Scalable approaches for uncertainty quantification are necessary for characterizing prediction confidence in large‐scale subsurface flow simulations with uncertain permeability. To this end we explore a multilevel Monte Carlo approach for estimating posterior moments of a particular quantity of interest, where we employ an element‐agglomerated algebraic multigrid (AMG) technique to generate the hierarchy of coarse spaces with guaranteed approximation properties for both the generation of spatially correlated random fields and the forward simulation of Darcy's law to model subsurface flow. In both these components (sampling and forward solves), we exploit solvers that rely on state‐of‐the‐art scalable AMG. To showcase the applicability of this approach, numerical tests are performed on two 3D examples—a unit cube and an egg‐shaped domain with an irregular boundary—where the scalability of each simulation as well as the scalability of the overall algorithm are demonstrated.

中文翻译：

---

在多级可伸缩框架中估算期望的后验数量

在表征具有不确定渗透率的大规模地下流动模拟中，预测表征的可信度是必要的，可扩展的不确定性量化方法是必要的。为此，我们探索了一种多级蒙特卡洛方法来估计特定数量的后矩，在此方法中，我们采用了元素凝聚的代数多重网格（AMG）技术来生成粗糙空间的层次结构，同时保证了近似空间的生成。空间相关的随机场和达西定律的正演模拟来模拟地下流动。在这两个组件（采样和正解）中，我们利用依赖于最新的可伸缩AMG的求解器。为了展示这种方法的适用性，