HIPAD LAB: HIGH PERFORMANCE SYSTEMS LABORATORY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING AND EARTH SCIENCES Iterative surrogate model development with applications to multi‐objective design under uncertainty [and stochastic sampling] Alexandros Taflanidis Associate Professor and Frank M. Freimann Collegiate Chair in Structural Engineering Department of Civil & Environmental Engineering & Earth Sciences Concurrent Associate Professor Department of Aerospace and Mechanical Engineering
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Iterative surrogate model development...Iterativesurrogate model development with applications to multi‐objective design under uncertainty [and stochastic sampling] Alexandros Taflanidis
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HIPAD LAB: HIGH PERFORMANCE SYSTEMS LABORATORY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING AND EARTH SCIENCES
and Frank M. Freimann Collegiate Chair in Structural Engineering Department of Civil & Environmental Engineering & Earth SciencesConcurrent Associate ProfessorDepartment of Aerospace and Mechanical Engineering
1 UQ tasks: design under uncertainty, stochastic sampling
• Metamodels (surrogate models): simple, data-driven approximations of the input/output relationship of complex numerical models
• Iterative, adaptive development of metamodels to support specific UQ1 tasks
• Goal is not to establish a globally accurate characterization of the input/output relationship, rather to accurately “perform” the UQ task
Seminar overview
Outline
• Motivation (why use metamodels?)
• Surrogate modeling overview
• Design under uncertainty using metamodels
• Iterative metamodel implementation for multi-objective design under uncertainty
• Iterative metamodel implementation for stochastic sampling
xnX x
ModelResponse
Performance evaluation
[ | , ] ( , )h h z x θ x θ
nΘ θ
θ1θ2
p(θ)
Design under uncertainty
Performance measure
( , ) znz x θ
Design problem under uncertainty
[ | , ] ( ) ( , ) (( )= [ ] )Θ Θp h p d h pH E h d z x θ θ θ x θ θ θx
arg min ( ) *
XH
xx x
Design under uncertainty [multi-objective]Design problem under uncertainty
( )= [ ( , )] ( , ) ( ) [ | , ] ( )i ip iΘi Θh pE d pH dh h x θ θ θ z x θx θ θx θ
arg min { ( )} P iX
H
x
X x
Design under uncertainty [multi-objective]Design problem under uncertainty
( )= [ ( , )] ( , ) ( ) [ | , ] ( )i ip iΘi Θh pE d pH dh h x θ θ θ z x θx θ θx θ
arg min { ( )} P iX
H
x
X x
Pareto Front
HP=H(XP) Feasible objective space
11 ( , ))= (( )Θ
h p dH x θ θ θx
22 ( , ))= (( )Θ
h p dH x θ θ θx
Design under uncertainty [multi-objective]Design problem under uncertainty
( )= [ ( , )] ( , ) ( ) [ | , ] ( )i ip iΘi Θh pE d pH dh h x θ θ θ z x θx θ θx θ
arg min { ( )} P iX
H
x
X x
H1(x)
H2(
x)
Model
nX xx
nΘ θθ~ ( )pθ θ
( , )z x θ[ | ],ih z x θPerformance
evaluation
Simulation-based optimization I
zn ( , )ih x θ
arg min { ( |{ })} jP i
XH
xX x θ
1
1 ( )( |{ }) ( , ) ; ~ ( ) ( )
jNj j j
i i jj
pH h qN q
θx θ x θ θ θθ
( , )( ) ( )ii Θh p dH x θ θ θx
arg min { ( )} P iX
H
x
X x
Simulation-based optimization II
Challenges Estimation error (accuracy of stochastic simulation)
needs to be addressed
Computational cost for a single evaluation significant since we need N model evaluations for each objective function calculation
Numerical differentiation might be only possibility for getting derivative information since we have assumed a black-box numerical model
Reduction of relative (common random
numbers) or absolute importance of error
(importance sampling)
Parallel computing
Algorithms for costly global optimization oralternative gradient
free approaches
1
1 ( ) ( |{ }) ( , ) ; ~ ( ) ( )
jNj j j
i jj
pH h qN q
θx θ x θ θ θθ
arg min { ( |{ })} jP i
XH
xX x θ
Outline
• Motivation (why use metamodels?)
• Surrogate modeling overview
• Design under uncertainty using metamodels
• Iterative metamodel implementation for multi-objective design under uncertainty
• Iterative metamodel implementation for stochastic sampling
-2
0
2
-2
0
20
0.1
0.2
0.3
0.4
Real model
Experiments Surrogate model approximationy1y2
z y1y2
z
y1y2
z
• Data-driven mathematical approximations of the input/output (y/z) relationship of complex numerical models (frequently references as process or computer code)
• Formulated based on a database of simulations for the complex process. This database is frequently referenced as experiments or training (or support) points
Surrogate modeling I
Gaussian process metamodel (GPM) I
( ) ( ) ( )Tz n y b y β yReal function is approximated as a realization of a stochastic process
(Gaussian metamodel or Gaussian process emulator)
* *
* *
( ) ( ) + ( )
p
T T
n n
z
y b y β r y α
β α
( ) ~ ( ( ), ( ))z N z y y y
* *
( ) [ ( ) ( ) ( ) ( ) ( )]
where ( ) ( ) ( )
( ) ( ) /
2 2 T T 1 1 T
T 1
2 T 1
1
n
y u y B R B u y r y Rr y
u y B R r y b y
F Bβ R F Bβ
Provides also local estimate for the predictive variance (“estimation error”)
10
20
30Exact function
Experiments
z(y)
Gaussian process metamodel (GPM) II
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y
10
20
30Exact function
Predictive mean
Experiments
z(y)
Gaussian process metamodel (GPM) II
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y
1
1( , | ) exp
nyysn I J
I J k ki
k k
Rs
y yy y s
Optimize hyper parameters of
correlation function
10
20
30Exact function
Predictive mean
Experiments
z(y)
Gaussian process metamodel (GPM) II
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
σ(y)
y
Predictive variance
Predictive mean ± σ
( ) ~ ( ( ), ( ))z N z y y y
10
20
30Exact function
Predictive mean
Experiments
z(y)
Gaussian process metamodel (GPM) II
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
σ(y)
y
Predictive variance
Predictive mean ± σ
( ) ~ ( ( ), ( ))z N z y y y
10
20
30Exact function
Experiments
z(y)
Design of experiments
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
σ(y)
y
Predictive variance
Predictive mean
10
20
30Exact function
Experiments
z(y)
Design of experiments
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
σ(y)
y
Predictive variance
Predictive mean
0
10
20
30Response
Predictive meanExperimentz(y)
Adaptive design of experiments
y0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
σ(y)
y
Select experiment in regions of low accuracy
2
2
1 ( )
( ) ([ |
)]z
GPMi
j
ni
jhjj z
hzz
y
yy y
Target only domain of interest based on some
preference function (density)0
0.2
0.4
0.6
Preference function
πp(y)
Utility metric U(y)
0
10
20
30Response
Predictive meanExperimentz(y)
Sample-based design of experiments I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
σ(y)
y
2
2
1 ( )
( ) ([ |
)]z
GPMi
j
ni
jhjj z
hzz
y
yy y
0
0.2
0.4
0.6
Preference function
πp(y)
• Simulate large number of samples from preference function
• Maintain only the samples that have larger associated error function
• Cluster them to avoid close proximity of experiments
Utility metric
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
60
σ(y)
y
0
0.2
0.4
0.6
DOE
0
10
20
30
Experiment (new)z(y)
previous errorcurrent error
Response
Predictive mean
Experiment
• Simulate large number of samples from utility function
• Maintain only the samples that have larger associated error function
• Cluster them to avoid close proximity of experiments
Preference function
πp(y)
Sample-based design of experiments II
-2
0
2
-2
0
20
0.1
0.2
0.3
0.4
Real model
Surrogate model approximationy1y2
z y1y2
z
y1y2
z
• Computationally VERY efficient (matrix manipulations that are easy to vectorize, no matrix inversion in implementation)
• Exact interpolation and accurate for approximating complex functions and can provide easily gradient information
• Local estimate for the predictive variance (can be used for adaptive DoE)
GPM
* *( ) ( ) + ( )T Tz y b y β r y α
* *( ) ( )+ ( )b rz y β J y α J y
2 2 1 1
1 1
( ) [1 ( ) ( ) ( )
( ) ( )]; ( ) ( ) ( )
T T
T T
y u y F R F u y
r y R r y u y F R r y b y
( ) ~( ( ), ( ))
zN z
yy y
Experiments (Support points)
Surrogate modeling II (GPM)
Outline
• Motivation (why use metamodels?)
• Surrogate modeling overview
• Design under uncertainty using metamodels
• Iterative metamodel implementation for multi-objective design under uncertainty
• Iterative metamodel implementation for stochastic sampling
Model
nX xx
nΘ θθ~ ( )pθ θ
( , )z x θ[ | ],ih z x θPerformance
evaluation
Surrogate model–aided optimization I
zn ( , )ih x θ
arg min { ( |{ })} jP i
XH
xX x θ
1
1 ( )( |{ }) ( , ) ; ~ ( ) ( )
jNj j j
i i jj
pH h qN q
θx θ x θ θ θθ
What space to formulate the metamodel in (input)?
What is output?
[ | ],ih z x θModel
nX xx
nΘ θθ~ ( )pθ θ
( , )z x θPerformance
evaluation
zn( , )ih x θ
[ ]y x θAugmented input space:Taflanidis, A.A. and J.L. Beck (2008).
“Stochastic Subset Optimization for problems with reliability objectives”. Probabilistic Engineering Mechanics, 23 (2-3): 324-338.
Zhang, J., Taflanidis A.A, and J.C. Medina (2017). “Sequential approximate optimization for design under uncertainty problems utilizing Kriging metamodeling in augmented input space”. Computer Methods in Applied Mechanics and Engineering. 31: 369-395
Surrogate model–aided optimization II
[ | ],ih z x θ
[ ]y x θAugmented input space:
GPMΘX
y
Model
nX xx
nΘ θθ~ ( )pθ θ
( , )z x θPerformance
evaluation
zn( , )ih x θ
( ( ), ( ))N z y σ y
( , )ih x θ Performance
evaluation
[ ( ), ( ) | , ]GPMih zz y σ y x θ
Surrogate model–aided optimization III
z(y1)z(y2)
z(y n)
...
Model
Model
Modely1
y2
y n
...
[ ]y x θ
Response
Metamodeling in augmented input space I
Get experiments covering variation in both x and θ
training points
GPMDevelop metamodel in
augmented space and use it to replace initial objective function with metamodel-
based approximation
Solve optimization using metamodel-based
approximation, exploit also the gradient information
We require each member in current Pareto set to stay close enough to at least one of
precedent members
Take account into all objectives
Stopping criteria III
Sample–based design of DoE
• Generate candidate DoE sampled for y=[x, θ] based on some preference density
• Keep (small) percentage of the candidate samples based on some utility metric U(y)
• Cluster retained samples to desired number of experiments and keep one experiment per cluster, the one that has the largest value of U(y)
Sample–based design of DoE
• Generate candidate DoE sampled for y=[x, θ] based on some preference density πp(y)= πp(x) πp(θ|x) with separate characteristics for x, θ
• Keep (small) percentage of the candidate samples based on some utility metric U(y)
• Cluster retained samples to desired number of experiments and keep one experiment per cluster, the one that has the largest value of U(y)
Iteration 2
H1(x)
H2(
x)
x2
x1
Pareto set iteration 2 XP2
Domains close to potential members of pareto set (exploration)
DoE preference density for x I
x2
x1
Iteration 2Iteration 1
H1(x)
H2(
x)
Pareto set iteration 2 XP2
Exclude current solutions ‘close to’ precedent ones
Pareto set iteration 1 XP1
XDoE = XP - {set of convergent points}
1
1( | ) ( ) DoE
jDoE DoE
n
pjDoE
Kn b
X xx x
( 1)
( 1)
1,21,2, ,( | ) min max | ' 0.5 |
kp
kconv p iir n
I
x X x x
Domains close to potential members of pareto set (exploration)
DoE preference density for x I
θ
Integrand( , ) ( )
ih pθx θ
1
1
Aggregating all objectives , , :1( | ) ( | ), where ( | ) ( , ) ( )
n
n
p pi i ij
h h
h pn
θ x θ x θ x x θ θ
Regions that contribute more to the integrand(importance sampling densities)
Preference density in augmented ( , ) ( |
space) ( | )p p DoE p x θ x X θ x
DoE preference density for θ
VARM[hi(x,θ)] (predicted) variance of ith performance measure (hi) under metamodel uncertainties for z
“Performance” Function
p[z(y)] p[h(y)]
2
1 ( )
VAR ( , ) ( )[ | ]z
j j
ni
M i ij j z z
hhz
x,θ
x,θx θ x,z θ
( )= max VAR ( , )M iiU hy x θ
Utility metric
Illustrative example IZhang, J. and A.A. Taflanidis (2017). “Multi-objective
optimization for design under uncertainty problems through surrogate modeling in augmented input
space”. Structural and Multidisciplinary optimization, under review.
Design of half-car nonlinear suspension system (4 design variables)
Random Road Surface (ISO 8608 spectral
characterization)
15 uncertain variables
1 2arg min{ ( ),
( , ) (
( )}
( ) )X
Θ
P
ii
H H
h p dH
x
X x x
x θ θ θx
2 ( , ) θx ftf ftrSD SDh
1ln( ) ln( ) (, )
x θ ac
b
R bh MS
~
1 2 2
ln( ) ln( )[ ( ), ( ) | , ]acb RMS
acGPM RMS bh
zz y σ y x θ
~ ~
2 [ ( ), ( ) | , ]GPMftf ftrh SD SD zz y σ y x θRoad holding: Average (RMS) force for
front & rear tire force
Passenger comfort: Probability (RMS) acceleration will exceed acceptable threshold
~~ ~
21 2 2 3/22 2 2 2
~ ~
2
ln( ) ln( ) (ln( )[ ( ), ( ) | , ](
[
ln
( ), ( ) |
( ) ln( ))
, ]
ac
acac ac
Racac acGPM
GPMft
MSb RMSb RMS b
f ft
RMS
r
RMSRMS b RMSh
h SD S
b
D
xx x
x x
z
z x
z y σ y x θ
z y σ y x θ
Illustrative example II
1 2arg min{ ( ), (
(
)}
( ) , ) ( )
GPM GPM
GP
PX
iM
iΘh p d
H H
H
x
x θ θ
x x
x θ
X
Illustrative results I
H1(x)
H2(
x)
Illustrative sample iteration
H1(x)
H2(
x)
Pareto front across iterations (performance evaluated by metamodel)
Illustrative results II
H1(x)
H2(
x)
Pareto front across iterations (performance evaluated by actual model)
Illustrative results III
H1(x)
H2(
x)
Comparison of pareto front to benchmark solution
3500 model evaluations
365000 model evaluations
Illustrative results IV
3500 model evaluations 2200 model evaluations
Illustrative results V
H1(x)
H2(
x)
Müller, J. (2017). SOCEMO: Surrogate Optimization of Computationally Expensive Multiobjective Problems. INFORMS Journal on Computing, 29(4), 581-596
3500 model evaluations
365000 model evaluations
Reference solutions
SOCEMOProposed approach
10000 model evaluations
Illustrative results VI
Outline
• Motivation (why use metamodels?)
• Surrogate modeling overview
• Design under uncertainty using metamodels
• Iterative metamodel implementation for multi-objective design under uncertainty
• Iterative metamodel implementation for stochastic sampling
Model
nΘ θ
θ1
θ2
Stochastic sampling I
PDF p(θ)
( )z θ
response
( )h θsystem function
“Performance” evaluation
Target probability density
( ) ( )( )= ( ) ( )( ) ( )
Θ
h p h ph p d
θ θθ θ θθ θ θ
θ1
θ2
Target PDF π(θ)
Examples
Importance Sampling (IS),
Posterior (Bayesian) analysis, Subset
Simulation (SS) …
How to sample from the target distribution when the response is evaluated under complex numerical model (simulator)?
Different approaches exist for stochastic sampling Rejection Sampling Markov chain Monte Carlo (MCMC) …
Higher efficiency requires the proposal density q(θ) being close to the target PDF π(θ)
Typically non-trivial task for tough (rare event, peaked posterior) target distributions
Use samples (trials) from some proposal density q(θ) to obtain
samples from target density π(θ)
EfficiencyIssue
[number of trialsneeded to obtain one equivalent independent sample]
Stochastic sampling II
• Construct a series of intermediate densities between p(θ) and π(θ)
• Smaller change between adjacent densities possible to efficiently sample from πj (θ) using πj-1(θ) as proposal density
Sequential sampling
π0(θ) ≡ p(θ)
πn(θ) ≡ π(θ)
π1(θ)
πj(θ)
πj+1(θ)
SubsetSimulation,CrossEntropyIS,…
TransitiveMCMC…
nΘ θ
θ1
θ2
Metamodel-aided stochastic sampling
PDF p(θ)
( )z θ
Predicted response
( )GPMh θ
Approximated target probability density
( ) ( )( )= ( ) ( )( ) ( )
GPMGPM G
Θ
PMGPM
h p h ph p d
θ θθ θ θθ θ θ
θ1
θ2
Approximated Target PDF πGPM(θ)
Surrogatemodel
θ1θ2
z
Predictive system function
“Performance” evaluation
( )z θ
Metamodelj
πn(θ) ≡ π(θ)
π1(θ)
πj(θ)
πj+1(θ)
π0(θ) ≡ p(θ)
Metamodel0
Metamodelj+1
Metamodeln
Global metamodel
Sequential sampling
1st iteration (k= 1)
InitializationGet initial experiments and
evaluate response
Utilize all available training points to formulate the GPM
Stochastic sampling to simulate samples {θ}s
(k)
from πGPM(k)(θ)
Iterative metamodel-aided stochastic sampling
Iterative implementation, to gradually converge to target
density
Preference for rejection sampling o Independent samples
o Exploit metamodel capability in providing vectorised predictions
1st iteration (k= 1)
InitializationGet initial experiments and
evaluate response
Utilize all available training points to formulate the GPM
Stochastic sampling to simulate samples {θ}s
(k)
from πGPM(k)(θ)
Iterative metamodel-aided stochastic sampling
Stochastic sampling to obtain samples from π(θ) using πGPM(θ)
as proposal density
Stopping criteriaSatisfied?
k=k+1no
Proceed to next iteration
Perform refinement DoE and evaluate
response
STOP
Iterative implementation, to gradually converge to target
density
yes
Zhang, J. and A.A. Taflanidis (2017). “Adaptive Kriging stochastic sampling and
density approximation and its implementation to rare-event estimation”. ASCE-ASME
Journal of Risk and Uncertainty in Engineering Systems, in press.
Stopping criteria
2( ) ( 1) ( ) ( 1) 21ˆ ( ), ( ) [ ( ) ( )]2
GPM k k k kGPM GPM GPMH Θ
D d θ θ θ θ θ Hellinger distance
Approximated target PDF in kth iteration πGPM(k)(θ)
Approximated target PDF in k-1th iteration πGPM(k-1)(θ)
Bounded symmetric metric ∈[0, 1], where higher value → higher discrepancy
Stopping criteria
2( ) ( 1) ( ) ( 1) 21ˆ ( ), ( ) [ ( ) ( )]2
GPM k k k kGPM GPM GPMH Θ
D d θ θ θ θ θ Hellinger distance
Approximated target PDF in kth iteration πGPM(k)(θ)
Approximated target PDF in k-1th iteration πGPM(k-1)(θ)
2( ) ( 1)
( ) ( 1) ( ) ( 1)1
1 1 ( ) ( )2 ( |{ } ,{ } ) ( |{ } ,{ } )
dN k r k r
r k k r k krd d s
GPM G
s d
P
s s
M
N q q
θ θ
θ θ θ θ θ θ
IS density qd for estimation based on samples from πGPM(k)(θ) [{θ}s
(k)] and πGPM(k-1)(θ) [[{θ}s(k-1)]
DoE strategy I: Target-based
Objective: enhance the metamodel accuracy around the sampling domain of interest (target PDF)
– Preference function: πp(θ) =πGPM(k)(θ)
– Utility metric: predicted variance of system function (h) under response prediction (z) metamodel uncertainties
“Performance” Function
Propagating the uncertaintiesp[z(θ)] p[h(θ)]
2
1 ( )
[VAR ( , ) ( )| ]z
j j
ni
M i ij j z z
hhz
θ
z θx θ θ
DoE strategy II: Response-based
Objective: enhance the metamodel accuracy around domains with insufficient information about metamodel output (response)
– Preference function: πp(θ) =πjGPM(k)(θ)
– Utility metric: differential entropy between prior/ posterior response prediction
Obtain the information gain by differential entropy DEz(θ)
Posterior (predictive distribution)Prior
(process variance) Lower DEz(θ)less information extracted from experiments for point (new
information might change predictions)
2 2
1 1
2 2
1
1DE ( ) [log(2 )] ( ( ))2
exp [log( ) log( ( ))]
z zz
z
n nn
i ii i
n
i ii
e
z θ θ
θ
• Combines the target-based and response-based design of experiments
•Simulate large number of samples from target-basedDoE PDF•Maintain only the samples that have high VARM [h(θ)]
Hybrid DoE
•Simulate large number of samples from response-based DoE PDF•Maintain only the samples that have small DEz(θ)
Combine, cluster and maintain only nearest neighbor to the centroids
Final Hybrid DoE
formulating an IS based on πGPM(θ) and relying again on established metamodel [AK-MCS (Echard et al. 2011)]
formulating an IS based on πGPM(θ) and relying on actual model [eg meta-IS (Dubourg et al. 2013), AM-SIS (Pedroni and Zio 2015)]
Once convergence to πGPM(θ) has been established we can calculate P(F)
1
1 ( ) ( )ˆ ( ) ; ~ ( )( )
cN GPM r rGPM r r
crrc c
h pP F qN q
θ θ θ θθ
1
1 ( ) ( )( ) ; ~ ( )( )
cN r rr r
crrc c
h pP F qN q
θ θ θ θθ
Based on πGPM(θ)
(samples)
IS-based rare event estimation
( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( | )( )( ) ( )
FΘ
F F
FΘ
P F I p d
I p I pp FP FI p d
θ θ θ
θ θ θ θθ θθ θ θ
[ ]Tlf lrC Cx
Illustrative examplesP1
P2 P3 P4 P5 P6A1, E1
A1, E1
A2, E2
ul um ur
AK-SSD applied with
addition of 2nθexperiments per
iteration
Case 1: Failure related to um
Case 2: Failure related to um,ur
and ul
server
uy kp
Isolator Restoring force
Isolator displacement
Ground Motion
(Stochastic model incorporating near fault effects)
δi
Fy,i
ki
-δy,i
Restoring force Fi for ith story as function of drift δi
δy,i
aiki
-Fy,i
Fim4=400 ton
k1 =460 MN/m
m3=500 ton
m2=500 ton
m1=500 ton
k2 =368 MN/m
k3 =276 MN/m
k1 =184 MN/m
kl
nθ=10 variables
Truss example
FIS-example
nθ=36 variables
Results IHellinger distance between approximated target PDFs in subsequent iterations
• Large initial improvement and then plateau reached in latter iterations small discrepancy between distributions less impactful refinement DoE
Results II
• Decreasing trend approximated distribution approaches actual one
Hellinger distance between approximated target PDFs and actual target PDF across k
• Large initial improvement and then plateau reachedconvergence of approximated PDFs, good indicator for convergence to target PDF
Results IIIFailure probability estimates using metamodel or exact numerical model
ˆ ( )GPMP F
ˆ ( )GPMP F
ˆ ( )GPMP F
Results IV
• Large initial improvement and then plateau reached as convergence establishedconvergence of approximated PDFs good indicator for efficiency of IS densities
Efficiency of second stage (N required to get cov of 5%) if samples from πkrig(k)(θ)were to be used to form IS densities
: estimated failure probability fully based on Kriging metamodel: estimated failure probability using stochastic simulation
n : total high-fidelity simulations needed for constructing Kriging metamodelNtot : total high-fidelity simulations needed to establish a stochastic-simulation
MCS: Direct Monte CarloPCE: Polynomial-Chaos Kriging approach (PCE) by (Schöbi et al. 2016)SS-AKSD: SS with adaptive kernel sampling densities (SS-AKSD) by (Jia et al. 2015)
ˆ( )P Fˆ ( )krigP Fˆ( )P Fˆ ( )krigP F
Results V
Conclusions
• Surrogate modelling can facilitate significant computational benefits for UQ applications.
• Gaussian process metamodels are especially relevant in this context due to their holistic treatment of uncertainty in metamodel predictions.
• Metamodel accuracy should be adaptively controlled (coupled with adaptive DoE) with goal to converge to accurate solutions leveraging (perhaps) inaccurate models (but accurate enough in domains of interest)
• For design uncertainty development of metamodels in augmented input space can facilitate additional computational benefits
• Sample-based DoE is an efficient approach for identifying new experiments and can provide significant accuracy improvement