标题： 基于多级蒙特卡罗方法的亨利问题不确定性量化 显示英文标题

标题： Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method

摘要： 我们研究了众所周知的多级蒙特卡洛（MLMC）方法在密度驱动流问题类中的适用性，特别是沿海含水层盐化问题。作为测试案例，我们求解了不确定的亨利盐水入侵问题。未知的孔隙度、渗透率和补给参数通过使用随机场进行建模。经典的确定性亨利问题是非线性的且随时间变化的，计算可能需要数小时。不确定设置需要求解确定性问题的多个实现，总计算成本急剧增加。而不是计算数百个随机实现，通常只计算均值和方差。标准方法如蒙特卡洛或代理方法是一个好的选择，但它们在相同且通常非常精细的网格上计算所有随机实现。它们也不平衡随机误差和离散化误差。这些事实促使我们应用MLMC方法。我们证明了通过在多级空间和时间网格上求解亨利问题，MLMC方法降低了整体计算和存储成本。为了进一步降低计算成本，在物理和随机空间中进行了并行化。为了解决每个确定性场景，我们以黑盒方式运行并行多重网格求解器ug4。 显示英文摘要

摘要： We investigate the applicability of the well-known multilevel Monte Carlo (MLMC) method to the class of density-driven flow problems, in particular the problem of salinisation of coastal aquifers. As a test case, we solve the uncertain Henry saltwater intrusion problem. Unknown porosity, permeability and recharge parameters are modelled by using random fields. The classical deterministic Henry problem is non-linear and time-dependent, and can easily take several hours of computing time. Uncertain settings require the solution of multiple realisations of the deterministic problem, and the total computational cost increases drastically. Instead of computing of hundreds random realisations, typically the mean value and the variance are computed. The standard methods such as the Monte Carlo or surrogate-based methods is a good choice, but they compute all stochastic realisations on the same, often, very fine mesh. They also do not balance the stochastic and discretisation errors. These facts motivated us to apply the MLMC method. We demonstrate that by solving the Henry problem on multi-level spatial and temporal meshes, the MLMC method reduces the overall computational and storage costs. To reduce the computing cost further, parallelization is performed in both physical and stochastic spaces. To solve each deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.