Blurb::
Local reliability method
Description::
Local reliability methods compute approximate response
function distribution statistics based on specified uncertain variable
probability distributions. Each of the local reliability methods can
compute forward and inverse mappings involving response, probability,
reliability, and generalized reliability levels.

The forward reliability analysis algorithm of computing
reliabilities/probabilities for specified response levels is called
the Reliability Index Approach (RIA), and the inverse reliability
analysis algorithm of computing response levels for specified
probability levels is called the Performance Measure Approach (PMA).

The different RIA/PMA algorithm options are specified using the
``mpp_search`` specification which selects among different limit state
approximations that can be used to reduce computational expense during
the MPP searches.
Topics::
uncertainty_quantification, reliability_methods
Examples::

Theory::
The Mean Value method (MV, also known as MVFOSM in
:cite:p:`Hal00`) is the simplest,
least-expensive method in that it estimates the response means,
response standard deviations, and all CDF/CCDF forward/inverse
mappings from a single evaluation of response functions and gradients
at the uncertain variable means. This approximation can have
acceptable accuracy when the response functions are nearly linear and
their distributions are approximately Gaussian, but can have poor
accuracy in other situations.

All other reliability methods perform an internal nonlinear
optimization to compute a most probable point (MPP) of failure. A
sign convention and the distance of the MPP from the origin in the
transformed standard normal space ("u-space") define the reliability
index, as explained in the :ref:`section on Reliability Methods on the Uncertainty Quantification page <uq:reliability>`.
Also refer to :ref:`topic-variable_support` for
additional information on supported variable types for transformations
to standard normal space.  The reliability can
then be converted to a probability using either first- or second-order
integration, may then be refined using importance sampling, and
finally may be converted to a generalized reliability index.
Faq::

See_Also::
