Reliability methods provide an alternative approach to uncertainty
quantification which can be less computationally demanding than
sampling techniques. Reliability methods for uncertainty
quantification are based on probabilistic approaches that compute
approximate response function distribution statistics based on
specified uncertain variable distributions. These response statistics
include response mean, response standard deviation, and cumulative or
complementary cumulative distribution functions (CDF/CCDF). These
methods are often more efficient at computing statistics in the tails
of the response distributions (events with low probability) than
sampling based approaches since the number of samples required to
resolve a low probability can be prohibitive.

The methods all answer the fundamental question: ``Given a set of
uncertain input variables, \f$\mathbf{X}\f$, and a scalar response
function, \f$g\f$, what is the probability that the response function is
below or above a certain level, \f$\bar{z}\f$?'' The former can be written
as \f$P[g(\mathbf{X}) \le \bar{z}] = \mathit{F}_{g}(\bar{z})\f$ where
\f$\mathit{F}_{g}(\bar{z})\f$ is the cumulative distribution function
(CDF) of the uncertain response \f$g(\mathbf{X})\f$ over a set of response
levels. The latter can be written as \f$P[g(\mathbf{X}) > \bar{z}]\f$ and
defines the complementary cumulative distribution function (CCDF).

This probability calculation involves a multi-dimensional integral
over an irregularly shaped domain of interest, \f$\mathbf{D}\f$, where
\f$g(\mathbf{X}) < z\f$ as displayed in Figure \ref figUQ05 for the
case of two variables. The reliability methods all involve the
transformation of the user-specified uncertain variables,
\f$\mathbf{X}\f$, with probability density function, \f$p(x_1,x_2)\f$, which
can be non-normal and correlated, to a space of independent Gaussian
random variables, \f$\mathbf{u}\f$, possessing a mean value of zero and
unit variance (i.e., standard normal variables). The region of
interest, \f$\mathbf{D}\f$, is also mapped to the transformed space to
yield, \f$\mathbf{D_{u}}\f$ , where \f$g(\mathbf{U}) < z\f$ as shown in
Figure \ref figUQ06. The Nataf transformation \cite Der86,
which is identical to the Rosenblatt transformation \cite Ros52 in
the case of independent random variables, is used in Dakota to
accomplish this mapping. This transformation is performed to make the
probability calculation more tractable. In the transformed space,
probability contours are circular in nature as shown in
Figure \ref figUQ06 unlike in the original uncertain variable
space, Figure \ref figUQ05 . Also, the multi-dimensional integrals
can be approximated by simple functions of a single parameter,
\f$\beta\f$, called the reliability index. \f$\beta\f$ is the minimum
Euclidean distance from the origin in the transformed space to the
response surface. This point is also known as the most probable point
(MPP) of failure. Note, however, the methodology is equally applicable
for generic functions, not simply those corresponding to failure
criteria; this nomenclature is due to the origin of these methods
within the disciplines of structural safety and reliability.
Note that there are local and global reliability methods. The majority 
of the methods available are local, meaning that a local optimization 
formulation is used to locate one MPP. In contrast, global methods
can find multiple MPPs if they exist.


\anchor figUQ05
\image html cdf_orig_graphic.png "Graphical depiction of calculation of cumulative distribution function in the original uncertain variable space."
\image html cdf_orig_graphic.eps "Graphical depiction of calculation of cumulative distribution function in the original uncertain variable space."

\anchor figUQ06

\image html cdf_tran_graphic.png "Graphical depiction of integration for the calculation of cumulative distribution function in the transformed uncertain variable space."
\image html cdf_tran_graphic.eps "Graphical depiction of integration for the calculation of cumulative distribution function in the transformed uncertain variable space."
