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, Kuo, and Sloan (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.

翻译：我们考虑对受域不确定性影响的Poisson问题进行不确定的量化。对于随机域的随机域的随机度参数化,我们使用由Kaarnioja、Kuo和Sloan(SIAM J. Numer. Anal.,2020年)最近推出的模式(SIM J. Numer. Anal,2020年),在这个模型中,将无限数量的独立随机变量作为定期函数输入随机域。我们开发了 lattice 准Monte Carlo(QMC) 幼稚规则,用于计算受域不确定性影响的Poisson问题解决方案的预期值。这些QMC规则可以显示为定期设置允许的更高顺序的合并速率,而该周期的设置与问题的随机维度无关。此外,我们通过将输入随机字段的切换成一定数目的术语和使用有限元素将空间域分解,对该问题进行了完全的近差分析。文件最后用数字实验来证明理论错误的估计。