Adaptive algorithms with a-posteriori Quasi-Monte Carlo estimation for parametric elliptic PDEs

Abstract

We introduce novel adaptive methods to approximate Partial Differential Equations (PDEs) with uncertain parametric inputs. A typical problem in Uncertainty Quantification is the approximation of the expected values of Quantities of Interest of the solution, which requires the efficient numerical approximation of high-dimensional integrals. We perform this task by a class of deterministic Quasi-Monte Carlo integration rules derived from Polynomial lattices, that allows to control a-posteriori the integration error without querying the governing PDE and does not incur in the curse of dimensionality. Based on an abstract formulation of Adaptive Finite Element methods for deterministic problems, we infer convergence of the combined adaptive algorithms in the parameter and physical space. We propose a selection of examples of PDEs admissible for these algorithms. Finally, we present numerical evidence of convergence.