DISTRIBUTION-BASED GLOBAL SENSITIVITY ANALYSIS BY

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

DISTRIBUTION-BASED GLOBAL SENSITIVITYANALYSIS BY POLYNOMIAL CHAOS

EXPANSION

Lukas Novak, Drahomır Novak

Novak.L@fce.vutbr.cz

Institute of Structural MechanicsFaculty of Civil Engineering

Brno University of TechnologyCzech Republic

24.-25. November 2020

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 1 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Uncertainty quantification

• the quantity of interest of a physical system is represented bya mathematical model (e.g. deflection) Y =M(X)

• necessity of the assumption of uncertain input variables(described by specific PDF) for real problems

• UQ of characteristics of Y =M(X): statistical moments,PDF, sensitivity analysis etc.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 2 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Distribution-based SA

• Sobol indices assume only first two statistical moments• it is not able to take the whole PDF into account

• distribution-based sensitivity analysis proposed by Borgonovo• based on difference (gray area) between original PDF/CDF

and conditional PDF/CDF

BORGONOVO E.: A new uncertainty importance measure. Reliability Engineering & System Safety, 92(6), p.

771-784, 2007.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 3 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Cramer–von Mises distance

• Let F be CDF of Y obtained by indicator function 1

F(t)

= P(Y ≤ t

)= E

[1Y≤t

], t ∈ R

• Let F u be conditional CDF of Y conditionally to Xu

F u(t)

= P(Y ≤ t

∣∣Xu) = E[1Y≤t |Xu

], t ∈ R

• Sensitivity measure based on CVM proposed by Gamboa et al.

Cu =

∫R E[(F u(t)− F

(t))2

]dF(t)∫

R F(t)(

1− F(t))dF(t)

GAMBOA F.; KLEIN T.; LAGNOUX, A.: Sensitivity Analysis Based on Cramer–von Mises Distance, SIAM/ASA

Journal on Uncertainty Quantification, 6(2), p. 522-548, 2018.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 4 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Polynomial Chaos Expansion

M(X) ≈ M(X) =∑α∈NM

βαΨα(X)

• deterministic coefficients to be computed - βα

• orthonormal basis of multivariate polynomials - Ψα(X)

• M represents number of input random variables

SOIZE, C.; GHANEM, R.: Physical systems with random uncertainties: Chaos representations with arbitrary

probability measure.J. Sci. Comput 26(2), 395– 410, 2004.

WIENER, N.: The Homogeneous Chaos. American Journal of Mathematics, 1938: p. 897–936.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 5 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Orthonormal basis

⟨ψα, ψβ

⟩=

∫ψα(ξ)ψβ(ξ)pξ(ξ)dξ = δαβ

• Multivariate basis functions are orthonormal with respect tothe joint PDF of germs pξ.

• Normalized Hermite polynomials are orthonormal to Gaussianprobability measure in the Wiener-Hermite PCE.

• transformation to standardized Gaussian space (Nataf)

• Common distributions can be associated to specific type ofpolynomial (Wiener-Askey scheme).

XIU, D.; KARNIADAKIS, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. J Sci.

Comput., 2002, 24(2):619-44.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 6 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Orthonormality of PCE

• generally statistical moment of any order is defined as:

⟨ym⟩

=

∫ [f(X)]m

pξ(ξ)dξ =

∫ [ ∑α∈NM

βαΨα(ξ)]m

pξ(ξ)dξ =

=

∫ ∑α1∈NM

...∑

αm∈NM

βα1 ...βαmΨα1 (ξ)...Ψαm(ξ)pξ(ξ)dξ =

=∑α1∈NM

...∑

αm∈NM

βα1 ...βαm

∫Ψα1 (ξ)...Ψαm(ξ)pξ

(ξ)dξ

• it might be computationally demanding to employ MC

• PCE leads to dramatical simplification of equation due to theorthonormality of basis polynomials

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 7 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

First two statistical moments

• Considering orthonormality of polynomials:∫Ψα(ξ)pξ

(ξ)dξ = 0 ∀α 6= 0 Ψ0 ≡ 1

• Mean value is given by the first term of the expansion:

µY =⟨y1⟩

= β0

• variance is defined as σ2Y =

⟨y2⟩− µ2

Y , thus assumingorthonormality, variance can be simply obtained as:

σ2Y =

∑α∈Aα 6=0

β2α

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 8 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Sobol’ indices

• Hoeffding-Sobol’ decomposition - Sobol’ indices (ANOVA)

• highly efficient derivation of Sobol’ indices from PCE• First order indices

Si =∑α∈Ai

β2α

σ2Y

Ai ={α ∈ NM : αi > 0, αj 6=i = 0

}• Total indices

STi =

∑α∈AT

i

β2α

σ2Y

ATi =

{α ∈ NM : αi > 0

}

SUDRET, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab Eng and System Safety, 2008,

93: p. 964-979.

SOBOL, I.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math

and Comput in Simulation 55, 2001, p. 271-280.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 9 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

PCE for CVM sensitivity

• PCE in combination with kernel density estimation → FPCE• CDF F u

PCE obtained by PCE reduced to selected terms f PCEu :

f PCEu (x) = β0 +∑A∼u

βαΨα(ξ) A∼u ={α ∈ AM,p : αk 6= 0↔ k ∈∼ u

}where ∼ u is a complement to any u ⊂ I = {1, 2, ...M}

• CVM based on PCE is calculated as:

τPCEu =

∫R

[F uPCE

(t)− FPCE

(t)]2

dFPCE(t)

• normalizing denominator can be simply summary of sensitivitymeasures for all possible τPCEu

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 10 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Proposed CVM-based indices

• finally, proposed sensitivity indices based on CVM and derivedfrom PCE:

CPCEu =

∫R

[F uPCE

(t)− FPCE

(t)]2

dFPCE(t)

∑∆=P(I )

∆ 6=I

τPCE∆

where ∆ = P(I ) is power set of I , i.e. ∆ contains all possiblesubsets of I

• it represents relative influence of each variable in 〈0, 1〉

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 11 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Validation of proposed method

The original mathematical model (Borgonovo et al. 2011) isrepresented by analytical function in the following form:

Y =n∏

i=1

X ai

• Xi ∼ LN (µ, σ) where µ and σ are the mean value andstandard deviation of ln(Xi )

• The reference solution is given for n = 3, a = 1, µ = 1 andσ1 = 16, σ2 = 4, σ3 = 1

BORGONOVO E.; CASTAINGS W.; TARANTOLA S., Moment independent importance measures: New results

and analytical test cases, Risk Analysis: An International Journal, 31(3), p. 404-428, 2011.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 12 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Comparison of results

• reference solution obtained by double-loop MC (ED 105)

• 1000 samples generated by LHS for PCE surrogate

Table : Comparison of proposed method with reference solution

Randomvariable

Referencesolution

Referencerelative

Proposedindices

X1 0.4720 0.68 0.68X2 0.1550 0.22 0.24X3 0.0071 0.10 0.08

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 13 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Deflection of simple beam

Variable Mean Standard deviation CoV

b (LN) 0.15 m 0.075 m 5 %h (LN) 0.3 m 0.015 m 5 %E (LN) 30000 Mpa 4500 Mpa 15 %q (LN) 10 kN/m 2 kN/m 20 %L (LN) 5 m 0.05 m 1 %

v(L/2) =5

384

qL4

EII =

bh3

12

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 14 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Deflection of simple beam

Comparison of statistical moments

Characteristic PCe (ED=100) Analytical

Mean value 8.3666 8.3687Variance 6.4294 6.4468

Comparison of sensitivity indices

Variable Sobol Total Proposed CVM

b 0.029 0.033h 0.263 0.164E 0.261 0.172q 0.428 0.629L 0.019 0.002

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 15 / 18

MotivationPolynomial Chaos Expansion

PCE for CVM sensitivityAcademic examples

Conclusions

Conclusion and further work

• PCE represents powerful UQ tool

• efficient statistical and sensitivity analysis due to theorthonormality of PCE (especially FEM)

• PCE can be used for distribution-based sensitivity analysis

• efficient calculation of indices in comparison to MC

• Furher work: more complicated examples, FEM, role ofcorrelation

Thank you for yourattention!

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 16 / 18

Appendix

Bibliography I

[1] Blatman G. and Sudret B.Adaptive sparse polynomial chaos expansion based on Least Angle RegressionJournal of Computational Physics 230, March 2011, p: 2345-2367DOI:10.1016/j.jcp.2010.12.021

[2] Sudret B.Global sensitivity analysis using polynomial chaos expansions.Reliab. Eng. and System safety, 93, 964–979. 2008

[3] Gamboa F., Klein T. and Lagnoux A.Sensitivity analysis based on Cramer-von Mises distanceSIAM/ASA Journal on UQ, 6(2), p: 522-548, 2018

[4] Novak, L. and Novak D.Polynomial chaos expansion for surrogate modelling: Theory and software.

Beton und Stahlbetonbau, 113: 27-32, 2018.

Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 17 / 18

Appendix

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Engineering Mechanics 2020 24.-25.11.2020 Lukas Novak, Drahomır Novak 18 / 18