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:我们考虑的是一组由密度驱动的流量问题。我们特别关心沿海含水层盐碱化问题。我们认为亨利盐水入侵问题是一个测试案例,其孔隙性、渗透性和补给参数不确定。不确定性的存在的原因是缺乏知识、测量不准确和无法测量每个空间或时间位置的参数。这是一个非线性和时间性的问题。解决办法是盐质量部分,这是不确定的,时间上的变化。孔隙性、渗透性、补给和质量部分的不确定性是用随机字段来模拟的。这项工作调查众所周知的多层次蒙特卡洛(MLMC)方法对此类问题的适用性。MLMC方法可以降低计算和储存的总成本。此外,MLMC方法在不同的空间和时间中运行多种情景,然后估算质量部分的平均值。在物理空间和沙变空间进行平行化。为了解决每一种确定性情景,我们在黑箱中运行平行的多格求解(MLMLMC)方法。我们从一个黑箱式的解决方案中运行了平行的多格解4。