Adaptive stochastic Galerkin FEM

Abstract

A framework for residual-based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic PDEs with random coefficients is presented. A parametric, deterministic elliptic boundary value problem on an infinite-dimensional parameter space is discretized by means of a Galerkin projection onto finite generalized polynomial chaos (gpc) expansions, and by discretizing each gpc coefficient by a FEM in the physical domain.An anisotropic residual-based a posteriori error estimator is developed. It contains bounds for both contributions to the overall error: the error due to gpc discretization and the error due to Finite Element discretization of the gpc coefficients in the expansion. The reliability of the residual estimator is established.Based on the explicit form of the residual estimator, an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic Galerkin discretization, and, embedded in the gpc adaptation loop, also the Finite Element mesh refinement of the gpc coefficients in the physical domain. Asynchronous mesh adaptation for different gpc coefficients is permitted, subject to a minimal compatibility requirement on meshes used for different gpc coefficients.Details on the implementation in the software environment FEniCS are presented; it is generic, and is based on available stiffness and mass matrices of a FEM for the deterministic, nonparametric nominal problem.Preconditioning of the resulting matrix equation and iterative solution are discussed. Numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented. They indicate substantial savings in total computational complexity due to FE mesh coarsening in high gpc coefficients.

Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

“…We consider only the SG approach in this paper, since its properties are advantageous in the case of linear dynamical systems, see [26]. However, we admit that in several cases the stochastic collocation method combines providing good results with flexibility in allowing the use of different simulation tools.…”

“…We consider the frequency interval ω ∈ [10 −2 , 10 6 ]. The maxima (26) are approximated on a fine grid within this frequency window. Table 3 shows the maximum differences of the approximations to the reference solution.…”

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

“…We also note that it is possible to implement SGFEM in a partially intrusive manner with iterative solution techniques, see e.g. [26]. Clearly the whether these requirements actually are intrusive depends on whether the user has models that can provide the required derivative information [14] or matrix-vector actions [26], respectively.…”

“…[26]. Clearly the whether these requirements actually are intrusive depends on whether the user has models that can provide the required derivative information [14] or matrix-vector actions [26], respectively. In the past, easily computing derivatives of complex forward models either by the tangent linear (forward mode) or adjoint (reverse mode) of differentiation required the use of automatic differentiation tools, e.g.…”

Adaptive algorithm for stochastic Galerkin method

“…They are based on the idea that the error estimates with respect to spatial and stochastic approximation spaces can be separated in some sense [4], [9], [13], [12]. Eigel et al in [12], [13] describe and prove residual based a posteriori error estimates derived from the adequate approaches for deterministic problems. A marking strategy for both physical and stochastic degrees of freedom is based on the Dorfler property [12].…”