## Preliminary list of abstracts

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*Keywords:*high dimensional polynomial approximation; sparse grids; PDEs with random coefficients

- Nobile, Fabio (Mathematics, MOX Laboratory, Politecnico di Milano, Italy)

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.

*SIAM Review*, vol. 52, pp. 317–355, June 2010.

*Proceedings of CANUM 2010*. To appear in ESAIM Proceedings.

*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.