We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case. The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements, and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction. The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

翻译：本文研究一类密度驱动流动问题，重点关注滨海含水层盐渍化问题。以亨氏盐水入侵问题为测试案例，考虑孔隙度、渗透率和补给参数存在不确定性的情形。产生不确定性的原因包括认知不足、测量不精确以及无法在每个空间或时间位置测量参数。该问题具有非线性和时变性特征，其解为随时间变化且具有不确定性的盐质量分数。采用随机场模型描述孔隙度、渗透率、补给量及质量分数的不确定性。本研究探讨了经典多层级蒙特卡洛方法（MLMC）在该类问题中的适用性。MLMC方法能有效降低总计算与存储成本，通过在不同空间和时间网格上运行多种情景，进而估算质量分数的均值。并行化过程同时在物理空间和随机空间展开。对于每个确定性情景求解，采用黑箱方式运行并行多重网格求解器ug4，并以拟蒙特卡洛方法的解作为参考解。