Abstract

In this talk we discuss a weighted approach for the reduction of parametrized partial differential equations with random input, focusing in particular on weighted approaches based on reduced basis (wRB) [Chen et al., SIAM Journal on Numerical Analysis (2013); D. Torlo et al., submitted (2017)] and proper orthogonal decomposition (wPOD) [L. Venturi et al., submitted (2017)]. We will first present the wPOD approach. A first topic of discussion is related to the choice of samples and respective weights according to a quadrature formula. Moreover, to reduce the computational effort in the offline stage of wPOD, we will employ Smolyak quadrature rules. We will then introduce the wRB method for advection diffusion problems with dominant convection with random input parameters. The issue of the stabilisation of the resulting reduced order model will be discussed in detail. This work is in collaboration with Francesco Ballarin (SISSA, Trieste), Davide Torlo (University of Zurich), and Luca Venturi (Courant Institute for Mathematical Sciences, NYC )