We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.

我们考虑受领域不确定性影响的泊松问题的不确定性量化。对于随机域的随机参数化，我们使用 Kaarnioja 等人最近引入的模型。（SIAM J. Numer. Anal.，2020），其中可数无限个独立随机变量作为周期函数进入随机场。我们开发了格准蒙特卡罗 (QMC) 体积规则，用于计算受域不确定性影响的泊松问题解的期望值。这些 QMC 规则可以显示出周期性设置允许的更高阶体积收敛速度，与问题的随机维度无关。此外，我们通过考虑将输入随机场截断为有限项并使用有限元离散空间域而产生的近似误差，对该问题进行了完整的误差分析。本文最后通过数值实验证明了理论误差估计。