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

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