Stochastic Galerkin finite element methods (SGFEMs) provide a powerful alternative to traditional sampling techniques for estimating the propagation of uncertainty in forward UQ models. This is particularly true for the models described by parametric PDEs where the inputs and solutions are sufficiently smooth functions of parameters. Appropriate multilevel construction of the underlying approximation spaces and adaptivity are the keys to computationally efficient SGFEMs, particularly in the case when PDE inputs depend on infinitely many uncertain parameters.
We consider a class of elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. Our focus in this talk is on estimating the propagation of uncertainty to a quantity of interest that targets a specific (e.g., localized) feature of the solution to the PDE and can be represented using a functional of the solution. We present a goal-oriented adaptive algorithm where the error in the goal functional (e.g., the expectation of the quantity of interest) is estimated by the product of computable estimates of the energy errors in Galerkin approximations of the primal and dual solutions. We will address in detail the marking and refinement steps of the algorithm, which both draw the information from spatial and parametric error indicators computed for the primal and dual Galerkin approximations. We will present theoretical results on the convergence of the goal-oriented adaptive algorithm and discuss the results of numerical experiments in the single-level and multilevel SGFEM settings.
Joint work with Dirk Praetorius (TU Wien) and Michele Ruggeri (U of Strathclyde).