Reliable Computing (2007) 13: 195–209
A Search Algorithm for Calculating Validated
The University of Texas at Austin, Department of Civil Engineering, 1 University Station C1792,
Austin, TX 78712–0280, USA, e-mail: email@example.com
(Received: 29 November 2004; accepted: 3 July 2005)
Abstract. The search algorithm presented allows the CDF of a dependent variable to be bounded with
100% conﬁdence, and allows for a guaranteed evaluation of the error involved. These reliability bounds
are often enough to make decisions, and often require a minimal number of function evaluations. The
procedure is not intrusive, i.e. it can be equally applied when the function is a complex computer
model (black box). The proposed procedure can handle input information consisting of probabilistic,
interval-valued, set-valued, or random-set-valued information, as well as any combination thereof.
The function as well as the joint pdf of the input variables can be of any type.
Determining validated bounds for the Cumulative Distribution Function (CDF) of
a function of random variables has attracted the attention of many scholars and a
recent literature review may be found in . R. E. Moore  and A. S. Moore
 were probably the ﬁrst ones to use interval analysis  to this end.
Forexample, Berleant and co-workers developed Statool –, a computer
program for obtaining bounds on the distributions of sums, products, and various
other functions of random variables where the dependency relationship of the
random variables need not be speciﬁed. Ferson  developed RiskCalc with
similar capabilities. Independently, Lodwick and Jamison  presented a method
for estimating and validating the cumulative distribution of a function of random
variables (independent or dependent).
Dubois and Prade  ﬁrstly indicated how Random Set Theory might be
used to bound the Cumulative Distribution Function (CDF) of a sum of two random
variables. Tonon et al.  and  generalized this idea to provide veriﬁed bounds
to the CDF of a general function y =
(u) where u is a generic random vector.
Random Set Theory allowed the abovementioned procedures developed by different
authors to be cast into a rigorous light. They also showed that their procedure can be
used equally well when some components of u are described as random variables,
some others as intervals or Cartesian products, and some others as random sets.
Additionally, a procedure was introduced to calculate the CDF of a particular value,
,ofy; this procedure is meant to be used in reliability analyses and yields veriﬁed
bounds on the reliability of a system. The motivation behind this procedure is