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

TL;DR

This paper proves the convergence of an adaptive stochastic collocation method for elliptic PDEs with affine diffusion coefficients, using residual-based error estimation and extending previous strategies from stochastic Galerkin methods.

Contribution

It introduces a convergence proof for an adaptive collocation algorithm for parametric elliptic PDEs, adapting techniques from stochastic Galerkin methods and exploring extensions to other collocation variants.

Findings

Convergence of the adaptive collocation method is established.

The residual-based a posteriori error estimator is effective for adaptivity.

Extensions to other adaptive collocation variants are discussed.

Abstract

Convergence of an adaptive collocation method for the stationary parametric 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 an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) by T. Gerstner and M. Griebel) is explored.