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：针对受域不确定性影响的泊松问题，本文开展不确定性量化研究。在随机域的随机参数化过程中，采用Kaarnioja、Kuo与Sloan（SIAM J. Numer. Anal., 2020）近期提出的模型，该模型将可数无穷多个独立随机变量以周期函数形式引入随机场。我们开发了用于计算受域不确定性影响的泊松问题解期望值的格点拟蒙特卡洛求积规则。这些QMC规则在周期设定下可呈现高阶求积收敛速率，且该速率独立于问题的随机维度。此外，我们通过考虑输入随机场截断至有限项导致的逼近误差以及使用有限元离散空间域带来的误差，给出了问题的完整误差分析。本文最后通过数值实验验证了理论误差估计结果。