Sparse grid stochastic collocation representations of parametric uncertainty, in combination with finite element discretization in physical space, have emerged as an efficient alternative to Monte-Carlo strategies, particularly in the context of nonlinear PDE models or linear PDE problems that are nonlinear in the parameterization of the uncertainty.
This talk will address the a posteriori error estimation and the design of adaptive algorithms for stochastic collocation finite element methods employing individually tailored spatial discretizations across collocation points (multilevel stochastic collocation). We will show how a general framework for a posteriori error estimation in the FEM-based discretizations can be applied in these approximation settings for PDEs with parametric or random inputs. The introduced hierarchical error estimates are useful not only for reliable error control, they also provide practical error indicators for guiding the adaptive refinement process. We will demonstrate the effectivity and robustness of the proposed error estimation strategy. We will also discuss the performance of the developed adaptive algorithms for solving PDE problems with non-affine parameterization of random inputs and those with parameter-dependent local spatial features.
Joint work with David Silvester and Feng Xu (University of Manchester).