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Strategies for optimal polynomial approximation of elliptic PDEs with random coefficients
Keywords: high dimensional polynomial approximation; sparse grids; PDEs with random coefficients
Tue, 11:10--11:35
  • Nobile, Fabio (Mathematics, MOX Laboratory, Politecnico di Milano, Italy)
  • Bäck, Joakim (Applied Mathematics and Computational Science, KAUST, Saudi Arabia)
  • Tamellini, Lorenzo (MOX, Department of Mathematics, Politecnico di Milano, Italy)
  • Tempone, Raul (Applied Mathematics and Computational Science, KAUST, Saudi Arabia)

Partial differential equations with stochastic coefficients are a suitable tool to describe systems whose parameters are not completely determined, either because of measurement errors or intrinsic lack of knowledge on the system. In this context, the goal is usually to compute an approximation of statistical moments of the state variables.

In the case of elliptic PDEs, this can be conveniently achieved using high order polynomial approximations of the state variables with respect to the random coefficients. We consider two different methods to build such approximations, namely the Stochastic Galerkin and Stochastic Collocation methods. Global polynomial approximations are sound since, for elliptic PDEs, the state variables usually exhibit high regularity in their dependence to the random parameters.

When the number of parameters is moderate, these methods can be remarkably more effective than classical sampling methods. However, contrary to the latter, the performance of polynomial approximations deteriorates as the number of random variables increases (curse of dimensionality); to prevent this, care has to be put in the construction of the approximating polynomial space.

We present theoretical analyses of the structure of the problem that lead to practical strategies to construct optimal polynomial spaces in the case of Stochastic Galerkin approximations, as well as optimal sparse grids in the case of Stochastic Collocation approximations. We apply the strategy to some practical examples and show its effectiveness.

[1] I. Babuska, F. Nobile, and R. Tempone, "A stochastic collocation method for elliptic partial differential equations with random input data," SIAM Review, vol. 52, pp. 317–355, June 2010.
[2] J. Bäck, F. Nobile, L. Tamellini, and R. Tempone. Implementation of optimal Galerkin and collocation approximations of PDEs with random coefficients. Proceedings of CANUM 2010. To appear in ESAIM Proceedings.
[3] J. Bäck, F. Nobile, L. Tamellini, and R. Tempone. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In Spectral and High Order Methods for Partial Differential Equations, J.S. Hesthaven and E.M. Ronquist editors, volume 76 of Lecture Notes in Computational Science and Engineering, pages 43–62. Springer, 2011.