This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain alongside finite element approximations in the spatial domain, and adaptivity is driven by recently proposed parametric and spatial a posteriori error indicators. We prove that for a general diffusion coefficient with finite-dimensional parametrization, the algorithm drives the underlying error estimates to zero. Thus, our analysis covers problems with affine and nonaffine parametric coefficient dependence.

翻译：本文针对具有参数系数的稳态扩散方程，对自适应随机配点算法的收敛性进行了分析。该算法在参数域采用稀疏网格配点法，同时在空间域采用有限元近似，其自适应性由最近提出的参数和空间后验误差指示器驱动。我们证明：对于具有有限维参数化的广义扩散系数，该算法能使底层误差估计趋近于零。因此，我们的分析涵盖了仿射及非仿射参数系数依赖性的问题。