Reduced Basis Methods for Parameterized Partial Differential Equations with Stochastic Influences Using the Karhunen-Loeve Expansion

We consider parametric partial differential equations (PPDEs) with stochastic influences, e. g., in terms of random coefficients. Using standard discretizations such as finite elements, this often amounts to high-dimensional problems. In a many-query context, the PPDE has to be solved for various instances of the deterministic parameter as well as the stochastic influences. To decrease computational complexity, we derive a reduced basis method (RBM), where the uncertainty in the coefficients is modeled using Karhunen-Lo` eve (KL) expansions. We restrict ourselves to linear coercive problems with linear and quadratic output functionals. A new a posteriori error analysis is presented that generalizes and extends some of the results by Boyaval et al. [Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 3187-3206]. The additional KL-truncation error is analyzed for the state, output functionals, and also for statistical outputs such as mean and variance. Error estimates for quadratic outputs are obtained using additional nonstandard dual problems. Numerical experiments for a two-dimensional porous medium demonstrate the effectivity of this approach.