Presentation by Mariana Clare, Imperial College London, UK, at the Delft3D and XBeach User Day: Coastal morphodynamics, during Delft Software Days - Edition 2019. Wednesday, 13 November 2019, Delft.

1. Mariana Clare1
Delft3D and XBeach User Day:
Coastal morphodynamics
Supervisors: Prof Matthew Piggott1,Dr.Colin Cotter1 and Dr.CatherineVillaret2
1Imperial College London 2East Point Geo Consulting
Using Multilevel Monte Carlo
Methods to assess erosion/flood
risk in the coastal zone
13th November 2019
m.clare17@imperial.ac.uk

2. February 2014 in Dawlish, Devon
2
This cost £35 million to fix and is estimated to have cost the Cornish economy £1.2 billion

3. 3
Assessing Risk
Problem:
Two main sources of uncertainty which propagate into prediction values:
1. Model Uncertainty
2. Uncertainty in Input Data
Possible Solution:
Use Monte Carlo simulation varying parameters e.g. Callaghan et al. (2013)
Problem with Monte Carlo:
XBeach is computationally expensive:
Callaghan et al. (2013) estimate to use Monte Carlo with XBeach with their
problem would take 4.5 millennia

4. POSSIBLE SOLUTION 2:
MULTILEVEL MONTE CARLO METHODS
4

5. 5
RECAP: Monte Carlo Simulation
Repeated random sampling of a variable from a distribution which are
then run through a model to produce a range of outcomes
Monte Carlo estimator for N sample points
e.g. ෣𝑋𝐵𝑒𝑎𝑐ℎ(𝑤) =
1
𝑁
σ𝑖=1
𝑁
𝑋𝐵𝑒𝑎𝑐ℎ(𝑤𝑖)

6. RECAP: Multilevel Grid
FINER grid
scales
MORE
Computationally
expensive
l = 0
l = 1
l = 2
l = 3
l = 4
6

7. Multilevel Monte Carlo (MLMC)
FINER
grid
scale
l = 0
l = 1
l = 2
l = 3
l = 4
7
we see
Using
same random number used to
generate Pi
l and Pi
l-1
Nl independent difference
estimators

8. Some MLMC Theory
8
where 𝑉𝑙 is the variance of the sample (𝑃𝑙
(𝑖)
− 𝑃𝑙
𝑖−1
) and 𝑁𝑙 the number of samples at each level
Variance is main source of error:
independent events
compared to O(Nl
-1/2)
for Monte Carlo

9. 9

10. 10
UNCERTAINTY IN BEACH BEDSLOPE ANGLE
Question: what is the maximum expected distance
along the x-axis the water reaches up the beach?
Angle Range
𝛼 ∈ [0, tan−1(0.5)]

11. 11
Does MLMC work with XBeach?
1D

12. 12
Does MLMC give a speed-up with XBeach?
Maximum expected surge distance: 873 m
1D Convergence Rate

13. 13
Distribution
1D

14. 14
Maximum expected surge distance: 869m
Runtime: 30 mins on 13 cores
2D
Convergence Rate

15. 15
Distribution
2D

16. 16
UNCERTAINTY INWAVEHEIGHT
Jonswap spectrum for
H0=2.5m, fp = 0.08s-1,
𝝲 = 3.3 and fynq = 0.3s-1
Waveheight Range
𝐻0 ∈ 1, 6 𝑚

17. 17
Wave Runup Height
Source: Manual, EurOtop. "Wave Overtopping of Sea Defences and Related Structures–Assessment Manual." UK: NWH Allsop, T. Pullen, T. Bruce. NL: JW van
der Meer. DE: H. Schüttrumpf, A. Kortenhaus. www. overtopping-manual. com (2007).

18. 18
Expected 𝑅 𝑢2%: 0.108m
Does MLMC work with XBeach?

19. 19

20. 20
Source: Gent, Marcel & Vroeg, J. (2008). Dune Erosion Prediction Method Based on Large-Scale Model Tests.
Erosion Volume
Use standard Delta Flume XBeach test case
Waveheight Range
𝐻0 ∈ 1, 6 𝑚

21. 21
Expected erosion volume: 0.0138
Does MLMC work with XBeach?

22. 22
• Introduce more complex wave profiles and coastal regions with
more complex shapes
• Investigate the effect of uncertainty in more than one parameter
for each test case simulation and for more than one quantity
• Investigate more parameters and more test cases – any ideas?
FURTHERWORK

23. 23

24. ▪ Giles, M. B. (2015). Multilevel Monte Carlo methods, Acta Numerica, 24:259-328.
▪ Callaghan, D. P., Ranasinghe, R., and Roelvink, D. (2013).Probabilistic estimation of storm
erosion using analytical, semi-empirical, and process based storm erosion models. Coastal
Engineering, 82:64-75.
▪ Cliffe, K. A., Giles, M. B., Scheichl, R., and Teckentrup, A. L. (2011). Multilevel Monte Carlo
methods and applications to elliptic PDEs with random coefficients. Computing and
Visualization in Science, 14(1):3-15.
▪ Roelvink, D.,Van Dongeren, A., McCall, R., Hoonhout, B.,Van Rooijen, A.,Van Geer, P., De Vet,
L., Nederhoff, K., and Quataert, E. (2015). XBeach technical reference: Kingsday release. Delft,
The Netherlands: Deltares,Technical report.
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