1. 1 Risk Assessment Via Monte Carlo Simulation: Tolerances Versus Statistics B. Ross Barmish ECE Department University of Wisconsin, Madison Madison, WI 53706 barmish@engr.wisc.edu

2. 2 A. C. Antoniades, UC Berkeley A. Ganesan, UC Berkeley C. M. Lagoa, Penn State University H. Kettani, University of Wisconsin/Alabama M. L. Muhler, DLR, Oberfaffenhofen B. T. Polyak, Moscow Control Sciences P. S. Shcherbakov, Moscow Control Sciences S. R. Ross, University of Wisconsin/Berkeley R. Tempo, CENS/CNR, Italy COLLABORATORS

3. 3 Overview Motivation The New Monte Carlo Method Truncation Principle Surprising Results Conclusion

4. 4 Monte Carlo Simulation Used Extensively to Assess System Safety Uncertain Parameters with Tolerances Generate “Thousands” of Sample Realizations Determine Range of Outcomes, Averages, Probabilities etc. How to Initialize the Random Number Generator High Sensitivity to Choice of Distribution Unduly Optimistic Risk Assessment

5. 5 It's Arithmetic Time Consider 1 + 2 + 3 + 4 + 5 +  + 20 = 210 Data and Parameter Errors Classical Error Accumulation Issues (1 § 1) + (2 § 1) + (3 § 1) +  (20 § 1) = 210 § 20

6. 6 Two Results Actual Result Alternative Result 190 · SUM · 230

7. 7 Circuit Example Output Voltage Range of Gain Versus Distribution

8. 8 More Generally Desired Simulation

9. 9 Two Approaches Interval of Gain Monte Carlo How to Generate Samples? Gain Histogram: 20,000 Uniform Samples

10. 10 Key Issue What probability distribution should be used? Conclusions are often sensitive to choice of distribution. Intractability of Trial and Error Combinatoric Explosion Issue

11. 11 Key Idea in New Research Classical Monte Carlo New Monte Carlo Random Number Generator F Samples of q System Model Samples of F(q) Statistics of q Random Number Generator F Samples of q System Model Samples of F(q) Statistics of q New Theory Bounds on q

12. 12 The Central Issue What probability distribution to use?

13. 13 Manufacturing Motivation

14. 14 Distributions for Capacitor

15. 15 Interpretation

16. 16 Interpretation (cont.) Random Number Generator F Samples of q System Model Samples of F(q) Statistics of q New Theory Bounds on q

17. 17 Truncation Principle

18. 18 Example 1: Ladder Network Study Expected Gain

19. 19 Illustration for Ladder Network

20. 20 Example 2: RLC Circuit Consider the RLC circuit

21. 21 Solution Summary For RLC

22. 22 Current and Further Research The Optimal Truncation Problem Exploitation of Structure Correlated Parameters New Application Areas; e.g., Cash Flows