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

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

Abstract

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

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?

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.

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 ·e_x|_x=0

= ˆ

q_in.

We model φ by a random field and assume:

K

=

KI, K

=

K

(

_φ

)

, and Kozeny–Carman-like dependence K

(

_φ

) =

_κ ^{· φ}

3

1 − φ², (1)

where κ is a scalar.

Discretisation: vertex-centered finite volume, implicit Euler

Porosity and solution of the Henry problem

ˆ

q_in= 6.6·10⁻²kg/s

c= 0 c= 1

p = −ρ₁gy 0

−1 m y 2 m

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

)

Expectation and variance of the mass fraction c

E

[

c

]

^∈

[

0,0.35

)

; Var

[

c

]

^∈

[

0.0,0.04

)

What can we compute?

QoIs: c in the whole domain, c at a point, or integral values (the freshwater/saltwater integrals):

Q_FW

(

t, ω

) :=

Z

x∈D

I

(

c

(

t,x, ω

)

^≤ 0.012178

)

dx, (2)

Q_s

(

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

]

.

Multi Level Monte Carlo (MLMC) method Spatial and temporal grid hierarchies

D₀,D₁, . . . ,D_L, T₀,T₁, . . . ,T_L;

n₀

=

512, n<sub>`</sub> ≈ n<sub>0</sub> · 16<sup>`</sup>,

τ_`+1

=

¹_{4τ`</sub>, r<sub>`+1}

=

4r_{`</sub> and r<sub>`}

=

r₀4^`.

Approx. error: kc − c_h,τk₂

=

^O

(

h

+

_τ

) =

^O

(

n^−1/2

+

r⁻¹

)

Computation complexity on level ` is

s_`

=

^O

(

n_{`</sub>r<sub>`}

) =

^O

(

4^3`γn₀ · r₀

)

Multi Level Monte Carlo (MLMC) method MLMC approximates E

[

g

]

^{≈ E}

[

g_L

]

using the following telescopic sum:

E

[

g_L

] =

^E

[

g₀

] +

L

X

`⁼1

E

[

g_{`</sub> − g<sub>`}−1

]

^≈

≈ m₀⁻¹

m₀

X

i=1

g₀^(0,i)

+

L

X

`⁼1









m_`⁻¹

m_`

X

i=1

(

g<sub>`</sub><sup>(`,i)</sup> − g<sub>`</sub><sup>(`,</sup>₋₁ⁱ⁾

)







 .

Let Y_`

:=

m_{`</sub><sup>−1</sup>P<sup>m</sup><sub>`}

i=1

(

g<sub>`</sub><sup>(`,i)</sup> − g<sub>`</sub><sup>(`,</sup>₋₁ⁱ⁾

)

, where g−1 ≡ 0, so that E

[

Y_`

] :=











E

[

g₀

]

_{, `}

=

0

E

[

g_{`</sub> − g<sub>`}−1

]

_{, ` >} 0 . (5)

MLMC notation

Denote by Y

:=

^PL_{`=0</sub>Y<sub>`} the multilevel estimator of E

[

g

]

based on L

+

1 levels and m<sub>`</sub> independent samples on level `, where

`

=

0, . . . ,L.

Denote V₀

:=

^V

[

g₀

]

and for ` ≥ 1, and V<sub>`</sub>

:=

^V

[

g_{`</sub> − g<sub>`}−1

]

, ` ≥ 1.

The standard theory states:

E

[

Y

] =

^E

[

g_L

]

_, ^V

[

Y

] =

^PL_{`=0</sub>m<sub>`}⁻¹V_`. The cost of the multilevel estimator Y is

S

:=

^PL_{`=0</sub>m<sub>`}s_`.

See details in Giles’18 or Teckentrup’s PhD Thesis, 2013

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, . . . ,m_L

) :=

L

X

`⁼0

m_{`</sub>s<sub>`}

+

_µ<sup>2 V`</sup> m<sub>`</sub> obtain

m_`

=

2ε⁻²

rV_{`</sub> s<sub>`}

L

X

i=0

pV_is_i

The total complexity is

S

:=

2ε⁻²











L

X

`⁼0

pV_{`</sub>s<sub>`}











2

The mean squared error (MSE)

Is used to measure the quality of the multilevel estimator:

MSE

:=

^Eh

(

Y − E

[

g

])

²ⁱ

=

^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

[

g_L − g

])

^{2 ≤ 1}_2ε²_. (7) and

V

[

Y

]

^{≤ 1}_2ε² (8)

Theorem

Consider a fixed t

=

t^∗. Suppose positive constants α, β, γ > 0 exist such that α ≥ ¹

2

min(

_{β, γd}

ˆ )

, and

|E

[

g_` − g

]

^{| ≤} c₁4<sup>−α`</sup> (9a) V<sub>`</sub> ≤ c₂4<sup>−β`</sup> (9b) s<sub>`</sub> ≤ c₃4^dˆγ`. (9c) Then, for any accuracy ε < e⁻¹, a constant c₄ > 0 and a

sequence of realizations {m_`}^L

`⁼0 exist, such that MSE

:=

^Eh

(

Y −E

[

g

])

²ⁱ _{< ε}²_, and the computational cost is

S

=















c₄ε⁻², β > _d

ˆ

_γ c₄ε⁻²

(log(

_ε

))

²_{, β}

= ˆ

dγ c₄ε⁻

2+^dˆγ_α^−β

, β < _d

ˆ

_γ.

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

ˆ

q_in

=

⁻6.6 · 10⁻²

(

1

+

0.5 · ξ₃

)(

1

+

sin₄₀^πt

)

_,

where ξ₁, ξ₂, and ξ₃ are sampled independently and uniformly in

[

⁻1,1

]

.

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

)

.

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.

MLMC: weak and strong convergence

(left) The mean value E

[

g_{`</sub> − g<sub>`}−1

]

and (right) the variance value V

[

g_{`</sub> − g<sub>`}−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.

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

]

.

Complexity on each mesh level `

` n<sub>`</sub>, (_n^n`

`−1) r<sub>`</sub>, (_r^r`

`−1) τ<sub>`</sub> Computing times (s<sub>`</sub>), (<sub>s</sub><sup>s`</sup>

`−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_{`</sub> is the number of time steps , τ<sub>`} 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

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

)

.

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

Comparison of MC and MLMC for different ε

Good news:

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

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., Pfl ¨uger, 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. Rosi ´c, 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

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.