## Applied Math Colloquium: Dr. Michal Béreš

#### VSB--Technical University of Ostrava (Czechia)

Friday, September 30, 2022 · 2 - 3:15 PM

Title: Reduced basis method for stochastic Galerkin system of equations

FE basis has orders of magnitude smaller size and the resulting reduced system is much easier to solve. In this talk, we discuss a Monte Carlo based approach for the construction of RB. We examine measures of error for stopping criteria, adaptive polynomial basis selection, and efficient solutions of reduced systems.

Abstract: The stochastic Galerkin method solves a partial differential equation with parameters or uncertainties in input data. It discretizes both the physical and the stochastic space. The resulting basis is assumed as a product of the two bases: finite elements (FE) for the discretization of physical space, and polynomials for the discretization of stochastic space. The basis is large and therefore resulting system of equations is extremely large and needs specialized tools for a feasible solution.

A very efficient tool for the solution is the reduced basis (RB) approach. The RB approach aims at a compression of the FE basis. The compressed FE basis has orders of magnitude smaller size and the resulting reduced system is much easier to solve. In this talk, we discuss a Monte Carlo based approach for the construction of RB. We examine measures of error for stopping criteria, adaptive polynomial basis selection, and efficient solutions of reduced systems.