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]