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

[email protected]

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

[email protected]



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.




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.




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.




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.




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.




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




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




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.




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




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〉




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.




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




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




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




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!


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.


Appendix

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distribution-based global sensitivity analysis by

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