论文标题
具有非线性目标功能的多级随机Galerkin FEM的目标适应性
Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals
论文作者
论文摘要
本文涉及与参数椭圆偏微分方程（PDES）解决方案相关的一定兴趣量的数值近似值。这项工作的关键新颖性在于关注不断可不同的非线性功能代表的利益量。我们考虑一类参数椭圆PDE，其中基础差异操作员对数量无限数量的不确定参数具有仿射依赖性。我们设计了一个面向目标的自适应算法，用于将解决方案的非线性功能近似为此类别的参数PDE。在算法中，使用多级随机Galerkin有限元方法（SGFEM）计算了针对原始问题和双重问题的参数解决方案的近似值，并且自适应改进过程由可靠的空间和参数误差指示器指导，这些误差指示可以识别出主要误差源。我们证明所提出的算法生成了多级SGFEM近似值，该算法的估计值估计目标功能中的误差将收敛至零。选择测试问题和非线性量的数值实验表明，提出的面向目标的自适应策略在基础多层近似空间的总体维度方面产生了最佳收敛速率（对于利率量的误差估计和参考误差）。
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 Gâteaux 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.
