, M. Talagrand: Probability in Banach Spaces. Isoperimetry and Processes. Springer-Verlag, Berlin, 1991.Google Scholar
  17. [17]
    P. Malliavin: Stochastic Analysis. Springer-Verlag, Berlin, 1997.Google Scholar
  18. [18]
    W. McLean: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, 2000.Google Scholar
  19. [19]
    J. C. Nédélec, J. P. Planchard: Une méthode variationelle d’éléments finis pour la résolution numérique d’un problème extérieur dans ℝ3. RAIRO Anal. Numér. 7 (1973), 105–129.Google Scholar
  20. [20]
    G. Schmidlin, C. Lage, and C. Schwab: Rapid solution of first kind boundary integral equations in ℝ 3. Eng. Anal. Bound. Elem. 27 (2003), 469–490.CrossRefGoogle Scholar
  21. [21]
    T. von Petersdorff, C. Schwab: Wavelet approximations for first kind boundary integral equations in polygons. Numer. Math. 74 (1996), 479–516.MathSciNetCrossRefGoogle Scholar
  22. [22]
    T. von Petersdorff, C. Schwab: Numerical solution of parabolic equations in high dimensions. M2AN, Math. Model. Numer. Anal. 38 (2004), 93–127.MathSciNetCrossRefGoogle Scholar
  23. [23]
    R. Schneider: Multiskalen-und Wavelet-Matrixkompression. Advances in Numerical Mathematics. Teubner, Stuttgart, 1998.Google Scholar
  24. [24]
    C. Schwab, R. A. Todor: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95 (2003), 707–734.MathSciNetCrossRefGoogle Scholar
  25. [25]
    C. Schwab, R. A. Todor: Sparse finite elements for stochastic elliptic problems—higher order moments. Computing 71 (2003), 43–63.MathSciNetCrossRefGoogle Scholar
  26. [26]
    S. A. Smolyak: Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4 (1963), 240–243.Google Scholar
  27. [27]
    V. N. Temlyakov: Approximation of Periodic Functions. Nova Science Publ., New York, 1994.Google Scholar
  28. [28]
    G.W. Wasilkowski, H. Wozniakowski: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complexity 11 (1995), 1–56.MathSciNetCrossRefGoogle Scholar
  29. [29]
    N. Wiener: The homogeneous Chaos. Amer. J. Math. 60 (1938), 987–936.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2006

Authors and Affiliations

  • Tobias von Petersdorff
    • 1
  • Christoph Schwab
    • 2
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Seminar for Applied MathematicsETH ZürichZürichSwitzerland

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