Risk Analysis: Efficient Computation of Failure Probability

Transcription

1 Risk Analysis: Efficient Computation of Failure Probability Nassim RAZAALY CWI - INRIA Bordeaux nassim.razaaly@inria.fr 20/09/2017 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

2 Motivation: Reliability v: Wind Speed [Random Variable] F c : Critical Load [Scalar] Load on the Blade x: F(v, x) [Expensive Function] Goal: Assess Blade Robustness p f = P v [F(v, x) > F c ] < 10 6? Blade x: Analysis Damaged Wind Turbine Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

3 Motivation: Very Low Failure Probability Industry Design Assessment Aeronautics Nuclear Power Plants = p f < 10 8 How to Compute p f? Accurately Low Number of Evaluations [CFD/Structural Analysis Models] Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

4 Contents 1 Introduction 2 Different Approaches 3 Some Examples 4 Conclusion Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

5 Section 1 Introduction

6 Motivation : Tail Probability Failure Risk/Tail Probability : P(J > J C ) Quantity of Interest : J : R d R J C R Critical value ξ R d Random variable Goal : P(J(ξ) > J C ) ξ pdf - Uncertain Input J(ξ) pdf - Quantity of Interest Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

7 Basic Strategy: Monte-Carlo In the Standard Space p f = P(G(X) < 0) = E[1 G<0 (X)] Standard Space: X N (0, I d ) p f 1 Disconnected Failure Regions Crude Monte-Carlo δ target = ˆσ f ˆp f = 1% = N MC 10 10!! MC points [Expensive Computations] p f = = 4 Failure points... Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

8 Section 2 Different Approaches

9 Strategy 1: Metamodel Substitution Metamodel Substitution No Error Control Requirements Few Expensive Calls Different Failure Branches No points Clustering p f 1 = Ñ MC very high 10 6 MC points [Cheap Computations] Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

10 Strategy 2: Metamodel Substitution + IS If p f 1 Metamodel Substitution + IS No Error Control Suitable for p f 1. Requirements Few Expensive Calls Different Failure Branches No points Clustering Suitable for very Low probability IS points [Cheap Computations] Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

11 Strategy 3: Metamodel + (quasi-optimal) IS Metamodel + IS Construct a Metamodel G Define a quasi-optimal Importance Sampling Density (ISD) based on G Run performance function G with IS Unbiased Estimation ˆp f. 200 IS points [Expensive Computations] Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

12 Section 3 Some Examples

13 Metamodel Construction 2D Example Two Disconnected Failure Regions Low Failure probability: p f = LHS Sampling in Standard Space Refined Metamodel: N calls = 66 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

14 Refined Metamodel: 2D examples Refined Metamodel ; N calls = 68 ; Refined Metamodel ; N calls = 112 ; p f = p f = Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

15 IS Metamodel: p f = E[1 G<0 (X)] ISD used on Metamodel Contour ; Ñ calls ; δ < 0.1% ; p f = Optimal ISD Contour Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

16 Quasi-Optimal IS: p f = E[1 G<0 (X)], using MCMC Quasi-opt IS Contour Sampled Points using MCMC ; N calls = ; δ < 1% Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

17 Section 4 Conclusion

18 Conclusion Tail Probability Accurate Metamodel Refinement Suitable for very low p f, multiple failure regions Sharp IS accuracy on metamodel Statistically consistent error Low number of performance function evaluations Perspectives Parallel Metamodel Refinement Extreme Quantile Evaluation: P(J(ξ) > q) = 10 6 Optimization under Failure Probability Constraint Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 13

19 References Q. Wang. Uncertainty Quantification for Unsteady Fluid Flow using Adjoint-based Approaches. PhD Thesis. Stanford University Echard B, Gayton N, Lemare M. AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation. Reliability Engineering and System Safety, 33 (2011) Echard B, Gayton N, Lemaire M, Relun N. A combined Inportance Sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models. Reliability Engineering and System Safety, 111 (2013) Cadini F, Gioletta A, Zio E. Improved metamodel-based importance sampling for the performance assessment of radiactive waste repositories. Reliability Engineering and System Safety, 134 (2015) Cadini F, Santos F, Zio E. An improved adaptive kriging-based importance technique for sampling multiple failure regions of low probability. Reliability Engineering and System Safety, 131 (2014) Dubourg V, Sudret B, Deheeger F. Metamodel-based importance sampling for structural reliability analysis. Probabilistic Engineering Mechanics, 33 (2013) Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7

20 Isoprobabilistic Transformation T From Physical Space to Standard Space p f = P(F (X) > F c ), with X = (X 1,..., X d ) independant. J(X) = F c F (X) = p f = P(J(X) < 0) U = T (X), U N (0, I d ) T (x) = [ φ 1 F Xk (x k ) ] k [[1,d]] G = J T 1 = G(U) = J T 1 (X) = J(X) = p f = P(G(U) < 0) If X not independent, use Rosenblatt Transformation. Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7

21 Two Failure Regions Example in 2D Equation c = 5, X = (X 1, X 2 ) N (0, I 2 ) { G(x 1, x 2 ) = min c 1 x 2 + e x ( x 1 5 ) 4 c 2 2 x 1 x 2 } Method N calls ˆp f ˆδ f 3 ˆσ f Interval Ñ calls ˆp G ˆδ p G 3 ˆσ p G Interval Crude MC 422, 110, < 5% [8.06, 11.9] 10 7 FORM Subset 700, < 5% [5.57, 7.53] 10 7 Meta-IS < 5% [7.80, 10.5] 10 7 MetaAK-IS < 5% [6.94, 9.38] 10 7 MetaAL-OIS % [8.58, 9.24] % [9.16, 9.21] 10 7 Perf + IS % [8.942, 8.996] 10 Results 7 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7

22 Metamodel G : Tricky Case G(x 1, x 2 ) = 10 2 i=1 (x 2 i 5 cos(2πx i )) Refined Metamodel: N calls = 135 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7

23 Metamodel G : Four Failure Case G(x 1, x 2 ) = min 3 + (x 1 x 2 ) 2 10 x 1+x (x 1 x 2 ) x 1+x 2 2 x 1 x (x 1 x 2 ) Refined Metamodel: N calls = 73 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7

24 Metamodel G : One Failure Case G(x 1, x 2 ) = 1 2 (x 1 2) (x 2 5) 3 3 Refined Metamodel: N calls = 27 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7

25 The End Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/ / 7