Distribution-Free Polynomial Chaos Expansion Surrogate ......Distribution-Free Polynomial Chaos Expansion Surrogate Models for Eﬃcient Structural Reliability Analysis HyeongUk Lim1

Distribution-Free Polynomial Chaos Expansion

Surrogate Models for Efficient Structural Reliability Analysis

HyeongUk Lim1 and Lance Manuel21PhD Candidate 2Texas Atomic Energy Research Foundation Professor of Engineering

Mechanics, Uncertainty and Simulation in Engineering (MUSE)The University of Texas at Austin

Engineering Mechanics Institute Conference 2019California Institute of Technology (June 18-21, 2019)

Dist-free PCE for Structural Reliability 1 / 14

Outline

1 Background: Structural Reliability

2 Structural Reliability by Distribution-free PCE

3 Benchmark Results

4 PCE on Reduced Dimension


Background: Structural Reliability


� Important: Accurate prediction of probability of failure for structural safety





� Limit state functions can involve the use of expensive computational models





� Limit state functions can involve the use of expensive computational models

� Can benefit from the use of “surrogate” functions that serve as approximations forthe “truth” limit state functions.


Structural Reliability by Distribution-free PCE Probability of Failure

Pf by Surrogate Models

Probability of failure using truth limit state function:

Pf = P [g ≤ 0] ≈1

N

N∑

i=1

I[g(x(i)) ≤ 0]





Pf = P [g ≤ 0] ≈1

N

N∑

i=1

I[g(x(i)) ≤ 0]

Probability of failure using surrogate limit state function:

P̂f = P [ĝ ≤ 0] ≈1

N

N∑

i=1

I[ĝ(x(i)) ≤ 0]





Pf = P [g ≤ 0] ≈1

N

N∑

i=1

I[g(x(i)) ≤ 0]

Probability of failure using surrogate limit state function:

P̂f = P [ĝ ≤ 0] ≈1

N

N∑

i=1

I[ĝ(x(i)) ≤ 0]

Appropriate development of ĝ is needed to yield:

Pf ≈ P̂f


Structural Reliability by Distribution-free PCE PCE: Polynomial Chaos Expansion

Generalized Spectral Approach

g can be an expansion of a series of the basis functions:



Generalized Spectral Approach

g can be an expansion of a series of the basis functions:

Any function of inputs can be represented by orthogonal basis functions in input-space



PCE: Askey scheme

A PCE surrogate for g is:

g(X) ≈ ĝ(X)︸︷︷︸

PCE surrogatelimit state function

=

P−1∑

i=0

ciΨi(T (X)︸︷︷︸

Q

)

Formal approach (Askey scheme): Pre-defined orthogonal polynomial family, Ψ(.),associated with independent Q

ex) Gaussian variables ⇒ Hermite polynomials



PCE: Askey scheme


g(X) ≈ ĝ(X)︸︷︷︸


=

P−1∑

i=0


Q

)



How about non-standard random variables?



PCE: Askey scheme


g(X) ≈ ĝ(X)︸︷︷︸


=

P−1∑

i=0


Q

)



How about non-standard random variables?How about dependent random variables?



PCE: Askey scheme


g(X) ≈ ĝ(X)︸︷︷︸


=

P−1∑

i=0


Q

)




ex) extreme value distributions, jointly distributed variables



PCE: Askey scheme


g(X) ≈ ĝ(X)︸︷︷︸


=

P−1∑

i=0


Q

)




ex) extreme value distributions, jointly distributed variables

We do “iso-probabilistic transformation”, T : X → Q, but it limits PCE use



Reasons Why T Limits PCE

g(X)truth modelin X domain

= g(Q)truth modelin Q domain

≈ ĝ(Q)surrogate

in Q domain






≈ ĝ(Q)surrogate

in Q domain

- T can be nonlinear (or given implicitly)






≈ ĝ(Q)surrogate

in Q domain


- g(Q) can be more complex than g(X)






≈ ĝ(Q)surrogate

in Q domain


- g(Q) can be more complex than g(X)

- ĝ(Q) aims to fit g(Q), not g(X)


Structural Reliability by Distribution-free PCE Refinement for non-standard and dependent variables

Arbitrary Polynomial Chaos Expansion (APCE)

Proposed:

ĝ(X) =

P−1∑

i=0

ciΨi(X)

To avoid T , basis polynomial functions, Ψi, in X space




Proposed:

ĝ(X) =

P−1∑

i=0

ciΨi(X)


How?




Proposed:

ĝ(X) =

P−1∑

i=0

ciΨi(X)


How? ⇒ Gram-Schmidt orthogonalization using joint moments



Toy Example with Non-standard and Dependent Variables

Consider g(x1, x2) = 12 + 4x1 + 4x2 + x1x2 + 2x1 sinx2 with

X1 : lognormal and Weibull combined

X2 : lognormal conditional on X1

jpdf : fX1,X2(x1, x2) = fX1(x1)fX2|X1(x2|x1)








0.0015 15

10 10

x1 x2

5 5

0 0

0.05

fX

1,X

2(x

1,x

2)

0.10








0.0015 15

10 10

x1 x2

5 5

0 0

0.05

fX

1,X

2(x

1,x

2)

0.10

0 100 200 300 400

b

10−5

10−4

10−3

10−2

10−1

100

P(g

>b)

TruthHPCE (Order= 2)

0 100 200 300 400

b

10−5

10−4

10−3

10−2

10−1

100

P(g

>b)

TruthAPCE (Order= 2)


Benchmark Results Example: Offshore floater - computational time-domain solver

Example: Offshore floater - extreme response

g = z − ZT (X)ZT is the T -year long-term extreme response, involving:

Image source: Bluewater, https://www.bluewater.com




g = z − ZT (X)ZT is the T -year long-term extreme response, involving:(1) long-term (environment) uncertainty(2) short-term time-domain simulations: ZT = max{u(t); 0 ≤ t ≤ T}





u(t) = uWF(t) + uLF(t) ≡ u(Hs, Tp, a1, ..., a960, θ1, ..., θ960, t)Dist-free PCE for Structural Reliability 10 / 14




0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

P[Z

T>

z]

MCSAPCE (p= 2)Mean of APCE

0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

P[Z

T>

z]

MCSHPCE (p= 2)Mean of HPCE





0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

P[Z

T>

z]


0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

P[Z

T>

z]


Selected X1 (≡ Hs) and X2 (≡ Tp) and build surrogate models





0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

P[Z

T>

z]


0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

P[Z

T>

z]


Selected X1 (≡ Hs) and X2 (≡ Tp) and build surrogate models

Can we include all the variables to make a PCE surrogate?


PCE on Reduced Dimension

Finding Important Direction in Inputs

How to do Dimension Reduction?



Finding Important Direction in Inputs

How to do Dimension Reduction?

C: covariance (uncentered) matrix of gradients

C ≡ E[(∂Z

∂x

)(∂Z

∂x

)T]

= WΛWT

W = [Wa, Wz]T; y = Wa

Tx



Example: Extreme response using Dimension Reduction

- The 1922-dimension variable vector is rotated to yield

the dominant 4-dimensional vector that best describes the response variability

- # of samples on reduced dimension: 105

- # of samples on original coordinate: 3.56× 109


Conclusions

Concluding Remarks

� Surrogates in structural reliability can be built using extensions to conventionalPolynomial Chaos Expansion (PCE)

� Distribution-Free PCE or Arbitrary PCE (APCE) - more efficient and accuratethan plain PCE when no obvious Askey scheme is available and avoids dependencestructure complications

� PCE on Reduced Dimension - useful for high-dimensional structural reliabilityproblems where Monte Carlo methods are inefficent


Conclusions

Concluding Remarks

� Surrogates in structural reliability can be built using extensions to conventionalPolynomial Chaos Expansion (PCE)

� Distribution-Free PCE or Arbitrary PCE (APCE) - more efficient and accuratethan plain PCE when no obvious Askey scheme is available and avoids dependencestructure complications

� PCE on Reduced Dimension - useful for high-dimensional structural reliabilityproblems where Monte Carlo methods are inefficent

Thank you very much for your attention! Any questions?


Backup slides

Framework: Structural reliability


Backup slides

Construction of Orthogonal Polynomials, ΨαConsider g(x1, x2) = 12 + 4x1 + 4x2 + x1x2 + 2x1 sinx2APCE: ĝ =

∑

α∈N2cαΨα(X), α = [α1, α2]T: multi-indices

Ψα=[0,0] = 1

Ψα=[1,0] = det[

1 E[X1]1 x1

]

= x1 − E[X1]

Ψα=[0,1] = det[

1 E[X2]1 x2

]

= x2 − E[X2]

Ψα=[2,0] =1

C2,0det

1 E[X1] E[X2] E[X21 ]E[X1] E[X21 ] E[X1X2] E[X

31 ]

E[X2] E[X1X2] E[X22 ] E[X21X2]

1 x1 x2 x21

Ψα=[1,1] =1

C1,1det

1 E[X1] E[X2] E[X1X2]E[X1] E[X21 ] E[X1X2] E[X

21X2]

E[X2] E[X1X2] E[X22 ] E[X1X22 ]

1 x1 x2 x1x2

Ψα=[0,2] =1

C0,2det

1 E[X1] E[X2] E[X22 ]E[X1] E[X21 ] E[X1X2] E[X1X

22 ]

E[X2] E[X1X2] E[X22 ] E[X32 ]

1 x1 x2 x22

Not just marginal moment products when dependent variables are considered to build Ψα


Backup slides

Steps for PCE on Active Subspace

(1) Estimate C using multiple sets of Morris’s elementary effect for each variable

(2) Identify dominant eigenvectors

(3) Build a PCE model using dataset, D = { yi︸︷︷︸

active variables=Wa

Txi

, Zi︸︷︷︸

extreme response

}NEi=1

- Order, p- Estimation of coefficients using D

(4) Model evaluations

- Comparison of P (Ẑ > z) against P (Z > z)


Backup slides Example: Correlated non-normal variables

Example: Correlated non-normal variables

gX(x) = b− (x1 − x2)X1: uniform [0 to 100], X2: unit-mean exponential, ρ12 = 0.5.


Backup slides Example: Correlated non-normal variables

Example: Correlated non-normal variables

gX(x) = b− (x1 − x2)X1: uniform [0 to 100], X2: unit-mean exponential, ρ12 = 0.5.

70 80 90 100

b

10−5

10−4

10−3

10−2

10−1

Pf

MCS

70 80 90 100

b

10−5

10−4

10−3

10−2

10−1

Pf

APCE (p= 1)

70 80 90 100

b

10−5

10−4

10−3

10−2

10−1

Pf

HPCE (p= 23)

Pf estimates for different b values from 10 MCS, APCE, and HPCE sets


Backup slides Example: Multimodal random variables

Example: Multimodal random variables

gX(x) = b− (sinx1 + 7 sin2 x2 + 0.1x43 sinx1)Xi (i = 1, 2, 3): a mixture distribution with pdf





f(x) =3∑

i=1

wiφi(x)

wi are set to 1/3 and φi are normal distribution pdfswith means and standard deviations: (µ, σ) = (2.0, 0.1), (2.5, 0.5), (3.5, 0.2)





f(x) =3∑

i=1

wiφi(x)

wi are set to 1/3 and φi are normal distribution pdfswith means and standard deviations: (µ, σ) = (2.0, 0.1), (2.5, 0.5), (3.5, 0.2)

10 20 30 40 50

b

10−5

10−4

10−3

10−2

10−1

Pf

MCS

10 20 30 40 50

b

10−5

10−4

10−3

10−2

10−1

Pf

APCE (p= 7)

10 20 30 40 50

b

10−5

10−4

10−3

10−2

10−1

Pf

JPCE (p= 10)

Pf estimates for different b values from 10 MCS, APCE, and JPCE sets


Backup slides Example: Mixed discrete-continuous support

Example: Mixed discrete-continuous support

gX(x) = b− (15 + 4x1x2 + 4x1x3 + 4x2x3 + 3x1 + 3x2 + 3x3 − x21 − x22 − x23)Xi (i = 1, 2, 3): a mixed discrete-continuous pdf





fX(x) = 0.7( 1√

2πexp(−x

2

2))+ 0.3δ(x− 2.0)





fX(x) = 0.7( 1√

2πexp(−x

2

2))+ 0.3δ(x− 2.0)

� Discrete at x = 2 ⇒ cumulative probability bumps up� HPCE cannot handle well the discontinuity





fX(x) = 0.7( 1√

2πexp(−x

2

2))+ 0.3δ(x− 2.0)

� Discrete at x = 2 ⇒ cumulative probability bumps up� HPCE cannot handle well the discontinuity

40 60 80 100 120

b

10−5

10−4

10−3

10−2

10−1

Pf

MCS

40 60 80 100 120

b

10−5

10−4

10−3

10−2

10−1

Pf

APCE (p= 2)

40 60 80 100 120

b

10−5

10−4

10−3

10−2

10−1

Pf

HPCE (p= 3)

Pf estimates for different b values from 10 MCS, APCE, and HPCE sets


Background: Structural ReliabilityStructural Reliability by Distribution-free PCEBenchmark ResultsPCE on Reduced Dimension

Documents

Distribution-Free Polynomial Chaos Expansion Surrogate ......Distribution-Free Polynomial Chaos Expansion Surrogate Models for Eﬃcient Structural Reliability Analysis HyeongUk Lim1