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

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

Objective To calculate automatically verified bounds on the CDF of y

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

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)

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

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: 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…

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

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

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

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

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

Case 2: CDF of a specific value y*

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

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

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

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

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

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

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

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

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

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

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

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)

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