An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

Abstract

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.

Semi-conservative high order scheme with numerical entropy indicator for intrusive formulations of hyperbolic systems

“…In the context of intrusive stochastic Galerkin methods, the functional dependence on the stochastic input is described a priori by a polynomial chaos (gPC) expansion and a Galerkin projection is used to obtain deterministic evolution equations for the coefficients in the series, called gPC modes. This approach has been successfully applied to diffusion [1,2,3], kinetic [4,5,6,7,8,9] and Hamilton-Jacobi equations [10,11]. In general, results for hyperbolic systems are not available [12,13], since desired properties like hyperbolicity and the existence of entropies are not transferred to the intrusive formulation.…”

Semi-Conservative High Order Scheme with Numerical Entropy Indicator for Intrusive Formulations of Hyperbolic Systems

“…In the context of intrusive stochastic Galerkin methods, the functional dependence on the stochastic input is described a priori by a polynomial chaos (gPC) expansion and a Galerkin projection is used to obtain deterministic evolution equations for the coefficients in the series, called gPC modes. This approach has been successfully applied to diffusion [1,2,3], kinetic [4,5,6,7,8,9] and Hamilton-Jacobi equations [10,11]. In general, results for hyperbolic systems are not available [12,13], since desired properties like hyperbolicity and the existence of entropies are not transferred to the intrusive formulation.…”