We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand-side functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces.

中文翻译：

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自适应多级随机Galerkin FEM的收敛性和速率最优性

我们分析了一种自适应算法，用于二维物理域中参数椭圆偏微分方程的数值解，其系数和右手边函数取决于无限多（随机）参数。该算法生成多级随机 Galerkin 近似；这些用稀疏广义多项式混沌展开来表示，系数驻留在与不同局部细化网格相关的有限元空间中。自适应性由两级后验误差估计器驱动，并在空间和参数误差指标的联合集上采用 Dörfler 类型标记。我们表明，在适当的饱和假设下，