Uncertainty Quantification in Coastal Aquifers Using the Multilevel Monte Carlo Method

Abstract

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.

Uncertainty quantification in coastal aquifers using the multilevel Monte Carlo method

“…By calculating the variance decay and computational cost at each level, we estimated the number of random samples required at each level. The estimates depend on the minimisation function used in the MLMC algorithm [3]. The difficulty observed is that the number of samples depends on the chosen point $$(t,\mathbf {x})$$.…”

“…To reduce the total computing and storage costs, we apply the standard MLMC method [12][13][14][15][16][17][18]. This method combines samples from different mesh levels in an efficient way (see our extended version [3]).…”

“…Here, it is important to see the decay of the error. Considerable variability is observed for 𝑡 ∈ [3,7] s and 𝑡 ∈ [8,25] s. Starting from 𝑡 ≈ 30 s, the variability between solutions decreases and stabilises. From these plots, we can try to estimate the decay rate.…”

“…Some auxiliary numerical tests and also QMC tests, which are not directly relevant to the MLMC method but are still useful for a better understanding of uncertainty propagation in the described physical phenomena, are provided in [3].…”

“…Since the length of this manuscript is very limited, we move the multi-level Monte Carlo description and some preliminary numerical uncertainty quantification tests to the arXiv version [3]. There, we have also considered some other quantities of interest.…”