This talk concerns the design and analysis of adaptive FEM-based solution strategies for partial differential equations (PDEs) with uncertain or parameter-dependent inputs. We present two conceptually different strategies: one is projection-based (stochastic Galerkin FEM) and the other is sampling-based (stochastic collocation FEM). These strategies have emerged and become popular as effective alternatives to Monte-Carlo sampling in the context of (forward) uncertainty quantification. Both stochastic Galerkin and stochastic collocation approximations are typically represented as finite (sparse) expansions in terms of a parametric polynomial basis with spatial coefficients residing in finite element spaces. The focus of the talk is on multilevel approaches where different spatial coefficients may reside in different finite element spaces and, therefore, the underlying spatial approximations are allowed to be refined independently from each other.

We start with a more familiar setting of projection-based methods, where exploiting the Galerkin orthogonality property and polynomial approximations in terms of an orthonormal basis facilitates the design and analysis of adaptive algorithms. We discuss a posteriori error estimation as well as the convergence and rate optimality properties of the generated adaptive multilevel Galerkin approximations for PDE problems with affine-parametric coefficients. We then show how these ideas of error estimation and multilevel adaptivity can be applied in a non-Galerkin setting of stochastic collocation FEM, in particular, for PDE problems with non-affine parameterization of random inputs and for problems with parameter-dependent local spatial features.

The talk is based on a series of joint papers with Dirk Praetorius (TU Vienna), Leonardo Rocchi (Birmingham), Michele Ruggeri (University of Strathclyde, Glasgow), David Silvester (Manchester), and Feng Xu (Manchester).