We propose and analyze a general goal-oriented adaptive strategy for approximating quantities of interest (QoIs) associated with solutions to linear elliptic partial differential equations with random inputs. The QoIs are represented by bounded linear or continuously G\^ateaux differentiable nonlinear goal functionals, and the approximations are computed using the sparse grid stochastic collocation finite element method (SC-FEM). The proposed adaptive strategy relies on novel reliable a posteriori estimates of the errors in approximating QoIs. One of the key features of our error estimation approach is the introduction of a correction term into the approximation of QoIs in order to compensate for the lack of (global) Galerkin orthogonality in the SC-FEM setting. Computational results generated using the proposed adaptive algorithm are presented in the paper for representative elliptic problems with affine and nonaffine parametric coefficient dependence and for a range of linear and nonlinear goal functionals.

翻译：本文提出并分析了一种通用的面向目标的自适应策略，用于逼近具有随机输入的线性椭圆偏微分方程解相关的感兴趣量。这些感兴趣量由有界线性或连续G\^ateaux可微非线性目标泛函表示，其近似解通过稀疏网格随机配置有限元方法计算。所提出的自适应策略依赖于在逼近感兴趣量时误差的新型可靠后验估计。我们误差估计方法的一个关键特征是在感兴趣量的近似中引入了修正项，以弥补SC-FEM框架中缺乏（全局）Galerkin正交性的不足。本文展示了采用所提自适应算法针对具有仿射与非仿射参数依赖性的典型椭圆问题、以及一系列线性和非线性目标泛函所生成的计算结果。