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A. C. Antoniades, UC Berkeley A. Ganesan, UC Berkeley C. M. Lagoa, Penn State University H. Kettani, University of

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. A. C. Antoniades, UC Berkeley A. Ganesan, UC Berkeley C. M. Lagoa, Penn State University

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A. C. Antoniades, UC Berkeley A. Ganesan, UC Berkeley C. M. Lagoa, Penn State University H. Kettani, University of

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  1. Risk Assessment Via Monte Carlo Simulation: Tolerances Versus Statistics B. Ross BarmishECE DepartmentUniversity of Wisconsin, MadisonMadison, WI 53706barmish@engr.wisc.edu

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

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

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

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

  8. More Generally • Desired Simulation

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

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

  12. The Central Issue What probability distribution to use?

  13. Manufacturing Motivation

  14. Distributions for Capacitor

  15. Interpretation

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

  17. Truncation Principle

  18. Example 1: Ladder Network • Study Expected Gain

  19. Illustration for Ladder Network

  20. Example 2: RLC Circuit Consider the RLC circuit

  21. Solution Summary For RLC

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

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