1. A Search Algorithm for Calculating Automatically Verified Reliability Bounds Fulvio Tonon, University of Utah, USA

2. Problem statement Vector of uncertain parameters u = (u 1,...,u p ) Joint PDF pro(u) System response y = f(u) CDF of y = ?

3. Objective To calculate automatically verified bounds on the CDF of y

4. Why automatically verified bounds? Speculative/theoretical reason Calculating bounds can be far more efficient than MC methods Bounds may be enough for making decisions

5. Two cases Case 1: The entire CDF of the response y is needed Case 2: Only the CDF of a particular value y* is needed (reliability analyses)

6. Vector of uncertain parameters u = (u 1,...,u p ) with joint PDF pro(u) The i-th parameter, u i, belongs to interval I i u is constrained within a p-dimensional box D = I 1 ,...,  I p Step 1: {A j, j = 1,...,N} = a partition of D and set. Case 1: entire CDF of the response

7. Case 1: entire CDF of the response (cont.) Step 2: Calculate the image f(A j ) of each set A j through function f. f Interval Analysis: http://cs.utep.edu/interval-comp/main.htmlhttp://cs.utep.edu/interval-comp/main.html Non-intrusive methods: f is a “black box” Intrusive methods: e.g., Modares and Mullen; Zhang and Muhanna, Neumaier, Corliss et al., Pownuk, many others…

8. Case 1: entire CDF of the response (cont.) Step 3: Calculate the upper, F y,upp, and lower, F y,low, bounds on the cumulative distribution function (CDF) of y, F y

9. Case 1: entire CDF of the response (cont.) Example u 1  N(10, 1) and u 2  N(100, 1) Of course, y  N(10009.55, 100) However, MC =>10 8 functional evaluations for error of  0.02 with a confidence of 95%

10. Case 1: entire CDF of the response, Example (cont.) 119 function evaluations Max error = 0.477 Max rel.error = 74%

11. Case 1: entire CDF of the response, Example (cont.) 943 function evaluations Max error=0.04% Max rel.error = 0.2%

12. Case 1: entire CDF of the response, Example (cont.) Functional eval. increases 8 times => 10-fold error decrease MC: 100 times increase => 10-fold error decrease

13. Case 2: CDF of a specific value y*

14. Case 2: CDF of a specific value y* (cont.) P*

15. Case 2: CDF of a specific value y* (cont.) What if the bounds are too large? If the error at y* is excessive, only refine the partition of S 3

16. Case 2: CDF of a specific value y* (cont.) Reliability analysis: f = safety margin z y*y* F upp (y*) F low (y*) P lim

17. Case 2: CDF of a specific value y* (cont.) Example z* = 0, P lim = 10 -5 First discretization, 48 functional evaluations

18. Case 2: CDF of a specific value y* (cont.) Example

19. Case 2: CDF of a specific value y* (cont.) Example y*y* P lim

20. Case 2: CDF of a specific value y* (cont.) Example

21. Case 2: CDF of a specific value y* (cont.) Example Second discretization, 58 functional evaluations

22. Case 2: CDF of a specific value y* (cont.) y*y* F upp (y*) F low (y*) P lim

23. Case 2: CDF of a specific value y* (cont.) Monte-Carlo: e= 2.8  10 -4 ;  = 10 -5 => nc = 90  10 6 >> 106

24. Advantages Verified bounds vs. confidence intervals Explicit evaluation of the error, not possible with num. meth. 106 functional evaluation vs. 10 8 for 99.999% confidence Variety of uncertainty descriptors: probabilistic, interval- valued, set-valued, and random-set-valued input

25. Disadvantages Number of functional evaluations, increases exponentially with the no. of uncertain variables; SOLUTION: use interval FE methods to map focal elements “Sophisticated methods of variance reduction appear to exhibit a dimensional effect and are probably ruled out in this range [>12 var.]. Some authors feel that the dimensional effect may even play a role in crude [sampling] methods inasmuch as it may occur in the constant in the asymptotic error term.” (Davis and Rabinowitz) Ian Sloan and Wozniakowski (2003)

26. THANK YOU !

27. Q & A