There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Le (Comput. Math. Appl., 2024, and SIAM J. Numer. Anal., 2024) as well as Harbrecht, Schmidlin, and Schwab (Math. Models Methods Appl. Sci., 2024) analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.

翻译：近年来，针对具有Gevrey正则输入的参数化偏微分方程（PDE）的不确定性量化研究兴趣激增。Gevrey函数类包含无穷光滑的函数，其高阶偏导数满足一定的增长条件，但一般而言这些函数并非解析函数。Chernov与Le（Comput. Math. Appl., 2024 及 SIAM J. Numer. Anal., 2024）以及Harbrecht、Schmidlin与Schwab（Math. Models Methods Appl. Sci., 2024）的最新研究分析了输入随机场相对于不确定参数一致有界的设定。在本文中，我们放宽了这一假设，允许参数依赖的界。参数化输入被建模为广义高斯随机变量，我们分析了应用拟蒙特卡洛（QMC）积分（使用随机平移的秩-1格点规则）来评估PDE响应统计量的方法。除了QMC误差分析外，我们还考虑了此设定下的维度截断误差和有限元误差。