uncertainty quantification in time- dependent...
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Uncertainty Quantification in Time-Dependent Reliability Analysis
1
Zhen Hua, Sankaran Mahadevana, and Xiaoping Dub
a. Department of Civil and Environmental Engineering
Vanderbilt University, Nashville, TN, USA b. Department of Mechanical and Aerospace Engineering
Missouri University of Science and Technology, Rolla, MO, USA
Outline
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• Background - Time-dependent reliability analysis - Epistemic uncertainty
• Uncertainty quantification in time-dependent reliability analysis - Modeling of epistemic uncertainty in random variables - Modeling of epistemic uncertainty in stochastic processes - Uncertainty quantification of time-dependent reliability
• Numerical examples • Conclusions and future work
Time-dependent reliability analysis
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0 0( , ) Pr{ ( , ( ), ) 0, [ , ]}e eR t t g t t t t t= ≤ ∀ ∈X Y
( , ( ), ) 0g t t >X YFailure:
Reliability:
Random variables
response
Stochastic processes
0 0
( , ) Pr{ ( ) ( , ( ), ) 0, [ , ]}f e e
p t t G g t tX Y
• The ability that a system performs its intended function over a time period of interest.
• The probability that the first time to failure (FTTF) is larger than a time period of interest.
• The longer the time period, the lower the reliability.
An example
Time-dependent reliability analysis
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State of the art • Composite limit-state function method (Singh and Mourelatos, 2010, Du, 2013) • Importance sampling approach (Mori and Ellingwood, 1993, Dey and Mahadevan, 2000, Singh and Mourelatos, 2011) • Upcrossing rate-based methods (Madsen, 1990, Zhang and Du, 2011, Hu and Du, 2013) • Extreme value response method (Wang and Wang, 2012, Hu and Du, 2015) • Other sampling-based methods (Hu and Du, 2014, Jiang, et al., 2014, Wang and Mourelatos, 2014) Only aleatory uncertainty
Challenges
• Representation of epistemic uncertainty in time-dependent reliability analysis • Time-dependent reliability analysis including epistemic uncertainty
Epistemic uncertainty
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Epistemic uncertainty
Data uncertainty Model uncertainty Surrogate model uncertainty
Distribution parameter uncertainty
Distribution type uncertainty
Model parameter uncertainty
Model discrepancy
Method error
Hyper-parameter uncertainty
Prediction uncertainty
Linearization error
• Epistemic uncertainty in random variables • Epistemic uncertainty in stochastic processes • How to efficiently perform uncertainty quantification in time-dependent reliability
analysis?
Modeling of Epistemic Uncertainty in Random Variables
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θ θθ
θ θ θ
( ) ( )( )
( ) ( )
Lp
L d
xx
x
θ θ θ( ) ( ) ( )p L x x
Data uncertainty Distribution type uncertainty Distribution parameter uncertainty
Bayes’ theorem
Posterior distribution
58 60 62 64 66 68 70 72 74 76 780
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
50 60 70 800
2
4
6
8
X2
Freq
uenc
y
Distribution parameters Mean Standard deviation
Prior distribution
An example
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Modeling of Epistemic Uncertainty in Stochastic Processes
Modeling of stochastic processes
• Karhunen-Loeve (KL) expansion method • Polynomial Chaos Expansion (PCE) • Linear Expansion (LE) method • Orthogonal Series Expansion (OSE) method • Expansion Optimal Linear Estimation (EOLE) method • ARMA or ARIMA model
(0) (1) (2) ( ) (1) ( )1 2 1
( ) ( ) ( ) ( ) ( ) ( ) ( )p qj i j j j i j j i j j i p i j i j i q
Y t Y t Y t Y t t t tϕ ϕ ϕ ϕ ε ω ε ω ε− − − − −= + + + + − − −
ARMA (p, q)
(0)
(1) ( )1j
j
Y pj j
(1) ( )1
(1) ( ) 21
( ) ( 1) ( ) 21
(1 ) , for 0
( ) , for 1, ,
0 for 1
pk j k j k p
qj j q
k k qj j j q k
k
k q
k q
(1) (2) ( )( ) ( 1) ( 2) ( )j j j j
pY j Y j Y j Y
k k k k p
Mean
Auto-covariance
Auto-correlation
Required in time-dependent reliability analysis
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Modeling of Epistemic Uncertainty in Stochastic Processes
Bayesian ARMA model
For given time series coefficients , the standard deviation in the noise term: φ ω,k k
2 2
1 1
1 ˆ[ ( ) ( )]( 1)
ts tn nj
k i k ij i pts
Y t Y tn n p
Observations Estimations of the time series
φ ω( , )k k k
L DThe likelihood function
Number of trajectories Number of time instants
φ ω φ ω1
( , ) ( , )tsn
ik k k k k ki
L L
D D
φ ω φ ω1 2
( , ) ( ( ), ( ), , ( ) , )i i i ik k k k k k nt k k
L L Y t Y t Y tD
φ ω φ ω φ ω
1 2( ( ), ( ), , ( ) , ) ( ( ) , , ) ( , )
nt nt
i i i ik k k nt k k k nt k k t t k k
L Y t Y t Y t L Y t LY Y
All trajectories
One trajectory
Time instants
φ ω φ ω φ ω( , ) ( , ) ( ) ( )k k k k k k k k
p L D D
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Modeling of Epistemic Uncertainty in Stochastic Processes
0 100 200 300 400 500-10
-5
0
5
10
15
t
Y(t)
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
1
2
3
4
5
6
7
8
9
φ1
PD
F
Prior DistributionPosterior Distribution
-0.4 -0.2 0 0.2 0.40
2
4
6
8
10
12
φ2
Prior Distribution
Posterior Distribution
-0.02 0 0.02 0.04 0.06 0.080
10
20
30
40
50
φ3
Prior DistributionPosterior Distribution
21 2 3
( ) 0.8231 ( ) 0.1256 ( ) 0.0812 ( ) (0, 2 )i i i i
Y t Y t Y t Y t N
An example
• Model the time series model based on collected data
• MCMC is used for calibration • Data from MCMC is recorded to
preserve the correlation between posterior distributions
MPP-based time-dependent reliability analysis
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MPP searches
Upcrossing rate
Random field modeling
Importance sampling
Time-dependent reliability
• The effects of epistemic uncertainty on MPP-based time-dependent reliability analysis • Problems with stationary stochastic processes • Two key elements: and
Uncertainty Quantification in Time-Dependent Reliability Analysis
*Yu
Affects failure threshold in U space and correlation over time
Affects Correlation over time
ρρ
ρ
0 0
0
* *T * *T02
( , ) ( , )
( , )
11 ( ( , ) )
T TL
T T T T
t t t t
t t
t t
X X Y Y
X X Y Y Y Y Y Y
Y Y Y Yu u u u
Uncertainty Quantification in Time-Dependent Reliability Analysis
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Quantify the uncertainty in the time-dependent reliability analysis result
A straightforward way
• Dimensionality of the surrogate model may be high: m(1+p+q)+n×nr
• Computationally expensive
Number of stochastic processes
Number of random variables
Uncertainty Quantification in Time-Dependent Reliability Analysis
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New approach
• Group one: -- random variables and stochastic processes that are exactly modeled • Group two: , -- random variables and stochastic processes with epistemic uncertainty
Classification of random variables and stochastic processes:
0 0( ) , Pr{ ( ) ( , , ) }a
fp t G t g e x y X x y
For given realization of variables with epistemic uncertainty:
aXX φ ω( , )
( )tY
For given value of epistemic parameters and :
0 0( ) ( ( ) , ) ( ) ( )
f fp t p t f f d d x y x y x y
( )f x ( )f y
( , )0
( , )0
2 2 20 ( )
( )
min ( )
( ( ), ( ), ( ))
a
a
t
t
t
g T T T e
ϕ ω
ϕ ω
XX Y
XX Y
u u u
u u u
Main task: how to efficiently obtain the key elements and for given value of the epistemic uncertainty
*Yu
Uncertainty Quantification in Time-Dependent Reliability Analysis
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Introducing an auxiliary variable:
0( ) Pr{ } ( ) ,
f fp a p fu U u p t x y
φ ω( , )10 0 0
( ) Pr{ ( ( ) , ( )) 0}f a f
p t U p t t X Y
φ ω
φ ω
( , )0
( , )0
0
( )
0 ( )
min ( )
[ , , ]
( ) ( ) ( ), ( )
a t
a f t
t
u
u p t T T
u
X Y
X Y
u
u u u
u u
MPP search
New approach (Cont.)
φ ω
φ ω
( , )0
( , )0
0
( )
( )
min ( )
[ , , ]
( ), ( )
a t
Ca t
t
u
u T T
u
X Y
X Y
u
u u u
u u
Conditional reliability index
• Independent from the distribution of variables with epistemic uncertainty
• Dependent on value of the variables with epistemic uncertainty
• Still applicable when the distribution changes
• Dimension: m(1+p+q)+n×nr m+n
Surrogate model of conditional reliability index
Numerical Examples
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1 2 1( ) ( )g t X X Y t
A mathematical example
0 1 2 1 0
( , ) Pr{ ( ) ( ) 0, [ , ]}f e e
p t t g X X Y t t
0 20 40 60 80 100140
150
160
170
180
Cycles
Y(t)
50 60 70 800
2
4
6
8
X2
Freq
uenc
y
is exactly modeled , and are modeled based on data. 1
X2
X1( )Y t
Experimental data
Numerical Examples
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0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
[0, t] cycles
Tim
e-de
pend
ent p
roba
bilit
y of
failu
re
0 10 20 300
0.1
0.2
0.3
0.4
pf(0
, t)
"True" pf curve
95% Prediction Interval
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
5
10
15
Time-dependent probability of failure
Prob
abili
ty D
ensit
y Fu
nctio
n (P
DF)
Aleatory+Epistemic"True" pfOnly aleatory
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
5
10
15
20
25
30
35
40
45
Time-dependent probability of failure
Prob
abili
ty D
ensit
y Fu
nctio
n (P
DF)
100 Cycles200 Cycles500 Cycles"True" pf
A mathematical example (Cont.)
UQ results of time-dependent reliability analysis
Last time instant
More data
Numerical Examples
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0 5 10 15 20 25 30 35 40 45 500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
[0, t] cycles
Tim
e-de
pend
ent p
roba
bilit
y of
failu
re
0 100 2000.02
0.04
0.06
pf(0
, t)
"True" pf curve
95% Prediction Interval
0.025 0.03 0.035 0.04 0.045 0.05 0.0550
20
40
60
80
100
120
140
pf (0, 50)
Aleatory+Epistemic"True" pfOnly aleatory
0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.0550
20
40
60
80
100
120
pf(0, 50)
1000 Cycles
"True" pf
200 Cycles
2000 Cycles
0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650
50
100
150
200
250
pf (0, 50)
1000 Cycles
2000 Cycles200 Cycles
2000 Cycles and 2000 Samples"True" pf
A beam example (Cont.)
• Uncertainty in the reliability analysis result is more sensitive to the epistemic uncertainty in the random variable
More stochastic load data More random variable data
Conclusions
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• A framework is developed for time-dependent reliability analysis in the presence of data uncertainty
• A new method is proposed to improve the efficiency of uncertainty quantification in time-dependent reliability analysis
• Numerical examples demonstrated the effectiveness of the proposed method
Future work
• Inclusion of other sources of epistemic uncertainty in time-dependent reliability analysis (i.e. model uncertainty)
• Time-dependent sensitivity analysis to quantify contributions of epistemic uncertainty sources
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• Air Force Office of Scientific Research (Grant No. FA9550-15-1-0018, Technical Monitor: Dr. David Stargel)
• National Science Foundation through grant CMMI 1234855
• The Intelligent Systems Center at the Missouri University of Science and Technology.
Acknowledgement