non-parametric surrogate models for uncertainty quantification … · 2020-02-24 · non-parametric...

Non-Parametric Surrogate Models forUncertainty Quantification in Structural Vibration

HyeongUk Lim (student of Dr. Lance Manuel)

The University of Texas at Austin

18th Dec 2018

HyeongUk Lim (UT Austin) UQ in Structural Vibration 18th Dec 2018 1 / 59

Outline

1 Introduction

2 Surrogate Model: Polynomial Chaos Expansion (PCE)

3 Engineering ApplicationsFatigue Damage in a Marine RiserLong-term Extreme Response of a Moored StructureUncertainty in Floor Vibration SystemStructural Reliability using Surrogate Models

4 Conclusions


About Me; Memory in KICT

� B.E., Kyung Hee University, 2006-2012

� M.S., KAIST, 2012-2014

� Research internship during 10/2014-05/2015

- Worked in Geotech Department- With many senior researchers- For on-line rock mass classification

� PhD candidate, UT Austin, 2015-2019 summer (expected)

Major: Civil Engineering, focused on numerical simulations using applied probabilities

Interests: Structural Vibration and Reliability via Uncertainty Quantification

Applications: Offshore structures, Wooden floors, and actively searching for others . . .


Introduction

Engineering Complex System Requires Prediction

Engineering complex system requires reliable prediction (e.g., by computer simulations)


Introduction

Purpose of Computer Simulation

� To inform some decision (for design, operations, or control)

- Quantities are predicted to inform decision- These are Quantities of Interest (QoIs)- In most cases, need to predict QoIs when confirming observational datais not available

� Computational model is not scientific theory

- Their validity depends on their purpose, such as- (1) nature of QoIs and (2) required accuracy

We need validation of computational models to predict QoIs well

How? ⇒ by Uncertainty Quantification (UQ)


Introduction

Validation Requires UQ

� Validation: systematic comparison of model outputs to measurements

- Confirm model is ”close” to observations- What about the unobserved QoI?

� Such comparisons are necessary but not sufficient for prediction

- Predictions are necessarily an extrapolation from available information- Need to assess confidence in the extrapolation

Validation requires UQ:

- How close between observations and model outputs?

- How strong is a validation test?

- Uncertainty in data, models and model parameters

(Ref: ”Validating Computational Predictions of Unobserved Quantities” by R. D. Moser)


Introduction

Uncertainty Quantification

� Uncertainties in “inputs” are modeled by PDFs:

X ∼ fX

� A computation model relates inputs to any “response” of interest:

M : y ≡ M(x)

� Uncertainty Quantification seeks statistics of Y

e.g., E[Y ]︸︷︷︸mean

, Var(Y )︸︷︷︸variance

, fY︸︷︷︸pdf

, FY︸︷︷︸cdf

, P (Y > b)︸︷︷︸exceedance

� Sensitivity analysis: to assess dominant sources of uncertainty


Introduction

Sources of Uncertainty

� Aleatory and Epistemic:

- natural randomness (e.g., ocean waves, material variability)- natural hazards (earthquakes, floods, landslides)- exceptional service loads- climate loads (hurricanes, snow storms, hail)- accidental human actions (explosions, fire)- lack of knowledge, limited data, imperfect models


Introduction

Computational Model: M

Complex physical systems assessed using computational models

Mathmodel

RealityComputationalmodel: M

error

uncertainty

error

(1) Validation + UQ (2) Verification

Two considerations:

� Mathematical idealization of physical phenomenon

� Computational implementation of numerical solution for math model


Introduction

Surrogate Model: M

A surrogate model, M, is an approximation of the computational model,M, that is constructed, based upon:

� some runs of the computation model (truth/original)

� assumed characteristics of the surrogate (e.g., polynomial form)

M(x)original

≈ M(x)surrogate

minutes or hours per a runvs.

seconds per 106 runs

Challenge: Surrogates require rigorous validation (against the “truth”)


Surrogate Model: Polynomial Chaos Expansion (PCE)

PCE: Formulation and Computation of Coefficients

Formation of PCE:

M(x) =P∑

i=0

ciΨi(x),

Parameters to be chosen carefully:ci, P , and also Ψi

Estimation of ci:

� Need a set of samples, D = {xj , yj}Nj=1

� Post-process of the set, {M(xj), yj}Nj=1

Estimation of P : Heuristics

Ψi: Askey scheme (parametric)


Surrogate Model: Polynomial Chaos Expansion (PCE)

PCE by Non-parametric Way

Formation of PCE:

M(x) =P∑

i=0

ciΨi(x),

When a limited number of data is available, Ψ can be determined by anon-parametric way, so called Arbitrary PCE (APCE):

� Using the sample raw moments of random variables

� And Gran-Schmidt orthogonalization

� Can be generalized for dependent random variables

Challenge:

� The number of available data matters


Engineering Applications Fatigue Damage in a Marine Riser

Application 1: Fatigue Damage in a Marine Riser



Introduction

� Fatigue damage in risers undergoing vortex-induced vibration (VIV)remains a challenging problem ⇒ great uncertainty associated with it.

� A distributed wake oscillator model (Srinil, 2010) has been proposedand its effectiveness demonstrated for different flow conditions andriser configurations.

� An Uncertainty Quantity (UQ) framework is presented – using up to 3random variables – to assess fatigue damage uncertainty.

� Case studies with different UQ surrogate models are compared vs.Monte Carlo simulation, while addressing efficiency and accuracy.



Structure Considered

V

L

X

Y Z

TR

Top-tensioned riser in uniform flow

Truth system:

� Governing PDEs ⇒ Standard Galerkinmethod

� Obtain displacement time series at xfrom the top

� Estimate fatigue damage usingpost-processing

Random variable inputs:

� ∆Amax, ∆ω: empirical coefficients

� V : current velocity

Validation:

� 20,000 runs of the truth system

� Exceedance probabilities



Runs of Truth System

Fatigue damage is estimated at x = L/4, L/2, and 3L/4 from the top:

0 10 20 30 40

Time (s)

-2

-1

0

1

2

v(x,t)

x = L/4x = L/2x = 3L/4

� Measurement data (used in PDEs) collected over several references

� Fit data to the empirical curves



Random Variables: ∆Amax, ∆ω

0.0 1.0 2.0 3.0 4.0

SG

0.0

0.5

1.0

1.5

2.0

Amax

MeasurementsFitted curve

0.0 1.0 2.0 3.0 4.0

SG

1.0

1.1

1.2

1.3

1.4

1.5

1.6

ωs,A/ω

n

MeasurementsFitted curve

� Measurement data (used in PDEs) collected over several references

� Fit data to the empirical curves



Random Variables: ∆Amax, ∆ω

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

∆A

0.0

1.0

2.0

3.0

4.0

pdf

DataSGLD

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

∆ω

0.0

1.0

2.0

3.0

4.0

pdf

DataSGLD

� Modeling the residuals by Shifted Generalized Lognormal Distribution(SGLD)



Monte Carlo Simulations

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80

∆Amax

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

∆ω

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80

∆Amax

0.30

0.35

0.40

0.45

0.50

V

-0.30-0.20-0.10 0.00 0.10 0.20 0.30 0.40

∆ω

0.30

0.35

0.40

0.45

0.50

V

� Monte Carlo simulations with(out) correlation between variables

� Dots go to red as fatigue damage is high

� Fatigue damage is obtained by the rainflow cycle counting



MCS vs. PCE

0 1 2 3 4 5 6 7 8 9 10

D

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

pdf

MCSPCE (Order= 5, N= 125)

0.1 1.0 10.0

Quantile of D (MCS)

0.1

1.0

10.0

Quan

tile

ofD

(PCE)

PCE (Order= 5, N= 125)

0 1 2 3 4 5 6 7 8 9 10

d

10−5

10−4

10−3

10−2

10−1

100

GD(d)

MCS (99% CI)PCE (Order= 5, N= 125)

� Comparison of pdf, QQ plots, and exceedance probability plots

� 10 sets of MCS and PCE

� PCE: order-5 (P ) and 53 quadrature points (for ci)


Engineering Applications Long-term Extreme Response of a Moored Structure

Application 2: Long-term Extreme Response of a MooredStructure



Introduction

� Long-term analysis using Monte Carlo Simulation (MCS) is expensive.

� Polynomial Chaos Expansion (PCE) framework for predictinglong-term surge motion extreme response of a moored floatingstructure.

� We consider uncertainty in (1) environmental variables and (2)time-varying short-term loads.

� Validation studies involve PCE vs. MCS long-term response on thesimple moored floating structure.



Problem Formulation

Mu(t) + 2ζ√KMu(t) +Ku(t) = FWF (t) + FLF (t)

F (1)(ω) = T (1)(ω)η(ω)

F (2)(ωr, ωs) = T (2)(ωr, ωs)η(ωr)η(ωs)

u(t) =R∑

r=1

FRF︷︸︸︷H(ωr)

Transfer︷︸︸︷T (1)(ωr)

Wave amplitude︷︸︸︷Are

iωrt +

R∑

r=1

R∑

s=1r 6=s

H(ωr − ωs)T(2)(ωr, ωs)ArA

∗se

i(ωr−ωs)t

QoI: max{u(t), T = 30 min}



Comparison of Exceedance Probability

� Lower probability level is of concern.

� More samples for PCE show small error in the estimation of PoEs.



Probability of Exceedance

0 2 4 6 8

z (m)

10−5

10−4

10−3

10−2

10−1

100

GZ(z)

MCSPCE

0 2 4 6 8

z (m)

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

GZ(z)

MCSPCEPCE-IS

� PCE model with order-3 and 100 samples estimate PoEs well.

� Importance sampling can be applied to PCE as well.


Engineering Applications Uncertainty in Floor Vibration System

Application 3: Uncertainty in Floor Vibration System



Introduction

� Wooden multi-story buildings have increased their market share inEurope during the last decades.

� Wooden buildings are more sensitive to dynamic loads at lowfrequencies due to the variability in the material properties of wood.

� For example, local variations occur even if wooden members are takenfrom the same batch, because wood is a natural material.

� The uncertainty in modal frequencies is studied.



Introduction



Mode shapes: simply supported at two edges

An wooden floor system with 7 joists:

U, Magnitude

+1.453e−03+8.467e−02+1.679e−01+2.511e−01+3.343e−01+4.175e−01+5.007e−01+5.839e−01+6.672e−01+7.504e−01+8.336e−01+9.168e−01+1.000e+00

Step: EigenfrequenciesMode 1: Value = 35454. Freq = 29.967 (cycles/time)Primary Var: U, MagnitudeDeformed Var: U Deformation Scale Factor: +2.470e−01

ODB: Inp_file_No_1.odb Abaqus/Standard 3DEXPERIENCE R2017x Mon Jul 30 00:47:26 Central Daylight Time 2018

X

Y

Z

U, Magnitude

+1.611e−04+8.348e−02+1.668e−01+2.501e−01+3.334e−01+4.168e−01+5.001e−01+5.834e−01+6.667e−01+7.501e−01+8.334e−01+9.167e−01+1.000e+00



X

Y

Z

U, Magnitude

+1.655e−03+8.486e−02+1.681e−01+2.513e−01+3.345e−01+4.177e−01+5.009e−01+5.841e−01+6.673e−01+7.505e−01+8.337e−01+9.169e−01+1.000e+00



X

Y

Z

U, Magnitude

+4.465e−04+8.374e−02+1.670e−01+2.503e−01+3.336e−01+4.169e−01+5.002e−01+5.835e−01+6.668e−01+7.501e−01+8.334e−01+9.167e−01+1.000e+00



X

Y

Z

30 Hz 31 Hz 35 Hz 40 HzU, Magnitude

+3.093e−04+8.468e−02+1.690e−01+2.534e−01+3.378e−01+4.222e−01+5.065e−01+5.909e−01+6.753e−01+7.596e−01+8.440e−01+9.284e−01+1.013e+00



X

Y

Z

U, Magnitude

+1.104e−05+9.064e−02+1.813e−01+2.719e−01+3.625e−01+4.531e−01+5.438e−01+6.344e−01+7.250e−01+8.156e−01+9.063e−01+9.969e−01+1.088e+00

Step: EigenfrequenciesMode 6: Value = 1.17269E+05 Freq = 54.502 (cycles/time)Primary Var: U, MagnitudeDeformed Var: U Deformation Scale Factor: +2.470e−01


X

Y

Z

U, Magnitude

+1.206e−03+1.052e−01+2.092e−01+3.132e−01+4.172e−01+5.212e−01+6.252e−01+7.292e−01+8.332e−01+9.372e−01+1.041e+00+1.145e+00+1.249e+00



X

Y

Z

U, Magnitude

+3.577e−04+1.146e−01+2.289e−01+3.431e−01+4.574e−01+5.716e−01+6.859e−01+8.001e−01+9.144e−01+1.029e+00+1.143e+00+1.257e+00+1.371e+00



X

Y

Z

46 Hz 55 Hz 66 Hz 76 Hz



UQ for modal frequencies: using SS (LHS) and MCS

SS = Systematic Sampling (Stratified using Latin Hypercube)

20 30 40 50 60 70 80

frequency (Hz)

0

1

2

3

pdf

20 30 40 50 60 70 80

frequency (Hz)

0

1

2

3

pdf

� (top) 40 SS vs.(bottom) ≈10,000 MCS

� densities from CDFs at0.0125, 0.0375, . . ., 0.9875

� SS - good match with MCS



Extremes - Tail Probabilities

26 28 30 32

f

10−4

10−3

10−2

10−1

100

P(f

1>

f)

p= 1, N= 24

MCSPCE

26 28 30 32

f

10−4

10−3

10−2

10−1

100

P(f

1>

f)

p= 2, N= 108

MCSPCE

26 28 30 32

f

10−4

10−3

10−2

10−1

100

P(f

1>

f)

p= 3, N= 360

MCSPCE

� (top) order-1, (mid) 2, (bottom) 3

� Samples for MCS & PCE: ≈ 10,000

� 10 sets of MCS & PCE(bootstrap resamplings for MCS)

� order-3 PCE suggests “convergence”


Engineering Applications Structural Reliability using Surrogate Models

Application 4: Structural Reliability using SurrogateModels



Summary for Examples

Ex 1: gX(x) = x1 + 2x2 + 2x3 + x4 − 5x5 − 5x6 + 0.0016∑

i=1

sin(100xi)

Ex 2: gX(x) = 1.1 − 0.00115x1x2 + 0.00157x22 + 0.00117x2

1 + 0.0135x2x3

−0.0705x2 − 0.00534x1 − 0.0149x1x3 − 0.0611x2x4

+0.071x1x4 − 0.226x3 + 0.0333x23 − 0.558x3x4 + 0.998x4 − 1.339x2

4

Ex 3: gX(x) = x1 − x2

Ex 4: gX(x) =6x1x3(x2−x3)x5

x2

(x6x3

5−(x6−x7)(x5−2x8)3) − x4

Ref Total Degree # of RVs Dependency Type of RVs Purpose

Ex 1 Liu and 1 6 no all lognormals noisy limit stateDer Kiureghian (1991)

Ex 2 Liu and 2 4 no X1 : type II extreme quadratic functionDer Kiureghian (1991) X2,3 : normal

X4 : lognormal

Ex 3 Der Kiureghian 1 2 yes, X1 : uniform dependent RVsand Liu (1986) ρX1X2

= 0.5 X2 : exponential

Ex 4 Huang 2 8 no all normalsand Du (2006)



Summary for Examples

Ex 5: gX (x) = 4 − x1/4 + sin(5x1) − x2

Ex 6: gX (x) =20∑

i=1xi − 8.951, gX (x) = −

20∑

i=1xi + 36.720

Ex 7: gX (x) = x7 − 3x4

√√√√ πx84x6A3

[Bx6

x5x6(4B2+C2)+DB2(x5E3+x6A3)E

4BF4

]

,

A,B,C,D,E, F are the functions of x

Ref Total Degree # of RVs Dependency Type of RVs Purpose

Ex 5 Sundar 1 2 no all normals sinusoidal functionand Shields (2016)

Ex 6 Engelund 1 20 no all exponentials multi-variablesand Rackwitz (1993)

Ex 7 Der Kiureghian 2 8 no all lognormals strong nonlinearityand De Stefano (1991)



Estimation of Probability of Failure

� Geometrical approximation of a limit state function (FORM/SORM):- Optimization algorithms (iterative search)- Can be considered as linear/quadratic limit state function approximation

� Advanced sampling:- Importance sampling (IS)- Subset simulation (SS)- and many variants . . .

� Surrogate limit state function: there are cases where each are useful- Polynomial Chaos Expansion (PCE)- Kriging- Support Vector Machine (SVM)- Active-subspace based surrogate- and many variants . . .

� Moment method:

- Pearson and Johnson system- Generalized Lambda distribution- Maximum entropy method (MEM)- and many variants . . .

The surrogate form should be chosen carefully depending on the nature of problems dealing with.



Probability of Failure using Surrogate Limit State

Probability of failureusing the “truth” limit state function : Pf = P [gX ≤ 0] =

∫

gX≤0

jpdf of X︷︸︸︷fX(x) dx ≈ 1

N

N∑

i=1

I[gX(x(i)) ≤ 0],

Probability of failureusing “surrogate” limit state function : Pf = P [gQ ≤ 0] =

∫

gQ≤0

jpdf of Q︷︸︸︷fQ(q) dq,≈ 1

N

N∑

i=1

I[gQ(q(i)) ≤ 0],

T : X ⇔ Q, mapping T is necessary

Propose Pf ≈ Pf ⇒ need appropriate validation of gQ, depending on problems



Example 1: Noisy Limit State Function

Given by Liu and Der Kiureghian (1991),

gX(x) = x1 + 2x2 + 2x3 + x4 − 5x5 − 5x6 + 0.001

6∑

i=1

sin(100xi),

Xi=1,2,3,4 ∼ LN(λ1, ζ1), µXi=1,2,3,4= 120, σXi=1,2,3,4

= 12,

X5 ∼ LN(λ2, ζ2), µX5= 50, σX5

= 15,

X6 ∼ LN(λ3, ζ3), µX6= 40, σX6

= 12,

Xi are independent each other.

Solution (given in Ref):

Pf = 1.23× 10−2, β = 2.3482.



Example 1: Noisy Limit State Function – PCE coefficients

For illustration purpose, 1 set of PCE coefficients (HPCE and APCE) is presented.

Truth:

gX(x) = x1 + 2x2 + 2x3 + x4 − 5x5 − 5x6 + 0.0016∑

i=1

sin(100xi)

HPCE:

gQ(q) = 269.9987 · 1 + 11.9692q1 + 23.9414q2 + 23.9378q3

+ 11.9738q4 − 73.3819q5 − 58.7016q6 + 0.5938(q21 − 1) + · · ·

APCE:

gX(x) = 269.9995 · 1 + 1 · (x1 − 120) + 2 · (x2 − 120) + 1.9999 · (x3 − 120)

+ 0.9999 · (x4 − 120)− 5 · (x5 − 50)− 5 · (x6 − 40)



Example 1 Results: APCE vs HPCE – Lower Tail10 sets of the truth and surrogates; LR; 3 ×

(6+pp

)samples.

Hermite polynomial (HPCE) is alternatively selected for lognormal variables.Arbitrary polynomial (APCE) is constructed using the raw moment sequence of a lognormal variable.

Truth APCE (p = 1) HPCE (p = 1) HPCE (p = 2) HPCE (p = 3) HPCE (p = 4)

RMSE 2.10 × 10−3 2.51 × 101 3.98 × 100 6.21 × 10−1 1.12 × 10−1

COV of Pf 2.68 × 10−2 2.68 × 10−2 5.04 × 10−2 2.89 × 10−2 2.69 × 10−2 2.68 × 10−2

σPf3.28 × 10−4 3.28 × 10−4 1.92 × 10−4 3.04 × 10−4 3.30 × 10−4 3.28 × 10−4

µPf1.23 × 10−2 1.23 × 10−2 3.82 × 10−2 1.06 × 10−2 1.22 × 10−2 1.23 × 10−2

# of samples 100,000 21* 21* 84* 252* 630*

P (g < b) = 10−3 [-148.017, -126.117][-148.016, -126.116] [-46.540, -37.658] [-133.088, -114.266][-146.744, -124.994][-148.164, -126.143]

P (g < b) = 10−4 [-293.892, -238.338][-293.889, -238.337] [-133.442, -96.513] [-259.843, -211.955][-284.256, -233.526][-293.353, -238.076]

P (g < b) = 10−5 [-498.822, -332.112][-498.821, -332.113][-217.616, -127.350][-437.862, -295.414][-478.828, -312.608][-497.078, -331.627]

Statistics are obtained using 100 sets; RMSE takes the mean value of RMSE of 100 sets.

* the number of samples that are used to make surrogates; the same 100,000 samples are used for theempirical PoEs.



Example 1 Results: APCE vs LPCE (Second Choice forAskey)Another Legendre polynomial (LPCE) is selected. The convergence looks bad; p = 4 is not enough;

LPCE is not appropriate for lognormal variables than HPCE.

Truth APCE (p = 1) LPCE (p = 1) LPCE (p = 2) LPCE (p = 3) LPCE (p = 4)

RMSE 2.10 × 10−3 4.25 × 101 3.02 × 101 2.14 × 101 2.37 × 101

COV of Pf 2.68 × 10−2 2.68 × 10−2 1.39 × 100 9.22 × 10−2 3.24 × 10−2 3.15 × 10−2

σPf3.28 × 10−4 3.28 × 10−4 8.18 × 10−6 9.83 × 10−5 3.26 × 10−4 3.34 × 10−4

µPf1.23 × 10−2 1.23 × 10−2 5.90 × 10−6 1.10 × 10−3 1.01 × 10−2 1.06 × 10−2

# of samples 100,000 21* 21* 84* 252* 630*

P (g < b) = 10−3 [-148.017 , -126.117][-148.016, -126.116] [16.985, 21.332] [-95.568, -88.236] [-65.415, -57.147] [-84.9292, -78.6656]

P (g < b) = 10−4 [-293.892 , -238.338][-293.889, -238.337] [-7.869, -0.0346] [-151.247, -135.693] [-126.485, -107.228] [-146.9516, -121.1396]

P (g < b) = 10−5 [-498.822 , -332.112][-498.821, -332.113][-32.690, -12.815][-235.621, -166.013][-202.552, -126.6234][-194.5709, -150.8808]



Example 2: Quadratic Limit State Function

Given by Liu and Der Kiureghian (1991),

gX(x) = 1.1− 0.00115x1x2 + 0.00157x22 + 0.00117x2

1

+ 0.0135x2x3 − 0.0705x2 − 0.00534x1 − 0.0149x1x3

− 0.0611x2x4 + 0.0717x1x4 − 0.226x3 + 0.0333x23

− 0.558x3x4 + 0.998x4 − 1.339x24,

X1 ∼ Type II Extreme, µX1 = 10, σX1 = 5,

X2 ∼ N(25, 52),

X3 ∼ N(0.8, 0.22),

X4 ∼ LN(λ4, ζ4), µX4 = 0.0625, σX4 = 0.0625,

Xi are independent each other.

Solution (given in Ref): β = 1.36, Pf = 8.69× 10−2 (by FORM, not MCS).



Example 2: Quadratic Limit State Function – PCEcoefficients

For illustration purpose, 1 set of PCE coefficients (HPCE and APCE) is presented.

Truth:

gX(x) = 1.1 − 0.00115x1x2 + 0.00157x22 + 0.00117x

21

+ 0.0135x2x3 − 0.0705x2 − 0.00534x1 − 0.0149x1x3

− 0.0611x2x4 + 0.0717x1x4 − 0.226x3 + 0.0333x23

− 0.558x3x4 + 0.998x4 − 1.339x24

HPCE:

gQ(q) = 269.9987 · 1 + 11.9692q1 + 23.9414q2 + 23.9378q3

+ 11.9738q4 − 73.3819q5 − 58.7016q6 + 0.5938(q21 − 1) + · · ·

APCE:

gX(x) = 0.1238 · 1 − 0.0066(x1 − 10.1327) + 0.0008(x2 − 24.8689) − 0.0357(x3 − 0.6991)

− 0.5185(x4 − 0.0694) + 0.0012(x21 − 27.9777x1 + 161.8982) + . . .




(4+pp

)samples.

Hermite polynomial (HPCE) is alternatively selected for Tyep II extreme and lognormal variables.Arbitrary polynomial (APCE) is constructed using the raw moment sequence of the variables.

Truth APCE (p = 1) APCE (p = 2) HPCE (p = 1) . . . HPCE (p = 10)

RMSE 3.78 × 10−1 6.90 × 10−16 3.57 × 100 1.40 × 10−1

COV of Pf 1.24 × 10−2 1.48 × 10−2 1.24 × 10−2 1.68 × 10−2 1.25 × 10−2

σPf6.89 × 10−4 8.09 × 10−4 6.89 × 10−2 7.22 × 10−4 . . . 6.96 × 10−4

µPf5.57 × 10−2 5.45 × 10−2 5.57 × 10−2 4.30 × 10−2 5.57 × 10−2

# of samples 100,000 15* 45* 15* 3003*

P (g < b) = 10−3 [-0.517, -0.465] [-0.191, -0.163] [-0.517, -0.465] [-0.0465, -0.0414] [-0.525, -0.473]

P (g < b) = 10−4 [-1.846, -1.276] [-0.722, -0.431] [-1.846, -1.276] [-0.0852, -0.0726] . . . [-1.703, -1.221]

P (g < b) = 10−5 [-12.563, -2.334] [-2.553, -1.021] [-12.563, -2.334] [-0.165, -0.0998] [-9.674, -2.656]





Example 3: Limit State Function with Dependent Variables

Given by Der Kiureghian and Liu (1986),

gX(x) = x1 − x2,

X1 ∼ Unif[0, 100], µX1 = 50, σX1 = 28.87,

X2 ∼ Exp(0.08), µX1 = 12.5, σX1 = 12.5,

ρX1X2 = 0.5

Solution : Pf = 4.86× 10−2 (by MCS).




(2+pp

)samples.

Hermite polynomial (HPCE) is alternatively selected.Arbitrary polynomial (APCE) is constructed using the raw moment sequence of the variables.

Truth APCE (p = 1) HPCE (p = 1) . . . HPCE (p = 23)

RMSE 3.62 × 10−15 2.29 × 101 1.19 × 101

COV of Pf 1.54 × 10−2 1.54 × 10−2 2.39 × 10−2 1.54 × 10−2

σPf7.47 × 10−4 7.47 × 10−4 4.18 × 10−4 . . . 7.48 × 10−4

µPf4.86 × 10−2 4.86 × 10−2 1.75 × 10−2 4.85 × 10−2

# of samples 100,000 9* 9* 900*

P (g < b) = 10−3 [-24.702, -22.884] [-24.702, -22.884] [-58.050, -55.812] [-29.665, -25.587]

P (g < b) = 10−4 [-46.322, -39.507] [-46.322, -39.507] [-79.211, -72.824] . . . [-156.423, -87.735]

P (g < b) = 10−5 [-84.501, -52.362] [-84.501, -52.362] [-135.081, -86.757] [-15491.774, -237.843]





Example 3 Results: APCE – Coefficients

For illustration purpose, the coefficients for 10 APCE sets (differentLR samples) are listed;

gX(x) = 30.123︸︷︷︸c0

× 1︸︷︷︸Ψ0

+ 1︸︷︷︸c1

× (x1 − 36.226)︸︷︷︸Ψ1=x1−E[X1]

−1︸︷︷︸c2

× (x2 − 6.103)︸︷︷︸Ψ2=x2−E[X2]

gX(x) = 29.048 + (x1 − 36.312) − (x2 − 7.264)

gX(x) = 42.315 + (x1 − 50.968) − (x2 − 8.654)

gX(x) = 43.153 + (x1 − 59.127) − (x2 − 15.974)

gX(x) = 45.758 + (x1 − 55.601) − (x2 − 9.843)

gX(x) = 26.434 + (x1 − 43.758) − (x2 − 17.325)

gX(x) = 39.118 + (x1 − 49.793) − (x2 − 10.675)

gX(x) = 34.669 + (x1 − 49.214) − (x2 − 14.545)

gX(x) = 20.141 + (x1 − 29.709) − (x2 − 9.568)

gX(x) = 34.766 + (x1 − 47.759) − (x2 − 12.993)

gX(x) = d0 + d1x1 + d2x2

gX(x) = d0 + d1x1 + d2x2

Errorbars of di by APCE (100sets)

Good approximation by APCE.



Reason Why HPCE Exhibits Bad ConvergencegX(x) = x1 − x2,X1 ∼ Unif[0, 100] and X2 ∼ Exp(0.08),ρX1X2

= 0.5.

Because Askey PCE scheme is based on the independence assumption of random variables,

we need to identify X′ = [X′

1, X′

2]T — independent physical space, though it is challenging.

⇒ but it is relatively easy to identify in normal space (by Nataf transformation),⇒ but gQ(q) may be more complicated than gX(x); lead to slower convergence of surrogate model.

By Nataf transformation, ρQ′

1,Q′

2can be found; Q′ = [Q′

1, Q′

2]T — dependent normal space.

And we can decorrelate a bivariate normal variable by Q = L−1Q′;

L : lower Cholesky triangle of a covariance matrix ΣX = LLT.

Thus, HPCE has to be done in Q space using nonlinear mappings as follows:.

X1 = F−1X1

(Φ(Q′

1)) = F−1X1

(Φ(Q1)),

X2 = F−1X2

(Φ(Q′

2)) = F−1X2

(

Φ

(ρQ′

1,Q′

2Q1 +

√1 − ρ2

Q′

1,Q′

2Q2

))

.



Example 4: Limit State Function: I-beam

Given by Huang and Du (2006),

gX(x) =6x1x3(x2 − x3)x5

x2

(x6x3

5 − (x6 − x7)(x5 − 2x8)3) − x4

X1 ∼ N(6070, 200)

X2 ∼ N(120, 6)

X3 ∼ N(72, 6)

X4 ∼ N(170000, 4760)

X5 ∼ N(2.3, 1/24)

X6 ∼ N(2.3, 1/24)

X7 ∼ N(0.16, 1/48)

X8 ∼ N(0.26, 1/48)

Solution (given in Ref): Pf = 8.711× 10−1 (by 106 MCS samples).

⇒ large failure probability; different thresholds (gX(x) < b) will be investigated here.



Example 4: Limit State Function: I-beam

Number of function evaluation: 45 (= 1×(8+22

))

HPCE and APCE give the same result (because Xi follow normal distributions).



Example 5: Limit State Function: Sinusoidal Function

Given by Sundar and Shields (2006),

gX(x) = 4− x1/4 + sin(5x1)− x2,

X1 ∼ N(0, 1),

X2 ∼ N(0, 1).

Solution (given in Ref): Pf = 4.5460× 10−4 (108 MCS samples).

� A sinusoidal function may be approximated by ahigher order polynomial function (Refer to the figin the right).

� However, sin(5x1) requires a quite higher orderpolynomial function (due to high frequency).

� Difficult to get an accurate surrogate by PCE (oraPCE) with a moderate number of samples.

� Even with a large number of samples, it is difficultto avoid overfitting issue.

A PCE surrogate might not be a good choice for oscillatory limit state function, but . . .



Example 5: Limit State Function: Sinusoidal FunctionAlternative—transforming to a favorable functional form to PCE:

g = g1 + g2,

g1 = 4 − x1/4 − x2,

g2 = sin(5x1),

� proceed with g1 and g2, instead of g

� note g2 and x1 are functionally dependent

� in general cases, the decomposition (e.g., g = g1 + g2) isnot easily obtained.

� but, assume that the decomposition is available.

Let’s define:

Pf = P (g < 0) =

∫P (g < 0 | g2)f(g2)dg2

- MCS: Inner loop

P (g < 0 | g2 = a)

= P (g1 < −a)

≈1

Nin

Nin∑

i=1

I(g(i)1 < −a)

=1

Nin

Nin∑

i=1

I(4 − x1/4 − x(i)2 < −a)

note sin(5x1) = a

- MCS: Outer loop

P (g < 0) ≈1

Nout

Nout∑

j=1

P (g1 < −a(j)

)

=1

Nout

Nout∑

j=1

[1

Nin

Nin∑

i=1

I(4 − x(j)1 /4 − x

(i)2 < −a

(j))

]

=1

Nout

Nout∑

j=1

[1

Nin

Nin∑

i=1

I(4 − x

(j)1 /4 − x

(i)2 < − sin(5x

(j)1 ))]




MCS: Truth vs. Alternative (nested MCS) MCS: Truth vs. Taylor series

� Conventional MCS (105 samples) vs.Alternative way (350 × 350 samples)

� A higher order Taylor series (p = 60) is

accurate downto P = 10−5 level.

� A higher order PCE may work or not.




- Recall alternative formulation:

P (g < 0)

≈1

Nout

Nout∑

j=1

[1

Nin

Nin∑

i=1

I(

g1︷︸︸︷4 − x

(j)1 /4 − x

(i)2 < −

g2︷︸︸︷sin(5x

(j)1 )

)]

- Samples for g1 and g2: 3N = 3 ×(2+11

)= 9

- PCE with g1 and Fourier series with g2:

PCE︷︸︸︷g1 ≈ g1 and

Fourierseries

︷︸︸︷g2 ≈ g2




�

PCE︷︸︸︷g1 and

Truth︷︸︸︷g2

g1 = 4.0000 × 1 − 0.2500 × x1 − 1.0000 × x2

g2 = sin(5x1)

�

PCE︷︸︸︷g1 and

Fourierseries (truncated)

︷︸︸︷g2

g1 = 4.0000 × 1 − 0.2500 × x1 − 1.0000 × x2

g2 = a0 + a1 cos(x1w) + b1 sin(x1w)

+ a2 cos(2x1w) + b2 sin(2x1w)

= 6.098 × 10−12

+ 2.197 × 10−12

cos(5x1) + 1 × sin(5x1)

+ 3.951 × 10−13

cos(10x1) + −3.017 × 10−12

sin(10x1)

g1 and g2 g1 and g2



Example 6: Limit State Function: 20 non-Gaussianvariables

Given by Engelund and Rackwitz (1993),

(1) : gX(x) =20∑

i=1

xi − 8.951,

(2) : gX(x) = −20∑

i=1

xi + 36.720,

Xi=1 to 20 ∼ Exp(1).

Solutions for the both (given in Ref): Pf = 1× 10−3.



Example 6: Limit State Function: 20 non-Gaussianvariables

(1) (2)

Number of function evaluations: 63 = 3×(20+1

1

)

With 63 evaluations to make a g, P (g < b) for any threshold value (b) can be estimated.



Example 7: Limit State Function: Strong Non-linearity

Given by Der Kiureghian and De Stefano (1991),

gX(x) = x7 − 3x4

√√√√ πx8

4x6A3

[Bx6

x5x6(4B2 + C2) + DB2

(x5E3 + x6A3)E

4BF 4

]

,

A =√

x4/x2, B = (x5 + x6)/2, C = (E − A)/F,

D = x2/x1, E =√

x3/x1, F = (E + A)/2,

Xi=1 to 20 ∼ Exp(1).

Solutions for the both (given in Ref): Pf = 4.79 × 10−3.

This limit state function exhibits strong non-linearity.

⇒ may not work with PCE or even APCE.


Conclusions

Conclusions

� Several engineering applications work well with PCE surrogate models.

� A surrogate limit state function by APCE can approximate the truth limitstate function well.

� APCE does not need any iso-probabilistic transformations, which can lead toslow convergence to the truth limit state function.

� HPCE with a higher order approximate the truth limit state function well,regarding accurate estimation of CoV of Pf ; however, other criteria such aslower tail behavior and RMSE is not satisfactory than APCE.

� It should be noted that there are cases where specific surrogate modelingmethod works best.


Thank You

Thank you very much for your attention!


non-parametric surrogate models for uncertainty quantification … · 2020-02-24 · non-parametric...

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