This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of interest represented by continuously G\^ateaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM) and the adaptive refinement process is guided by reliable spatial and parametric error indicators that identify the dominant sources of error. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest demonstrate that the proposed goal-oriented adaptive strategy yields optimal convergence rates (for both the error estimates and the reference errors in the quantities of interest) with respect to the overall dimension of the underlying multilevel approximations spaces.

翻译：本文关注与参数化椭圆型偏微分方程（PDEs）解相关联的感兴趣量的数值逼近。本工作的关键创新在于其聚焦于由连续Gâteaux可微非线性泛函表示的感兴趣量。我们考虑一类参数化椭圆型PDEs，其中底层微分算子对可数无穷多个不确定参数具有仿射依赖性。我们设计了一种面向目标的自适应算法，用于逼近此类参数化PDEs解的非线性泛函。该算法中，原始问题和对偶问题的参数化解的逼近采用多层随机伽辽金有限元法（SGFEM）计算，且自适应细化过程由可靠的空间和参数误差指示器引导，以识别主导误差源。我们证明了所提算法生成的多层SGFEM逼近能使目标泛函的误差估计收敛到零。针对选定测试问题和非线性感兴趣量的数值实验表明，所提出的面向目标自适应策略在底层多层逼近空间的总维度下，能实现（误差估计和感兴趣量参考误差的）最优收敛速率。