Blurb::
Specify which MPP search option to use
Description::
The \c x_taylor_mean MPP search option performs a
single Taylor series approximation in the space of the original
uncertain variables ("x-space") centered at the uncertain variable
means, searches for the MPP for each response/probability level using
this approximation, and performs a validation response evaluation at
each predicted MPP. This option is commonly known as the Advanced
Mean Value (AMV) method. The \c u_taylor_mean option is identical to
the \c x_taylor_mean option, except that the approximation is
performed in u-space. The \c x_taylor_mpp approach starts with an
x-space Taylor series at the uncertain variable means, but iteratively
updates the Taylor series approximation at each MPP prediction until
the MPP converges. This option is commonly known as the AMV+ method.
The \c u_taylor_mpp option is identical to the \c x_taylor_mpp option,
except that all approximations are performed in u-space. The order of
the Taylor-series approximation is determined by the corresponding \c
responses specification and may be first or second-order. If
second-order (methods named \f$AMV^2\f$ and \f$AMV^2+\f$ in
\cite Eld06a), the series may employ 
analytic, finite difference, or quasi Hessians (BFGS or SR1).
The \c x_two_point MPP search option uses an x-space Taylor series 
approximation at the uncertain variable means for the initial MPP 
prediction, then utilizes the Two-point Adaptive Nonlinear 
%Approximation (TANA) outlined in \cite Xu98 
for all subsequent MPP predictions. The \c u_two_point approach is 
identical to \c x_two_point, but all the approximations are performed 
in u-space. The \c x_taylor_mpp and \c u_taylor_mpp, \c x_two_point 
and \c u_two_point approaches utilize the \c max_iterations and 
\c convergence_tolerance method independent controls to control the
convergence of the MPP iterations (the maximum number of MPP 
iterations per level is limited by \c max_iterations, and the MPP
iterations are considered converged when 
\f$\parallel {\bf u}^{(k+1)} - {\bf u}^{(k)} \parallel_2\f$ < 
\c convergence_tolerance). And, finally, the \c no_approx option 
performs the MPP search on the original response functions without 
the use of any approximations. The optimization algorithm used to 
perform these MPP searches can be selected to be either sequential 
quadratic programming (uses the \c npsol_sqp optimizer) or nonlinear 
interior point (uses the \c optpp_q_newton optimizer) algorithms 
using the \c sqp or \c nip keywords.

In addition to the MPP search specifications, one may select among
different integration approaches for computing probabilities at the
MPP by using the \c integration keyword followed by either \c
first_order or \c second_order. Second-order integration employs the
formulation of \cite Hoh88 
(the approach of \cite Bre84 and the correction 
of \cite Hon99 are also implemented, but are not active). 
Combining the \c no_approx option of the MPP search with first- and 
second-order integrations results in the traditional first- and 
second-order reliability methods (FORM and SORM). These integration 
approximations may be subsequently refined using importance sampling.
The \c refinement specification allows the seletion of basic
importance sampling (\c import), adaptive importance sampling (\c
adapt_import), or multimodal adaptive importance sampling (\c
mm_adapt_import), along with the specification of number of samples
(\c samples) and random seed (\c seed). Additional details 
on these methods are available in \cite Eld04 
and \cite Eld06a and in the Uncertainty 
Quantification Capabilities chapter of the Users Manual 
\cite UsersMan.

Topics::        uncertainty_quantification, reliability_methods	
Examples::
Theory::
Faq::
See_Also::	
