We investigated the applicability and efficiency of the MLMC approach for the Henry-like problem with uncertain porosity, permeability, and recharge. These uncertain parameters were modeled by random fields with three independent random variables. The numerical solution for each random realization was obtained using the well-known ug4 parallel multigrid solver. The number of required random samples on each level was estimated by computing the decay of the variances and computational costs for each level. We also computed the expected value and variance of the mass fraction in the whole domain, the evolution of the pdfs, the solutions at a few preselected points $(t,\bx)$, and the time evolution of the freshwater integral value. We have found that some QoIs require only 2-3 of the coarsest mesh levels, and samples from finer meshes would not significantly improve the result. Note that a different type of porosity may lead to a different conclusion. The results show that the MLMC method is faster than the QMC method at the finest mesh. Thus, sampling at different mesh levels makes sense and helps to reduce the overall computational cost.

1. Center for Uncertainty
Quantification
Salinization of coastal aquifers under uncertainties
Alexander Litvinenko1
, Dmitry Logashenko2
, Raul Tempone1,2
, Ekaterina Vasilyeva2
, Gabriel Wittum2
1
RWTH Aachen, Germany, 2
KAUST, Saudi Arabia
litvinenko@uq.rwth-aachen.de
Center for Uncertainty
Quantification
Center for Uncertainty
Quantification
Abstract
Problem: Henry saltwater intrusion (nonlinear and time-dependent)
Input uncertainty: porosity, permeability, and recharge (model by random fields)
Solution: the salt mass fraction (uncertain and time-dependent)
Method: Multi Level Monte Carlo (MLMC) method
Deterministic solver: parallel multigrid solver ug4
Questions:
1. How long can water wells be in
use?
2. Where is the largest uncertainty?
3. What are the exceedance proba-
bilities?
4. What is the mean scenario (and
its variations)?
5. What are the extreme scenarios?
6. How do the uncertainties change
with time?
Figure 1: Henry problem, taken from https://www.mdpi.com/2073-4441/10/2/230
1. Henry problem settings
The mass conservation laws for the entire liquid phase and salt yield the following equations
∂t(ϕρ) + ∇ · (ρq) = 0,
∂t(ϕρc) + ∇ · (ρcq − ρD∇c) = 0,
where ϕ(x, ξ) is porosity, x ∈ D, is determined by a set of RVs ξ = (ξ1, . . . , ξM, ...).
c(t, x) mass fraction of the salt, ρ = ρ(c) density of the liquid phase, and D(t, x) molecular
diffusion tensor.
For q(t, x) velocity, we assume Darcy’s law:
q = −
K
µ
(∇p − ρg),
where p = p(t, x) is the hydrostatic pressure, K permeability, µ = µ(c) viscosity of the liquid
phase, and g gravity. To compute: c and p.
Comput. domain: D × [0, T]. We set ρ(c) = ρ0 + (ρ1 − ρ0)c, and D = ϕDI,
I.C.: c|t=0 = 0, B.C.: c|x=2 = 1, p|x=2 = −ρ1gy. c|x=0 = 0, ρq · ex|x=0 = q̂in.
We model the uncertain ϕ using a random field and assume: K = KI, K = K(ϕ).
We use a Kozeny–Carman-like dependence: K(ϕ) = κKC ·
ϕ3
1 − ϕ2
.
Methods: Newton method, BiCGStab, preconditioned with the geometric multigrid method
(V-cycle), ILUβ-smoothers and Gaussian elimination.
2. Solution of the Henry problem
q̂in = 6.6 · 10−2
kg/s
c = 0 c = 1
p = −ρ1gy
0
−1 m
2 m
y
x
D := [0, 2] × [−1, 0]; a realization of c(t, x); ϕ(ξ∗
) ∈ [0.18, 0.59]; mass fraction
c(T, x, ϕ(ξ∗
)) ∈ [0, 0.35] with isolines {x : |c(T, ϕ(ξ∗
)) − c(T)| = 0.1 · i}, i = 1, 2, 3,
ξ∗
= (0.5898, 0.7257, 0.9616); variance Var[c] ∈ [0.0, 0.04], t = 6016 s.
QoIs: c in the whole domain, c at a point, or an integral value (the freshwater integral):
QFW (t, ω) :=
Z
x∈D
I(c(t, x, ω) ≤ 0.012178)dx,
2.1 Multi Level Monte Carlo (MLMC) method
Hierarchy D0, D1, . . . , DL, Temporal grid hierarchy T0, T1, . . . , TL; n0 = 512, nℓ ≈ n0 · 2dℓ
, d = 2,
τℓ+1 = 1
2τℓ, rℓ+1 = 2rℓ and rℓ = r02ℓ
.
Computation complexity is sℓ = O(nℓrℓ), sℓ = O
1
h0τ0
2(d+1)ℓγ
.
MLMC approximates E [gL] ≈ E [g] using the following telescopic sum:
E [gL] ≈ m−1
0
m0
X
i=1
g
(0,i)
0 +
L
X
ℓ=1
m−1
ℓ
mℓ
X
i=1
(g
(ℓ,i)
ℓ − g
(ℓ,i)
ℓ−1 )
!
.
Minimize F(m0, . . . , mL) :=
PL
ℓ=0 mℓsℓ + µ2 Vℓ
mℓ
, obtain mℓ = ε−2
q
Vℓ
sℓ
PL
i=0
√
Visi.
100 realizations of QFW (t); Evolution of the pdf of c(t, x), t = {3τ, . . . , 48τ}; pdf of the earliest
time point when c(t, x) 0.9, x = (1.85, −0.95); mean values E [c(t, x9, y9)]; variances
Var[c](t, x9, y9) on levels 0,1,2,3.
E [gℓ − gℓ−1] (t, x9, y9); V [gℓ − gℓ−1] (t, x9, y9), ℓ = 1, 2, 3, QoI is the integral value over D9 ; 100
realisations of g1 − g0 (left), g2 − g1 (center), g3 − g2 (right), QoI gℓ is the integral value
Qs(t, ω) :=
R
x∈D9
c(t, x, ω)ρ(t, x, ω)dx computed over a subdomain around 9th point, t ∈ [τ, 48τ].
Level ℓ nℓ, ( nℓ
nℓ−1
) rℓ, ( rℓ
rℓ−1
) τℓ = 6016/rℓ
Computing times (sℓ), ( sℓ
sℓ−1
)
average min. max.
0 153 94 64 0.6 0.5 0.7
1 2145 (14) 376 (4) 16 7.1 (14) 6.9 8.7
2 33153 (15.5) 1504 (4) 4 252.9 (36) 246.2 266.2
3 525825 (15.9) 6016 (4) 1 11109.8 (44) 9858.4 15506.9
#ndofs nℓ, number of time steps rℓ, time step τℓ; average, minimal, and maximal computing
times on each level ℓ.
ε2
0.1 1 10 100
total cost of MC, SMC 9.5e + 6 9.5e + 5 9.5e + 4 ‘9.5e + 3
total cost of MLMC, S 4.25e + 4 4.25e + 3 4.25e + 2 4.25e + 1
{m0, m1, m2, m3} {7927, 946, 57, 2} {793, 95, 6, 0} {79, 9, 1, 0} {8, 1, 0, 0}
Comparison of MC and MLMC and the number of samples on each level vs. ε2
.
ε2
m0 m1 m2 m3
1 73 8 1 0
0.5 290 32 3 0
0.1 7258 811 68 1
0.05 29031 3245 274 5
MLMC: the number of samples mℓ on level ℓ
Weak and strong convergences at (t, x, y) = (14, 1.60, −0.95). Decay of absolute and rela-
tive errors between the mean values computed on a fine mesh via QMC and via MLMC at
(t, x, y) = (12, 1.60, −0.95).
Acknowledgements: KAUST HPC and the Alexander von Humboldt foundation.
References
1. A. Litvinenko, D. Logashenko, R. Tempone, E. Vasilyeva, G. Wittum, Uncertainty quantification in coastal aquifers using the multilevel Monte Carlo method,
arXiv:2302.07804, 2023
2. A. Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Solution of the 3D density-driven groundwater flow problem with uncertain porosity and perme-
ability, GEM-International Journal on Geomathematics 11, 1-29, 2020
3. A. Litvinenko, A.C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen, R. Tempone, Computation of electromagnetic fields scattered from objects with uncertain
shapes using multilevel Monte Carlo method, IEEE Journal on Multiscale and Multiphysics Computational Techniques 4, 37-50, 2019
4. A. Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Propagation of Uncertainties in Density-Driven Flow, In: Bungartz, HJ., Garcke, J., Pflüger,
D. (eds) Sparse Grids and Applications - Munich 2018. LNCSE, vol. 144, pp 121-126, Springer, 2018