bayesian calibration of model uncertainty
TRANSCRIPT
Bayesian calibration of model uncertaintyLaurent van den Bos, Benjamin Sanderse, Wim Bierbooms, Gerard van Bussel
Uncertainties in wind energy? Why?
To model loads on wind turbines, we need to consider both the uncertainconditions and the errors in the model, e.g.:1. Aleatory (irreducible): wind, waves, humidity, temperature, etc.
Influences total production of turbine2. Epistemic (reducible): model error, measurement errors, bias, etc.
Influences reliability of models under consideration
How to formulate this?
One comprehensive model for aleatory and epistemicuncertainty
Four steps:1. Development of robust and adaptive aleatory uncertainty quantification
based on collocation methods2. Development of epistemic uncertainty quantification and error
estimation tools3. Coupling aleatory, epistemic, and numerical tools in small-scale
applications4. Large-scale demonstration to OWT applications
Model ?(with hidden error)
Ingredients: data, model and their relation
Given are:I Measurement data d;I Parameters θ;I A model m(θ);I The relation between the model and the data.
Simple example (see Kennedy and O’Hagan (2001)):
d = m(θ) + ε, ε ∼ N(0,σ2),
where ε given on beforehand.
Bayesian model calibration
Determine likelihood (function of θ):
p(d|θ) ∝ exp[−
12σ2 ‖d − m(θ)‖2
]Law of Bayes:
p(θ|d) ∝ p(d|θ)p(θ)Here:1. p(θ) describes prior knowledge;2. p(d|θ) describes likelihood of observing data;3. p(θ|d) describes information of the parameter θ, with a pdf
Information about uncertainty is captured in p(θ|d)
Extension: Surrogate modeling
Problem: Sampling from p(θ|d) is very expensive
Solution: Do not use the full model, but a simplified surrogate (seeMarzouk et al. (2007); Birolleau et al. (2014))
Physics
Epstemic: θ
Aleatory: ξ Model: m(ξ,θ) → p(θ|d)
Surrogate: m(ξ,θ) → p(θ|d)
Data: d depending on ξ
Evaluations at θ1,θ2, . . . Estimate p of p
Example: Wake parameter calibration
Given are:I d: Data from Navier–Stokes code (from ECNS, Sanderse (2011));I Use Jensen wake model, i.e. θ = (α, r0) with
I r0: initial wake radius,I α: linear wake expansion coefficient.
I d = m(θ) + ε, ε ∼ N(0,0.12)
Model is linear → No surrogate is needed in this case
-2 -1 0 1 2
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
y
0.8
0.85
0.9
0.95
1
1.05
True flow
0.06 0.08 0.1 0.12 0.14
α
0.4
0.42
0.44
0.46
0.48
0.5
r0
MAP
Bayesian mean
Posterior
-2 -1 0 1 2
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
y
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Calibrated flow
Conclusions & Future workWe saw:I Convert all model uncertainty to distributions on the parametersI Expensive model? Use a surrogate!
Future work:I Use more complex model for surrogate modelingI Convergence proof for surrogate modelingI Incorporate aleatory uncertainties
Literature & AcknowledgmentsA. Birolleau, G. Poette, and D. Lucor. Adaptive Bayesian inference for discontinuous inverse problems,application to hyperbolic conservation laws. Communications in Computational Physics, 16(1):1–34, 2014.doi: 10.4208/cicp.240113.071113a.
M. C. Kennedy and A. O’Hagan. Bayesian calibration of computer models. Journal of the Royal StatisticalSociety: Series B (Statistical Methodology), 63(3):425–464, 2001. doi: 10.1111/1467-9868.00294.
Y. M. Marzouk, H. N. Najm, and L. A. Rahn. Stochastic spectral methods for efficient Bayesian solution ofinverse problems. Journal of Computational Physics, 224(2):560–586, 2007. doi: 10.1016/j.jcp.2006.10.010.
B. Sanderse. Energy-Conserving Navier–Stokes solver. Verification of steady laminar flows. Technical ReportECN-E–11-042, Energy research Centre of the Netherlands, 2011.
This research is supported by the Dutch Technology Foundation STW, which is part of the NetherlandsOrganization for Scientific Research (NWO), and which is partly funded by the Ministry of Economic Affairs(project P14-03 EUROS).
CWI is the national research intitute for mathematics and computer science in the Netherlands. http://www.cwi.nl