In interval analysis, one assumes that nothing is known about 
an epistemic uncertain variable except that its value lies 
somewhere within an interval. In this situation, it is NOT 
assumed that the value has a uniform probability of occuring 
within the interval. Instead, the interpretation is that 
any value within the interval is a possible value or a potential 
realization of that variable. In interval analysis, the 
uncertainty quantification problem is one of determining the 
resulting bounds on the output (defining the output interval) 
given interval bounds on the inputs. Again, any output response 
that falls within the output interval is a possible output 
with no frequency information assigned to it.

We have the capability to perform interval analysis using either
\c global_interval_est or \c local_interval_est.  In
the global approach, one uses either a global optimization method or a
sampling method to assess the bounds.  \c global_interval_est
allows the user to specify either \c lhs, which performs Latin
Hypercube Sampling and takes the minimum and maximum of the samples as
the bounds (no optimization is performed) or \c ego. In the case
of \c ego, the efficient global optimization method is used to
calculate bounds. The ego method is described in
Section <!-- ~\ref{opt:methods:gradientfree:global} -->.  If the problem is
amenable to local optimization methods (e.g. can provide derivatives
or use finite difference method to calculate derivatives), then one
can use local methods to calculate these
bounds. \c local_interval_est allows the user to specify
either \c sqp which is sequential quadratic programming, or
\c nip which is a nonlinear interior point method.

Note that when performing interval analysis, it is necessary to define
interval uncertain variables as described in
Section <!-- ~\ref{variables:uncertain} -->. For interval analysis, one must
define only one interval per input variable, in contrast with
Dempster-Shafer evidence theory, where an input can have several
possible intervals. Interval analysis can be considered a special
case of Dempster-Shafer evidence theory where each input is defined by
one input interval with a basic probability assignment of one. In
Dakota, however, the methods are separate and semantic differences
exist in the output presentation. If you are performing a pure
interval analysis, we recommend using either
\c global_interval_est or \c local_interval_est
instead of \c global_evidence or \c local_evidence, for
reasons of simplicity. %An example of interval estimation is found in
%the \c dakota/share/dakota/examples/users/cantilever_uq_global_interval.in, 
%and also in Section <!-- ~\ref{uq:examples:interval} -->.

%Note that we have kept separate implementations of interval analysis and
%Dempster-Shafer evidence theory because our users often want to couple
%interval analysis on an ``outer loop'' with an aleatory, probabilistic
%analysis on an ``inner loop'' for nested, second-order probability
%calculations. See Section <!-- ~\ref{adv_models:mixed_uq} --> for additional
%details on these nested approaches.
These interval methods can also be used as the outer loop within an
interval-valued probability analysis for propagating mixed aleatory
and epistemic uncertainty -- refer to
Section <!-- ~\ref{adv_models:mixed_uq:ivp} --> for additional details.

<!-- TODO %\subsubsection{Interval Analysis for Cantilever}\label{uq:examples:interval} -->

%Interval analysis is often used to model epistemic uncertainty. 
%In interval analysis, the 
%uncertainty quantification problem is one of determining the 
%resulting bounds on the output (defining the output interval) 
%given interval bounds on the inputs. 

%We can do interval analysis using either
%\c global_interval_est or \c local_interval_est.
%In the global approach, one uses either a global optimization 
%method or a sampling method to assess the bounds, whereas the 
%local method uses gradient information in a derivative-based 
%optimization approach. 
 
An example of interval estimation 
is shown in Figure <!-- ~\ref{uq:examples:interval_input} -->, with example results in 
Figure <!-- ~\ref{uq:examples:interval_out} -->. This example is a demonstration 
of calculating interval bounds for three outputs of the cantilever beam 
problem. The cantilever beam problem is described in detail in 
Section <!-- ~\ref{additional:cantilever} -->. Given input intervals of [1,10] on 
beam width and beam thickness, we can see that the interval estimate of 
beam weight is approximately [1,100].

<!-- TODO
\begin{figure}[htbp!]
  \centering
  \begin{bigbox}
    \begin{small}
      \verbatimtabinput[8]{cantilever_uq_global_interval.in}
    \end{small}
  \end{bigbox}
\caption{Dakota input file for performing UQ using interval analysis --
see \c dakota/share/dakota/examples/users/cantilever_uq_global_interval.in} }
\label{uq:examples:interval_input}
\end{figure}
-->

\anchor uq:examples:interval_out
\verbatim
------------------------------------------------------------------
Min and Max estimated values for each response function:
weight:  Min = 1.0000169352e+00  Max = 9.9999491948e+01
stress:  Min = -9.7749994284e-01  Max = 2.1499428450e+01
displ:  Min = -9.9315672724e-01  Max = 6.7429714485e+01
-----------------------------------------------------------------
\endverbatim
<!-- \caption{Excerpt of UQ output for interval example.} -->



