Steps in Uncertainty Quantification
1
6. Quantification of Model Discrepancy – Thin Beam“Essentially all models are wrong, but some are useful” George E.P. Box
with
Example: Thin beam driven by PZT patches
Euler-Bernoulli Model:
Statistical Model:
Note: 7 parameters, 32 states
For all � 2 V
ZL
0
⇢(x)
@2w
@t
2 + �@w
dt
��dx +
ZL
0
YI(x)
@2w
@x
2 + cI(x)@3
w
@x
2@t
�� 00
dx
= k
p
V (t)
Zx2
x1
� 00dx
⇢(x) = ⇢hb + ⇢p
h
p
b
p
�p
(x) , YI(x) = YI + Y
p
I
p
�p
(x)
cI(x) = cI + c
p
I
p
�p
(x)y(t
i
, q) = w(ti
, x , q)
ti
YiYi = y(ti , q) + "i 2
Quantification of Model Discrepancy – Thin BeamExample: Good model fit
Note: Observation errors not iid
Model Fit to Data
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
x 10−7
Frequency
Fre
quency
Conte
nt
Mathematical Model y(ti;q)
Data
Reference: Additive observation errors
Yi = y(ti , q) + "i
Yi = y(ti , q) + �(ti , eq) + "i
• M.C. Kennedy and A. O’Hagan, Journal of the Royal Statistical Society, Series B, 2001.
Quantification of Model Discrepancy – Thin BeamExample: Good model fit
Problem: Observation errors not iid
Result: Prediction intervals wrong
Prediction Intervals and Data
Model Fit to Data
Approaches: • GP Model: Inaccurate for extrapolation• Control-based approaches: difficult to extrapolate.• Problem: correct physics or biology required for extrapolation!
Quantification of Model Discrepancy – Thin BeamProblem: Measurement errors not iid
Result: Prediction intervals wrong
Prediction Intervals and Data
Model Fit to Data
One Approach: • Determine components of model you trust (e.g., conservation laws) and don’t trust (e.g., closure relations). Embed uncertainty into latter.• T. Oliver, G. Terejanu, C.S. Simmons, R.D. Moser, Comput Meth Appl Mech Eng, 2015.
2018-19 SAMSI Program: Model Uncertainty: Mathematical and Statistical (MUMS)
Quantification of Model Discrepancy – Thin Beam
Our Solution: “Optimize” calibration interval
• Use damping/frequency domain results to guide.
0.75 0.8 0.85 0.9 0.95 1−60
−40
−20
0
20
40
60
Time (s)
Dis
pla
cem
en
t (µ
m)
2 2.1 2.2 2.3 2.4 2.5−50
0
50
Time (s)
Dis
pla
cem
ent (µ
m)
3 3.1 3.2 3.3 3.4 3.5−40
−30
−20
−10
0
10
20
30
40
Time (s)
Dis
pla
cem
en
t (µ
m)
0 1 2 3−30
−20
−10
0
10
20
30
Time (s)
Dis
pla
cem
ent (µ
m)
0 1 2 3−40
−30
−20
−10
0
10
20
30
Time (s)
Dis
pla
cem
en
t (µ
m)
Note: We have substantially extended calibration regime.Calibrate on [0,1] Calibrate on [0.25,1.25]