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

Dmitry Logashenko, Alexander Litvinenko, Raul Tempone, Ekaterina Vasilyeva, Gabriel Wittum

TL;DR

This paper investigates applying multilevel Monte Carlo (MLMC) to the Henry density-driven salinity intrusion problem under uncertainty in porosity, permeability, and recharge modeled as random fields. It couples MLMC with a parallel UG4 multigrid solver on a hierarchy of space–time meshes to estimate the mean and variance of the salt mass fraction $c$ and related QoIs, achieving substantial computational savings over standard Monte Carlo. Key findings include QoI-dependent convergence rates and significant cost reductions (up to several orders of magnitude) when using MLMC, enabling efficient uncertainty quantification for coastal aquifer management. The work demonstrates practical viability and provides a framework for extending MLMC to more complex, multiscale porosity/permeability models and data-informed parameter calibration.

Abstract

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.

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

TL;DR

This paper investigates applying multilevel Monte Carlo (MLMC) to the Henry density-driven salinity intrusion problem under uncertainty in porosity, permeability, and recharge modeled as random fields. It couples MLMC with a parallel UG4 multigrid solver on a hierarchy of space–time meshes to estimate the mean and variance of the salt mass fraction and related QoIs, achieving substantial computational savings over standard Monte Carlo. Key findings include QoI-dependent convergence rates and significant cost reductions (up to several orders of magnitude) when using MLMC, enabling efficient uncertainty quantification for coastal aquifer management. The work demonstrates practical viability and provides a framework for extending MLMC to more complex, multiscale porosity/permeability models and data-informed parameter calibration.

Abstract

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.
Paper Structure (10 sections, 1 theorem, 41 equations, 11 figures, 3 tables)

This paper contains 10 sections, 1 theorem, 41 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Henry Problem with Uncertain Parameters
  3. Problem setting
  4. Modeling porosity, permeability, and recharge
  5. Numerical methods for the deterministic problem
  6. Multilevel Monte Carlo
  7. Numerical Experiments
  8. Computation via QMC
  9. Numerical tests for the MLMC method
  10. Conclusion

Key Result

Theorem 4

Consider a fixed $t=t^*$. Suppose positive constants $\alpha,\beta,\gamma > 0$ exist such that $\alpha \geq \frac{1}{2} \hbox{min}(\beta, \gamma \hat{d})$, and Then, for any accuracy $\varepsilon < e^{-1}$, a constant $c_4>0$ and a sequence of realizations $\{m_{\ell}\}_{\ell=0}^L$ exist, such that $\text{MSE} < \varepsilon^2$, where $\text{MSE}$ is defined in eq:MSE, and the computational cost i

Figures (11)

  • Figure 1: (left) Computational domain $\mathcal{D}:=[0,2]\times [-1,0]$; (right) the mass fraction $c\in [0,1]$ and the streamlines of the velocity field ${\mathbf{q}}$ for the undisturbed Henry problem at $t = 6016$$\mathrm{s}$. The dark red colour corresponds to $c=1$ and the blue colour to $c=0$.
  • Figure 2: Positions of the 15 pre-selected points with small subdomains around them. The size of the subdomain around point $(x_i,y_i)$ is $[x_i-0.1,x_i+0.1]\times [y_i-0.1,y_i+0.1]$.
  • Figure 3: (left and center) A realisation of porosity $\phi(\bm{\xi}^*) \in [0.18, 0.59]$ and permeability $K \in [1.77e-10, 4.35e-9]$. (right) Corresponding mass fraction $c(T,\mathbf{x},\phi(\bm{\xi}^*)) \in [0,0.35]$ with isolines $\{\mathbf{x}:\; |c(T,\phi(\bm{\xi}^*)) - \overline{c}(T)|=0.1\cdot i\}$, $i=1,2,3$, $t=T=6016$ s.
  • Figure 4: (left) Mean value $\overline{c} \in [0,1]$ and (right) variance ${\mathrm{Var}\mspace{-2mu}\left[c\right]} \in [0.0,0.04]$ of the mass fraction, with contour lines $\{\mathbf{x}:\; {\mathrm{Var}\mspace{-2mu}\left[ c \right]}=0.01\cdot i\}$, $i=1..3$, $t=T=6016$ s.
  • Figure 5: (left) Mean values $\mathbb{E}\left[{c}(t,x_9,y_9)\right ]$ and (right) variances ${\mathrm{Var}\mspace{-2mu}\left[c\right]}(t,x_9,y_9)$ of the mass fraction computed on levels 0,1,2,3 vs. time $t$.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8