Polynomial approximation of PDEs with stochastic coefficients

Tamellini, Lorenzo

Polynomial approximation of PDEs with stochastic coefficients

Monday 26th March 2012

Nobile, F.

Quarteroni, A.

In this thesis we focus on PDEs in which some of the parameters are not known exactly but affected by a certain amount of uncertainty, and hence described in terms of random variables/random fields. This situation is quite common in the engineering practice. A common goal in this framework is to compute statistical indices, like mean or variance, for some quantities of interest related to the solution of the equation at hand (“uncertainty quantifi- cation”). The main challange in this task is represented by the fact that in many applications tens/hundreds of random variables may be necessary to obtain an accurate representation of the solution. The numerical schemes adopted to perform the uncertainty quantification should then be designed to reduce the degradation of their performance whenever the number of parameters increases, a phenomenon known as “curse of dimensionality”. Two methods that seem promising in this sense are the Stochastic Galerkin method and the Stochastic Collocation method. Such methods have therefore recently attracted the interest of the uncertainty quantification community, and have proved to be more effective than sampling methods like Monte Carlo, at least for problems with a moderate number of random parameters. We will compare in detail these methods, and then propose for both suitable generalizations that have shown to be optimal in terms of accuracy per cost for particular problems. We will also introduce the idea of Generalized Spectral Decomposition for the Stochastic Galerkin method, and explore its application in the context of non scalar equations, focusing on the case of the stationary Navier-Stokes equations. Finally, we will show two applications of the Stochastic Collocation method in the geological and hydraulic engineering. Keywords: Uncertainty Quantification, Stochastic Galerkin Method, Stochastic Collocation Method, best-M-terms approximation, Generalized Spectral Decomposition