On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion

Convergence of an adaptive collocation method for the parametric stationary diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with a hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the now classical approach proposed in [T. Gerstner and M. Griebel, Computing, 71 (2003), pp. 65-87]) are explored.