Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Multi-objective Optimization under Uncertainty using Polynomial Chaos Expansion J.Lebon∗,⊤ , R. Filomeno Coelho⊤ , P. Breitkopf∗ , P. Villon∗
⊤
Polytechnic school of Brussels, Université Libre de Bruxelles B
[email protected] – i http://batir.ulb.ac.be ∗
Laboratoire Roberval, Université de Technologie de Compliègne i http://www.umr6253roberval.fr
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Optimization in civil engineering
Nimègue’s bridge (Netherlands)
www.ney.be
Reducing costs Reducing environmental impact Reducing delays Improve safety PCE for multi-objective optimization
2
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Introduction Structural optimization of sizing (x=cross sections, ...) and shape optimization (x=nodes coordinates) of truss structures, ⇒ f (x) (Mass, Maximal Axial Stress, Compliance, ....)
Maritim Museum or Amsterdam
Lille Football Stadium
Multi-objective optimization (Mass Vs Structural Strength, Cost Vs Safety ...) Antagonist behaviors to be taken into account simultaneously! ⇒ f(x) = [f1 (x), f2 (x), . . . , fm (x)] Random uncertainties treatment (structural properties, cross sections, loading,...) ⇒ f(x, ξ) = [f1 (x, ξ), f2 (x, ξ), . . . , fm (x, ξ)] PCE for multi-objective optimization
3
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Introduction Structural optimization of sizing (x=cross sections, ...) and shape optimization (x=nodes coordinates) of truss structures, ⇒ f (x) (Mass, Maximal Axial Stress, Compliance, ....)
Maritim Museum or Amsterdam
Lille Football Stadium
Multi-objective optimization (Mass Vs Structural Strength, Cost Vs Safety ...) Antagonist behaviors to be taken into account simultaneously! ⇒ f(x) = [f1 (x), f2 (x), . . . , fm (x)] Random uncertainties treatment (structural properties, cross sections, loading,...) ⇒ f(x, ξ) = [f1 (x, ξ), f2 (x, ξ), . . . , fm (x, ξ)] PCE for multi-objective optimization
3
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Introduction Structural optimization of sizing (x=cross sections, ...) and shape optimization (x=nodes coordinates) of truss structures, ⇒ f (x) (Mass, Maximal Axial Stress, Compliance, ....)
Maritim Museum or Amsterdam
Lille Football Stadium
Multi-objective optimization (Mass Vs Structural Strength, Cost Vs Safety ...) Antagonist behaviors to be taken into account simultaneously! ⇒ f(x) = [f1 (x), f2 (x), . . . , fm (x)] Random uncertainties treatment (structural properties, cross sections, loading,...) ⇒ f(x, ξ) = [f1 (x, ξ), f2 (x, ξ), . . . , fm (x, ξ)] PCE for multi-objective optimization
3
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Optimizer
Deterministic Variables x - Cross sections - Nodes coordinates
Compliance b b b b
b b b b b b
Function evaluation
b b
Mass Deterministic FEM Analysis - Mass - Compliance - Maximal Axial Stress - Maximum displacement
PCE for multi-objective optimization
4
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Shape optimization
Academic example1
Variables {x} Nodes coordinates
Objectives {f(x)} Mass
Submitted to {g(x)}
xinf ≤ x ≤ xsup
Axial stress
PCE for multi-objective optimization
5
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
6
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
7
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
8
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
9
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
10
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
11
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
12
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
13
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization
Multi-objective optimization of light truss structures
PCE for multi-objective optimization
14
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Shape optimization
Academic example 2
20 kN Variables {x} Nodes coordinates
Objectives {f(x)} Mass
Submitted to {g(x)}
xinf ≤ x ≤ xsup
Compliance
PCE for multi-objective optimization
15
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Multi-objective optimization of light truss structures
PCE for multi-objective optimization
16
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Multi-objective optimization of light truss structures
PCE for multi-objective optimization
17
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Multi-objective optimization of light truss structures
PCE for multi-objective optimization
18
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Multi-objective optimization of light truss structures
PCE for multi-objective optimization
19
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Multi-objective optimization of light truss structures
PCE for multi-objective optimization
20
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Deterministic Multi-Objective Optimization Multi-objective optimization of light truss structures
PCE for multi-objective optimization
21
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Incorporating uncertainties
On material properties
On loadings (Seisms, ...)
Million Dollar Bridge (Alaska, Etats-Unis), fr.structurae.de
On assembly quality
Welding of a subway bridge www.otua.org
PCE for multi-objective optimization
22
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Problematic Taking into account of uncertainties in multi-objective optimization
Maritime Museum of Amsterdam (Netherlands)
Industrial issues Structural safety Structural optimization Non intrusive methods (SAMCEF, ABAQUS, ANSYS,...) Scientific issues Coupling multiobjective optimization and uncertainties with reasonnable computational costs PCE for multi-objective optimization
23
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Non intrusive tools Multi-objective optimization Evolutionary algorithms [Deb, 2002] [Fonseca and Flemming, 1995] [Barakat et al., 2004]
Multi-objective optimization under uncertainties
[Abdullah Konak, 2006]
+ Uncertainties
=
Robust multi-objective optimization [Erfani and Utyuzhnikov, 2011] [Deb and Gupta, 2005]
Simulations methods
Reliable multi-objective optimization
Spectral methods
[Filomeno Coelho et al., 2011]
[Ghanem, 1998]
[Crespo et al., 2010] [Bruno Sudret, 2000]
[Barakat et al., 2004]
[Blatman and Sudret, 2008a]
PCE for multi-objective optimization
24
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Incorporating uncertainty
Academic example Variables {x}
submited to {g(x)}
Nodes coordinates Objectifs {f(x, ξ)} Variables {ξ} Perturbation of nodes coord.
xinf ≤ x ≤ xsup
Mass Maximum Axial Stress Compliance
Loading, Material properties
Submited to {g(x, ξ)} Failure probability Pf : P(σ > σu ) Robustness measure R
PCE for multi-objective optimization
25
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Uncertainty propagation [Filomeno Coelho and Bouillard, 2010, Lebon et al., 2010, Filomeno Coelho et al., 2011]
x
2
Optimizer Deterministic Variables
x - Cross sections - Nodes coordinates
Stochastic Variables
ξ
b b b
- Perturbations - Material properties - ...
b
b
b
b b b
b
b
b b b b
b b
b
b b
b b
b
b b
b b
b b
b b
b
b
b
b
b
b
b b
b
b b
b
b b b
b
b
b b
b b
b b b b
b b
b b
b
b
b
b b b b b b b b b b b b
b b b
b
x 1
Pareto Set Compliance b
Deterministic FEM Analysis
(x,ξ) 7→ f(x,ξ) - Mean - Std. Dev. - Sensitivity indices - Probability of failure
Expensive
BLACK-BOX FUNCTION
b b b
? b
b b b b b
? PCE for multi-objective optimization
b
Pareto Front
b
Mass
26
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Uncertainty propagation with metamodel [Filomeno Coelho and Bouillard, 2010, Lebon et al., 2010, Filomeno Coelho et al., 2011]
x
2
Optimizer Deterministic Variables
Stochastic Variables
ξ
x - Cross sections - Nodes coordinates
b b b
- Perturbations - Material properties - ...
b
b
b
b b b
b
b
b b b b
b b
b
b b
b b
b
b b
b b
b b
b b
b
b
b
b
b
b
b b
b
b b
b
b b b
b
b
b b
b b
b b b b
b b
b b
b
b
b
b b b b b b b b b b b b
b b b
b
x 1
Pareto Set Compliance
(x,ξ) 7→ ˜f (x,ξ) - Mean - Std. Dev. - Sensitivity indices - Probability of failure
Unexpensive
METAMODEL
b b b b
? b
b b b b b
? PCE for multi-objective optimization
b
Pareto Front
b
Mass
27
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Metamodel for uncertainty propagation
?
METAMODEL (x, ξ) 7→ ˜f(x, ξ) FORM SORM Neural Networks Support Vector Machine Spectral decomposition
Simulations based techniques - Crude Monte Carlo Simulation ...- Importance sampling
Analytical derivation
Post-treatment Statistical measures • Robust Based Design Optimization - Mean - Std. Dev. - Sensitivity indices
- From Polynomial Coefficients
• Reliability Based Optimization - Probability of failure - Probability of non dominance
PCE for multi-objective optimization
28
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Polynomial Chaos Expansion as a stochastic metamodel
Sampling in θ space b
fj (x, θ)
θ2 b
b
b
b
b
b
b
b b
b b
θ1
Polynomial Chaos Expansion fj (x, θ) ≈ γ0 (x) +
P−1 X
¯ γi (x)Ψαi (θ)
i=1
PCE for multi-objective optimization
29
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Polynomial Chaos Expansion as a stochastic metamodel Sampling in θ space b
fj (x, θ)
θ2 b
b
b
b
b
b
b
b b
b b
θ1
Polynomial Chaos Expansion fj (x, θ) ≈ γ0 (x) +
P−1 X
¯ γi (x)Ψαi (θ)
i=1
Post-treatment Robust Optimization: E[fi ](x)
=
σ2 (fi )(x)
=
γ0 (x) P−1 X
E[Ψ2i ]γi (x)2
i=1
Reliability Optimization: Z Pfailure
=
p(θ)dθ
g(θ) 106 ≈ 5 days
5000
5500
Nb Simulations = 2200≈ 20 min
PCE for multi-objective optimization
31
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
32
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
33
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
34
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
35
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
36
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
37
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
38
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
39
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
40
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
41
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Stochastic Multi-Objective Optimization Multi-objective optimization of light truss structures under uncertainty
PCE for multi-objective optimization
42
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Polynomial approximation limitation Polynomial chaos f (x, ξ) ≈ γ0 (x) +
P−1 X
γi (x)Ψi (ξ)
i=1
Hermitian Basis Degree N
Ψ0 (ξ) Ψ1 (ξ) Ψ2 (ξ) Ψ3 (ξ)
Ψ4 (ξ) Ψ (ξ) 5 .. .
=
1 ξ1 ξ2 1 − ξ12 ξ1 × ξ2 2 1 − ξ 2 .. .
Variables M
Terms P
2
10
11
10
286
287
20
1771
1772
3
Calls
PCE for multi-objective optimization
43
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Filtering the basis elements Polynomial chaos f (x, ξ) ≈ γ0 (x) +
P−1 X
γi (x)Ψi (ξ)
i=1
Hermitian Basis Degree N
Ψ0 (ξ) Ψ1 (ξ) Ψ2 (ξ) Ψ3 (ξ) Ψ4 (ξ) Ψ5 (ξ) .. .
=
1 ξ1 ξ2 1 − ξ12 ξ1 × ξ2 2 1 − ξ 2 .. .
Variables M
Terms P
Calls
2
↓↓↓
↓↓↓
↓↓↓
↓↓↓
3
↓↓↓
10 20
↓↓↓
PCE for multi-objective optimization
44
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Incorporating uncertainty
Academic example Variables {x}
submited to {g(x)}
Nodes coordinates Objectifs {f(x, ξ)} Variables {ξ}
xinf ≤ x ≤ xsup
Mass Maximum Axial Stress
Young modulus of each bars
Submited to {g(x, ξ)} Failure probability Pf : P(σ > σu )
(22 Random variables)
Robustness measure R PCE for multi-objective optimization
45
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Results 7
4.5
x 10
Deterministic PARETO SET Stochastic PARETO SET with MC α=0,95 Stochastic PARETO SET with LARS α=0,95
Maximal Axial Stress (Pa)
4
3.5
3
2.5
2 3200
3400
3600
3800 4000 Mass (kg)
4200
4400
Number of calls to the exact model: With Monte Carlo Simulation: > 107 calls With metamodels: < 104 calls PCE for multi-objective optimization
46
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Conclusions Conclusions Metamodelling strategy are needed to treat real industrial cases Non intrusive methods may be developed on this purpose. Bi-level multi-objective formulation successfully applied for a reasonnable number of stochastic variables. Sparse strategies show promising results to decrease the computational cost
PCE for multi-objective optimization
47
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
Future Prospects Challenges: Coupling sparse strategies to multi-objective optimization under uncertainties in a smarter way to keep on decreasing the computational cost Development of efficient learning strategies for stochastic metamodels Design of experiment for sparse stochastic metamodels Application to multi-objective optimization ... with adaptive scheme Treatment of large scale structures (higher dimension) What if the stochastic variables are non gaussian? What if the stochastic variables are correlated?
Maritim Museum or Amsterdam
Lille Football Stadium PCE for multi-objective optimization
48
Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
References I [Abdullah Konak, 2006] Abdullah Konak, David W.Coit, A. E. (2006). Multi-objective optimization using genetic algorithm: A tutorial. Reliability Engineering and System Safety, 91:992–1007. [Anthony Chen, 2010] Anthony Chen, Juyoung Kim, S. L. Y. K. (2010). Stochastic multi-objective models for network design problem. Expert Systems with Applications, 37:1608–1619. [Barakat et al., 2004] Barakat, S., Bani-Hani, K., and Taha, M. Q. (2004). Multi-objective reliability-based optimization of prestressed concrete beams. Structural Safety, 26(3):311 – 342. [Berveiller et al., 2006] Berveiller, M., Sudret, B., and Lemaire, M. (2006). Stochastic finite element : a non intrusive approach by regression. Revue Européenne de Mécanique Numérique, 15:81–92. [Blatman and Sudret, 2008a] Blatman, G. and Sudret, B. (2008a). Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes-Rendus Mécanique, 336:518–523. [Blatman and Sudret, 2008b] Blatman, G. and Sudret, B. (2008b). Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus Mécanique, 336:518–523. [Breitkopf et al., 2002] Breitkopf, P., Rassineux, A., and Villon, P. (2002). An introduction to moving least squares meshfree methods. Revue européenne des éléments finis, 11(7-8):826–867. [Bruno Sudret, 2000] Bruno Sudret, A. D. K. (2000). Stochastic finite elements methods and reliability- a state of the art. Technical report, Department of civil and environmental engineering. [Busacca et al., 2001] Busacca, P. G., Marseguerra, M., and Zio, E. (2001). Multiobjective optimization by genetic algorithms: application to safety systems. Reliability Engineering System Safety, 72(1):59 – 74.
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MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
References II [Crespo et al., 2010] Crespo, O., Bergez, J., and Garcia, F. (2010). Multiobjective optimization subject to uncertainty: Application to irrigation strategy management. Computers and Electronics in Agriculture, 74(1):145 – 154. [Deb, 2002] Deb, K. (2002). A fast and elitist multiobjective genetic algorithm: Nsga ii. IEEE Transactions on Evolutionary Computation, 6(2). [Deb and Gupta, 2005] Deb, K. and Gupta, H. (2005). Searching for robust pareto-optimal solutions in multi-objective optimization. In Coello Coello, C. A., Hernández Aguirre, A., and Zitzler, E., editors, Evolutionary Multi-Criterion Optimization, volume 3410 of Lecture Notes in Computer Science, pages 150–164. Springer Berlin / Heidelberg. 10.1007/978-3-540-31880-41 1. [Deb et al., 2007] Deb, K., Padmanabhan, D., Gupta, S., and Kumar Mall, A. (2007). Reliability-based multi-objective optimization using evolutionary algorithms. EMO, 4403:66–80. [DorigoDicaro, 2004] DorigoDicaro (2004). Ant colony optimization: a new meta-heuristic. [Eldred, 2011] Eldred, M. (2011). Design under uncertainty employing stochastic expansion methods. International Journal for Uncertainty Quantification, 1(2):119–146. [Erfani and Utyuzhnikov, 2011] Erfani, T. and Utyuzhnikov, S. V. (2011). Controll of robust design in multiobjective optimization under uncertainties. Structural and Multidisciplinary Optimization. [Fasshauer, 2005] Fasshauer, G. E. (2005). Dual bases and discrete reproducing kernels: A unified framework for rbf and mls approximation. Journal of Engineering Analysis with Boundary Elements, 29:313–325. [Fasshauer and Zhang, 2003] Fasshauer, G. E. and Zhang, J. G. (2003). Recent results for moving least squares approximation. In Lucian, M. L. and Neamtu, M., editors, Geometric Modeling and Computing.
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PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
References III [Filomeno Coelho and Bouillard, 2010] Filomeno Coelho, R. and Bouillard, P. (2010). Multiobjective reliability-based optimization with stochastic metamodels. Evolutionary Computation. [Filomeno Coelho et al., 2011] Filomeno Coelho, R., Lebon, J., and Bouillard, P. (2011). Hierarchical stochastic metamodels based on moving least squares and polynomial chaos expansion - application to the multiobjective reliability-based optimization of 3d truss structures. Structural and Multidisciplinary Optimization, 43(5):707–729. [Fonseca and Flemming, 1995] Fonseca, C. and Flemming, P. (May 19, 1995). An overview of evolutionnary algorithmms in multiobjective opimization. Evolutionnary Computation. [Fragiadakis et al., 2006] Fragiadakis, M., Lagaros, N. D., and Papadrakakis, M. (2006). Performance-based multi-objective optimum design design of steel structures considering life-cycle cost. Structural and Multidisciplinary Optimization, 32:1–11. [Ghanem, 1991] Ghanem, R. (1991). Stochastic finite Elements: A Spectral Approach. Springer. [Ghanem, 1998] Ghanem, R. (1998). Adaptative polynomial chaos expansions applied to statistics of extremes in non linear random vibration. Probabilistic engineering mechanics, 12(2):125–136. [Ghanem and Spanos, 1991] Ghanem, R. and Spanos, P. D. (1991). Spectral stochastic finite-element formulation for reliability analysis. Journal of Engineering Mechanics, 117(10):2351–2372. [Gil and Andreau, 00] Gil, L. and Andreau, A. (00). Shape and cross-section optimisation of a truss structure. Computers and Structures, 79:681–689. [James Kennedy, 1995] James Kennedy, R. E. (1995). Particule swarm optimization. Proceedings of IEEE international conference on . . . .
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PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
References IV [Kang et al., 2010] Kang, S.-C., Koh, H.-M., and Choo, J. F. (2010). An efficient response surface method using moving least squares approximation for structural reliability analysis. Probabilistic Engineering Mechanics, 25(4):365 – 371. [Kleijnen, 2009] Kleijnen, J. P. (2009). Kriging metamodeling in simulation: A review. European Journal of Operational Research, 192(3):707 – 716. [Lancaster and Salkaukas, 1981] Lancaster, P. and Salkaukas, K. (1981). Surfaces generated by moving least squares methods. Mathematics of Computation, 37(155):141–158. [Lebon et al., 2010] Lebon, J., Filomeno Coelho, R., and Bouillard, P. (2010). Adaptive stochastic metamodel for truss optimisation. Avignon. GdR MASCOT NUM 2010. [Lebon et al., 2011a] Lebon, J., Filomeno Coelho, R., Breitkopf, P., and Bouillard, P. (2011a). Model selection techniques applied to structural reliability using bilevel hierarchical metamodels. Minneapolis, Minnesota. 11th National Congress on Computational Mechanics. [Lebon et al., 2011b] Lebon, J., Filomeno Coelho, R., Breitkopf, P., and Bouillard, P. (2011b). Sélection de modèle appliquée à l’optimisation fiabiliste par métamodèles stochastiques hiérarchiques. Giens (France). CSMA 2011. [Lebon et al., 2011c] Lebon, J., Filomeno Coelho, R., Breitkopf, P., and Bouillard, P. (2011c). Validation of metamodelling strategies for probability assessment in a non-intrusive stochastic finite element context. V International Conference on Adaptive Modeling and Simulation, ADMOS 2011. [Nayroles et al., 1992] Nayroles, B., Touzot, G., and Villon, P. (1992). Generalizing the finite element method: Diffuse approximation and diffuse elements. Computational Mechanics, 10:307–318. [Rajan Filomeno Coehlo, 2009] Rajan Filomeno Coehlo, P. B. (2009). Multiobjective reliability-based optimization with stochastic metamodels. Evolutionnary computation.
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Context
MOO under uncertainty
PCE as a Stochastic Metamodel
Sparse PCE for MOO under uncertainty
Conclusions
Prospects
References V [Rangavajhala and Sankaran, 2010] Rangavajhala, S. and Sankaran, M. (2010). A joint probability approach to multo-objective optimization under uncertainty. In 36th design Automation Conference, Parts A and B, volume 1, pages 985–996. ASME. [Sahinidis, 2004] Sahinidis, N. V. (2004). Optimization under uncertainty: state of the art and opportunities. Computers and chemical engineering, 28:971–983. [Salazar et al., 2006] Salazar, D., Rocco, C. M., and Galvan, B. J. (2006). Optimization of constrained multiple-objective reliability problems using evolutionary algorithms. Reliability Engineering System Safety, 91. [T, 1996] T, B. (1996). Evolutionnary Algorithms in Theory and Practice. Oxford University Press New York. [T. Hastie et al., 2009] T. Hastie, R., Tibshirani, J., and Friedman, G. (2009). The Elements of Statistical Learning, Data Mining, Inference and Prediction. Springer. [Zhaowang et al., 2011] Zhaowang, L., Yong Seog, K., and Chen, A. (2011). Multi-objective alpha reliable path finding in stochastic network with correlated link costs: A simulation-based multi-objective algorithm approach (smoga). Expert Systems with Applications, 38:1515–1528.
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