We investigated the applicability and efficiency of the MLMC approach to the Henry-like problem with uncertain porosity, permeability and recharge. These uncertain parameters were modelled by random fields with three independent random variables. Permeability is a function of porosity. Both functions are time-dependent, have multi-scale behaviour and are defined for two layers. The numerical solution for each random realisation was obtained using the well-known ug4 parallel multigrid solver. The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. The MLMC method was used to compute the expected value and variance of several QoIs, such as the solution at a few preselected points $(t,\bx)$, the solution integrated over a small subdomain, and the time evolution of the freshwater integral. We have found that some QoIs require only 2-3 mesh levels and samples from finer meshes would not significantly improve the result. Other QoIs require more grid levels.

1. Uncertainty quantification in the coastal aquifers using Multi Level Monte Carlo
Alexander Litvinenko,
joint work with D. Logashenko, R. Tempone, E. Vasilyeva, G. Wittum
RWTH Aachen and KAUST

2. Overview
Problem: Henry saltwater intrusion (nonlinear and
time-dependent, describes a two-phase subsurface flow)
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
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3. Henry problem
1. How long can wells be used?
2. Where is the largest uncertainty?
3. Freshwater exceedance probability?
4. What is the mean scenario and its variations?
5. What are the extreme scenarios?
6. How do the uncertainties change over time?
taken from https://www.mdpi.com/2073-4441/10/2/230
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4. 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,
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.
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5. Henry problem settings
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 φ by a random field and assume:
K = KI, K = K(φ), and Kozeny–Carman-like dependence
K(φ) = κ ·
φ3
1 − φ2
, (1)
where κ is a scalar.
Discretisation: vertex-centered finite volume, implicit Euler
Methods: Newton method, BiCGStab preconditioned with the
geometric multigrid method (V-cycle), ILUβ-smoothers.
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6. Porosity and 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) with streamlines of
the velocity field q; porosity φ(ξ∗
) ∈ (0.18,0.59); permeability
K ∈ (1.8e − 10,4.4e − 9)
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7. Expectation and variance of the mass fraction c
E[c] ∈ [0,0.35); Var[c] ∈ [0.0,0.04)
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8. What can we compute?
QoIs: c in the whole domain, c at a point, or integral values (the
freshwater/saltwater integrals):
QFW(t,ω) :=
Z
x∈D
I(c(t,x,ω) ≤ 0.012178)dx, (2)
Qs(t,ω) :=
Z
x∈D
c(t,x,ω)ρ(t,x,ω)dx, (3)
Q9(t,ω) :=
Z
x∈∆9
c(t,x,ω)ρ(t,x,ω)dx, (4)
where ∆9 := [x9 − 0.1,x9 + 0.1] × [y9 − 0.1,y9 + 0.1].
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9. Multi Level Monte Carlo (MLMC) method
Spatial and temporal grid hierarchies
D0,D1,...,DL,
T0,T1,...,TL;
n0 = 512, n` ≈ n0 · 16`,
τ`+1 = 1
4τ`, r`+1 = 4r` and r` = r04`.
Approx. error: kc − ch,τk2 = O(h + τ) = O(n−1/2 + r−1)
Computation complexity on level ` is
s` = O(n`r`) = O(43`γ
n0 · r0)
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10. Multi Level Monte Carlo (MLMC) method
MLMC approximates E[g] ≈ E[gL]
using the following telescopic sum:
E[gL] = E[g0] +
L
X
`=1
E[g` − g`−1] ≈
≈ m−1
0
m0
X
i=1
g
(0,i)
0 +
L
X
`=1






m−1
`
m
X̀
i=1
(g
(`,i)
` − g
(`,i)
`−1 )






.
Let Y` := m−1
`
Pm`
i=1(g
(`,i)
` − g
(`,i)
`−1 ), where g−1 ≡ 0, so that
E[Y`] :=







E[g0], ` = 0
E[g` − g`−1], ` > 0
. (5)
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11. MLMC notation
Denote by Y:=
PL
`=0 Y` the multilevel estimator of E[g] based
on L + 1 levels and m` independent samples on level `, where
` = 0,...,L.
Denote V0 := V [g0] and for ` ≥ 1, and V` := V [g` − g`−1], ` ≥ 1.
The standard theory states:
E[Y] = E[gL], V [Y] =
PL
`=0 m−1
` V`.
The cost of the multilevel estimator Y is
S :=
PL
`=0 m`s`.
See details in Giles’18 or Teckentrup’s PhD Thesis, 2013
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12. Minimization problem
For a fixed variance V [Y] = ε2/2, the cost S is minimized by
choosing as m` the solution of the optimization problem:
F(m0,...,mL) :=
L
X
`=0
m`s` + µ2 V`
m`
obtain
m` = 2ε−2
r
V`
s`
L
X
i=0
p
Visi
The total complexity is
S := 2ε−2








L
X
`=0
p
V`s`








2
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13. The mean squared error (MSE)
Is used to measure the quality of the multilevel estimator:
MSE := E
h
(Y − E[g])2
i
= V [Y] + (E[Y] − E[g])2
, (6)
where Y is what we computed via MLMC, and E[g] what
actually should be computed. To achieve
MSE ≤ ε2
for some prescribed tolerance ε, we ensure that both
(E[Y] − E[g])2
= (E[gL − g])2
≤ 1
2ε2
. (7)
and
V [Y] ≤ 1
2ε2
(8)
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14. Theorem
Consider a fixed t = t∗. Suppose positive constants α,β,γ > 0
exist such that α ≥ 1
2 min(β,γd̂), and
|E[g` − g]| ≤ c14−α`
(9a)
V` ≤ c24−β`
(9b)
s` ≤ c34d̂γ`
. (9c)
Then, for any accuracy ε < e−1, a constant c4 > 0 and a
sequence of realizations {m`}L
`=0 exist, such that
MSE := E
h
(Y − E[g])2
i
< ε2
,
and the computational cost is
S =













c4ε−2, β > d̂γ
c4ε−2 (log(ε))2
, β = d̂γ
c4ε
−
2+
d̂γ−β
α
, β d̂γ.
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15. Modeling of porosity and recharge:
We assume two horizontal layers: y ∈ (−0.8,0] (the upper layer)
and y ∈ [−1,−0.8] (the lower layer).
The porosity inside each layer is uncertain and is modeled as:
φ(x,ξ) = 0.35 · C0(ξ1) · C1(ξ1,ξ2) · C2(ξ1,ξ2)
where C0(ξ1) =
(
1.2 · (1 + 0.2ξ1) if y −0.8
1 if y ≥ −0.8
C1(ξ1,ξ2) = 1 + 0.15(ξ2cos(πx/2) − ξ2sin(2πy) + ξ1cos(2πx))
C2(ξ1,ξ2) = 1 + 0.2(ξ1sin(64πx) + ξ2sin(32πy))
Recharge
q̂in = −6.6 · 10−2
(1 + 0.5 · ξ3)(1 + sinπt
40),
where ξ1, ξ2, and ξ3 are sampled independently and uniformly
in [−1,1].
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16. Examples: two porosity and two permeability realisations
1st row: porosity φ1 ∈ (0.29,0.49) and φ2 ∈ (0.21,0.54).
2nd row: permeability K1 ∈ (5.9 · 10−10,3.25 · 10−9) and
K2 ∈ (1.97 · 10−10,4.5 · 10−9).
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17. Mean and variance on different levels
Comparison of mean values E[c(t,x9,y9)];
and variances Var[c](t,x9,y9) computed on levels 0,1,2,3.
We observe:
(on the left) that the results obtained on the coarsest scale are
not so accurate. All other scales produce more or less similar
results.
(on the right) Each new finer scale gives better and better
results.
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18. MLMC: weak and strong convergence
(left) The mean value E[g` − g`−1] and (right) the variance value
V [g` − g`−1] as a function of time for t ∈ [τ,48τ], ` = 1,2,3.
For every time point we observe convergence in the mean and
in the variance. The amplitude is decreasing.
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19. QoI is the integral value over D9 ; 100 realisations of g1 − g0,
g2 − g1, g3 − g2, QoI g` is the integral value Q9(t,ω) computed
over a subdomain around 9th point, t ∈ [τ,48τ].
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20. Complexity on each mesh level `
` n`, ( n`
n`−1
) r`, ( r`
r`−1
) τ`
Computing times (s`), ( s`
s`−1
)
average min. max.
0 153 94 64 0.6 0.5 0.7
1 2145 (14) 376 (4) 16 7.1 (14) 7 9
2 33153 (15) 1504 (4) 4 253 (36) 246 266
3 525825 (16) 6016 (4) 1 11110 (44) 9860 15507
Here:
r` is the number of time steps ,
τ` is a time step
τ` = 6016/r`,
#ndofs= n`,
average, minimal, and maximal computing times on each level
`.
The numbers in brackets (column 2 and 3) confirm the theory
that the method has order one w.r.t to h and order one w.r.t. the
time step τ.
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21. Rates of the weak and strong convergences
(left) Weak (α = 0.94, ζ1 = 3.2) and (right) strong (β = 1.7,
ζ2 = 4.8) convergences in log-scale computed for levels 0,1,2,3
(horizontal axis). The QoI is a subdomain integral of c over D9,
a domain around point (x,y)9 = (1.65,−0.75).
Now the identified convergence rates (red color) can be used to
estimate L and all m`.
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22. Comparison MLMC vs MC
ε 0.1 0.05 0.01
ε2 0.01 0.0025 0.0001
MC cost 2.0 · 103 2.8 · 105 3.1 · 108
MLMC cost 6.4 · 101 1.06 · 103 8.9 · 104
required L 2 3 4
m0,m1,m2,m3 44,5,0,0 362,43,3,0 16672,1990,120,4
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23. Comparison of MC and MLMC for different ε
Good news:
1. MLMC (red line) is much faster than MC (blue line)
2. MLMC theory (dashed violet line) fits to the MLMC numerics
(red line) 22 / 27

24. Conclusion
1. Investigated efficiency of MLMC for Henry problem with
uncertain porosity, permeability, and recharge.
2. Uncertainties are modeled by random fields.
3. MLMC could be much faster than MC, 3200 times faster !
4. The time dependence is challenging.
Remarks:
1. Check if MLMC is needed.
2. The optimal number of samples depends on the point (t,x)
3. An advanced MLMC may give better estimates of L and m`.
Future work:
1. Consider a more complicated/multiscale/realistic porosity
and geometry
2. Incorporate known experimental and measurement data to
reduce uncertainties.
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25. More advanced and realistic computing domain
A. Schneider et al., Modeling saltwater intrusion scenarios for a coastal aquifer at the German North Sea, E3S Web of Conferences 54, 00031 (2018)
Sandelermöns: 3d hydrogeological model (30x vertically exaggerated) with coarse grid, rivers, pumping wells and recharge
map (right).
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26. More advanced and realistic computing domain
A. Schneider et al., Modeling saltwater intrusion scenarios for a coastal aquifer at the German North Sea, E3S Web of Conferences 54, 00031 (2018)
Situation of the Sandelermöns model area including the wells of the three waterworks and area of saline groundwater.
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27. Literature
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, 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, Cham. https://doi.org/10.1007/978-3-030-81362-8_52023
3. A .Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability
GEM-International Journal on Geomathematics Vol. 11, pp 1-29, 2020
4. A Litvinenko, AC 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, Vol. 4, pp 37-50, 2019.
5. H.G. Matthies, E. Zander, B.V. Rosić, et al. Parameter estimation via conditional expectation: a Bayesian inversion. Adv. Model. and Simul. in Eng. Sci. 3, 24 (2016).
https://doi.org/10.1186/s40323-016-0075-7
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28. Acknowledgments
We thank the KAUST HPC support team for assistance with
Shaheen II and for the project k1051.
This work was supported by the Alexander von Humboldt
foundation.
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