Targeted Random Sampling for Reliability Assessment: A Demonstration of Concept

Transcription

1 Illinois Institute of Technology; Chicago, IL Targeted Random Sampling for Reliability Assessment: A Demonstration of Concept Michael D. Shields Assistant Professor Dept. of Civil Engineering Johns Hopkins University V.S. Sundar Postdoctoral Fellow Dept. of Civil Engineering Johns Hopkins University May 2014

2 Motivation of Present Work Monte Carlo Simulation (MCS) is the most robust and accurate (and sometimes the only) means of reliability analysis MCS is often (rightfully so) criticized for its high computational cost The cost of MCS has been greatly reduced through the advent of sophisticated sampling and simulation methods We can still do better!

3 Outline of Presentation Overview of Reliability Methods Emphasis on Monte Carlo Simulation based methods Refined Stratified Sampling Concept / Theory Algorithms / Applications Targeted Random Sampling Concept / Theory Algorithm Examples Challenges & Future Directions

4 Reliability Analysis Reliability Methods Approximate Methods Exact Methods FORM SORM Response Surface Monte Carlo Numerical Int. - Low accuracy - Very efficient - Extensible to multiple failure modes - Moderate dimensions - Moderate accuracy - Low efficiency - Single failure mode only - Moderate dimensions - Moderate accuracy - Moderate efficiency - Multiple failure modes - Moderate to high dimensions - Very Robust - Accurate - Numerous methods available - Computationally expensive - Computationally intractable for many cases - Only applicable in low dimension with special limit states. Direct MC Importance Sampling Subset Simulation 1 Line Sampling 2 Targeted Sampling - Robust - Accurate - Very Expensive - High dimensions - Multiple failure modes - Moderate accuracy - Moderate efficiency - Widely used - Many variants - Accurate - Efficient - High dimensions - MCMC can cause problems - Accurate - Efficient - High dimensions - May still require hundreds of calculations - Very accurate & efficient - High dimensions - Multiple failure modes 1. Au, S.-K. and Beck, J.L. (2001). Estimation of small failure probabilities in high dimension by subset simulation Probabilistic Engineering Mechanics 2. Koutsourelakis, P.S., Pradlwarter, H.J., Schueller, G.I. (2004). Reliability of structures in high dimension, part I: algorithms and applications Probabilistic Engineering Mechanics

5 Stratified Sampling Random Sampling Stratified Sampling P[X 2 ] 0.5 P[X 2 ] P[X ] 1 Sample size extension is straightforward No control over where new samples are placed No need to compute sample weights P[X ] 1 Almost limitless control over how samples are added Carefully select, based on the available information, which strata to add samples to Is there a better way?

6 Stratified Sampling Random Sampling Stratified Sampling P[X 2 ] 0.5 P[X 2 ] P[X ] 1 Sample size extension is straightforward No control over where new samples are placed No need to compute sample weights P[X ] 1 Almost limitless control over how samples are added Carefully select, based on the available information, which strata to add samples to Is there a better way?

7 Stratified Sampling Random Sampling Stratified Sampling P[X 2 ] 0.5 P[X 2 ] P[X ] 1 Sample size extension is straightforward No control over where new samples are placed No need to compute sample weights P[X ] 1 Almost limitless control over how samples are added Carefully select, based on the available information, which strata to add samples to Is there a better way?

8 Refined Stratified Sampling Y" (1,1)" w 1 (x 1,y 1 )" (x 2,y 2 )" w 2 (x 1,y 1 )" 2. Randomly divide probability space into strata Ω i ; i=1,2 w 1 (x 1,y 1 )" (x 3,y 3 )" w 3 w 2 (x 2,y 2 )" 6. Divide largest stratum Ω k to obtain strata Ω i ; i=1,2,3,4 (0,0)" X" 1. Randomly"sample" w 1 (x 1,y 1 )" w 2 (x 2,y 2 )" 4. Randomly divide stratum Ω k to obtain strata Ω i ; i=1,2,3 w 1 (x 1,y 1 )" w 4 (x 4,y 4 )" (x 3,y 3 )" w 3 w 2 (x 2,y 2 )" 8. Randomly divide stratum Ω k to obtain strata Ω i ; i=1,2,3,4,5 New"stratum." Sample"carried"from"previous"step." New"sample." X"~"Probability"space"for"RV1" Y"~"Probability"space"for"RV2" 3. Randomly sample in empty stratum Ω i and assign weights: w i =P(Ω i ) w 1 (x 1,y 1 )" (x 3,y 3 )" w 3 w 2 (x 2,y 2 )" 5. Randomly sample in empty stratum Ω i and assign new weights: w i =P(Ω i ) 7. Randomly sample in empty stratum Ω i and assign new weights: w i =P(Ω i ) 9. Randomly sample in empty stratum Ω i and assign new weights: w i =P(Ω i ) 1. Shields, M.D., Teferra, K., Hapij, A. and Daddazio, R. (2014) Refined stratified sampling for efficient Monte Carlo based uncertainty quantification Reliability Engineering & System ty (forthcoming)

9 Why Refine Strata? Consider a N samples from stratum Ω 1 : Var! 1 T # " S $ = p 1 N σ M j=1 p j 2 2 n j σ j Ω 1 Consider stratum refinement with a single sample in each stratum: 2 Var! 2 T # " S $ = p 1 N 2 Ω 1 N σ 2 i + i=1 Other strata M j=1 p j 2 2 n j σ j Ω 1 Evaluate the difference in the variances: Var! 1 T # " S $ Var! " T S 2 Ω 1 Other strata # $ = p 2 ' 1 N ) σ ( N N i=1 (Assumes a balanced sub-stratification) σ i 2 *, > 0 +

10 Variance Savings by RSS: Example Consider samples drawn from x~n(0,1) with T S = E[x]: σ 1 2 = σ 2 2 = p 1 = 0.5 p 2 = 0.5 σ 1 2 = p 1 = 0.5 σ 2 2 = p 2 = 0.25 σ 3 2 = p 2 = 0.25 σ 3 2 = p 2 = 0.25 σ 2 2 = p 2 = 0.25 ( ) 2 1 Var! b " T S # $ = 2 2 σ ( 1 2 ) 2 2 σ 2 2 Var! b " T S # $ = Var! " T S ( ) 2 1 Var! u " T S # $ = 2 2 u # $ = σ ( 1 4 ) 2 σ ( 1 4 ) 2 2 σ 3 Var! u " T S # $ = 1 4 Var! u " T S # $ = ( ) 2 σ ( 1 4 ) 2 σ ( 1 4 ) 2 σ ( 1 4 ) 2 2 σ 4 >30% reduction in variance by adding one stratum >60% reduction in variance by fully stratifying

11 RSS Algorithms 1. Random Stratum Division 2. Variance Minimization 3. Sensitivity-Based Stratum Division 4. Enhanced Space-Filling 5. Targeted Random Sampling 6. Others (Stochastic Search, Stochastic Field Simulation, etc.)

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13 Random (iid) sampling

14 Stratified Sampling/ Refined Stratified Sampling

15 Targeted Random Sampling The biggest problem with Monte Carlo methods is that they are wasteful Many simulations in regions that are, statistically, of little or no use This is especially problematic for reliability analysis where probability of failure is very low

16 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat

17 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Identify Strata Adjacent to Surface

18 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample

19 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample Sandia National Laboratories

20 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample

21 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample

22 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample

23 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample

24 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Determine stratum pair to split & add sample

25 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Repeat until convergence criteria satisfied

26 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Repeat until convergence criteria satisfied

27 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Repeat until convergence criteria satisfied

28 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Repeat until convergence criteria satisfied

29 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Repeat until convergence criteria satisfied

30 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat

31 1. Define an initial stratification of the domain & sample 2. Identify point pairs on opposite sides of the failure surface 3. Identify the pair with the largest separation and divide the stratum at a point in between (alternatively, pair with largest associated probability) 4. Sample in the new stratum 5. Refine the target space 6. Repeat Notes: TRS constructs a piecewise approximation of the failure surface Need to identify at least one failure point per disjoint failure region Uses available information to inform stratification Repeat until convergence criteria satisfied

32 Limit State: Example 1: Initial Sample G(U) = U sin 5U 1 Probability of : p f ( ) + 4 U 2 ; U ~ N (0,1) e e 7 1.0e 12 4 U e 7 G(U)= e e e 7 1.0e U 1

33 Limit State: Probability of : Example 1: 200 Samples G(U) = U sin 5U 1 p f ( ) + 4 U 2 ; U ~ N (0,1) TRS MCS U 2 3 G(U)=0 P F U Added samples

34 Limit State: Example 1: 500 Samples G(U) = U sin 5U 1 Probability of : p f ( ) + 4 U 2 ; U ~ N (0,1) U G(U)= U 1 P F Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 MCS Added samples

35 Limit State: Example 1: Comparison G(U) = U sin 5U 1 Probability of : p f ( ) + 4 U 2 ; U ~ N (0,1) Method Mean P F COV MCS (10 8 samples) FORM 4.154e- 4 N/A 4 U G(U)=0 SORM Importance Sampling (IS) SS(500; 50; 10) (~1000 samples) N/A N/A e- 4 (100) 66% U 1 TRS (9+491) (500 samples) e- 4 (100) 4%

36 P F Limit State*: Probability of : Example 2: Comparison Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 MCS Added samples 5 G(X) = X i C; X i ~ Exp(λ i ); λ i =1 i; C = i=1 p f Method Mean P F COV MCS (10 8 samples) FORM SORM IS (1000 samples) SS(500; 50; 10) (~1000 samples) TRS ( ) (1000 samples) e- 6 3% e e e e- 6 (100) e- 6 (100) 1. Engelund, S. and Rackwitz, R. (1993). A benchmark study on importance sampling techniques in structural reliability. Structural ty. 12: % 20%

37 Challenges & Future Directions Defining an appropriate initial stratification in high dimension 3 20 ~ 3.5e+9 Use MCMC to explore the space and post-stratify Dynamic Problems & Stochastic Processes/Fields Other RSS designs Variance Minimization for UQ Sensitivity-based Sampling Space-filling sample designs Etc.