.. _method-local_reliability-mpp_search:

""""""""""
mpp_search
""""""""""


Specify which MPP search option to use



**Topics**


uncertainty_quantification, reliability_methods


.. toctree::
   :hidden:
   :maxdepth: 1

   method-local_reliability-mpp_search-x_taylor_mean
   method-local_reliability-mpp_search-u_taylor_mean
   method-local_reliability-mpp_search-x_taylor_mpp
   method-local_reliability-mpp_search-u_taylor_mpp
   method-local_reliability-mpp_search-x_two_point
   method-local_reliability-mpp_search-u_two_point
   method-local_reliability-mpp_search-x_multi_point
   method-local_reliability-mpp_search-u_multi_point
   method-local_reliability-mpp_search-no_approx
   method-local_reliability-mpp_search-sqp
   method-local_reliability-mpp_search-nip
   method-local_reliability-mpp_search-integration


**Specification**

- *Alias:* None

- *Arguments:* None

- *Default:* No MPP search (MV method)


**Child Keywords:**

+-------------------------+--------------------+--------------------+-----------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword     | Dakota Keyword Description                    |
|                         | Group              |                    |                                               |
+=========================+====================+====================+===============================================+
| Required (Choose One)   | MPP Approximation  | `x_taylor_mean`__  | Form Taylor series approximation in "x-space" |
|                         |                    |                    | at variable means                             |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `u_taylor_mean`__  | Form Taylor series approximation in "u-space" |
|                         |                    |                    | at variable means                             |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `x_taylor_mpp`__   | X-space Taylor series approximation with      |
|                         |                    |                    | iterative updates                             |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `u_taylor_mpp`__   | U-space Taylor series approximation with      |
|                         |                    |                    | iterative updates                             |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `x_two_point`__    | Predict MPP using Two-point Adaptive          |
|                         |                    |                    | Nonlinear Approximation in "x-space"          |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `u_two_point`__    | Predict MPP using Two-point Adaptive          |
|                         |                    |                    | Nonlinear Approximation in "u-space"          |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `x_multi_point`__  | MPP search for local reliability based on     |
|                         |                    |                    | QMEA multi-point approximation in u-space     |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `u_multi_point`__  | MPP search for local reliability based on     |
|                         |                    |                    | QMEA multi-point approximation in x-space     |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `no_approx`__      | Perform MPP search on original response       |
|                         |                    |                    | functions (use no approximation)              |
+-------------------------+--------------------+--------------------+-----------------------------------------------+
| Optional (Choose One)   | Optimization       | `sqp`__            | Use a sequential quadratic programming method |
|                         | Solver             |                    | for solving an optimization sub-problem       |
|                         |                    +--------------------+-----------------------------------------------+
|                         |                    | `nip`__            | Use a nonlinear interior point method for     |
|                         |                    |                    | solving an optimization sub-problem           |
+-------------------------+--------------------+--------------------+-----------------------------------------------+
| Optional                                     | `integration`__    | Integration approach                          |
+----------------------------------------------+--------------------+-----------------------------------------------+

.. __: method-local_reliability-mpp_search-x_taylor_mean.html
__ method-local_reliability-mpp_search-u_taylor_mean.html
__ method-local_reliability-mpp_search-x_taylor_mpp.html
__ method-local_reliability-mpp_search-u_taylor_mpp.html
__ method-local_reliability-mpp_search-x_two_point.html
__ method-local_reliability-mpp_search-u_two_point.html
__ method-local_reliability-mpp_search-x_multi_point.html
__ method-local_reliability-mpp_search-u_multi_point.html
__ method-local_reliability-mpp_search-no_approx.html
__ method-local_reliability-mpp_search-sqp.html
__ method-local_reliability-mpp_search-nip.html
__ method-local_reliability-mpp_search-integration.html



**Description**


The ``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 ``u_taylor_mean`` option is identical to
the ``x_taylor_mean`` option, except that the approximation is
performed in u-space. The ``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 ``u_taylor_mpp`` option is identical to the ``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
``responses`` specification and may be first or second-order. If
second-order (methods named :math:`AMV^2`  and :math:`AMV^2+`  in
:cite:p:`Eld06a`), the series may employ
analytic, finite difference, or quasi Hessians (BFGS or SR1).
The ``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:p:`Xu98`
for all subsequent MPP predictions. The ``u_two_point`` approach is
identical to ``x_two_point``, but all the approximations are performed
in u-space. The ``x_taylor_mpp`` and ``u_taylor_mpp``, ``x_two_point``
and ``u_two_point`` approaches utilize the ``max_iterations`` and
``convergence_tolerance`` method independent controls to control the
convergence of the MPP iterations (the maximum number of MPP
iterations per level is limited by ``max_iterations``, and the MPP
iterations are considered converged when
:math:`\parallel {\bf u}^{(k+1)} - {\bf u}^{(k)} \parallel_2`  <
``convergence_tolerance``). And, finally, the ``.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 ``npsol_sqp`` optimizer) or nonlinear
interior point (uses the ``optpp_q_newton`` optimizer) algorithms
using the ``sqp`` or ``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 ``integration`` keyword followed by either
``first_order`` or ``second_order``. Second-order integration employs the
formulation of :cite:p:`Hoh88`
(the approach of :cite:p:`Bre84` and the correction
of :cite:p:`Hon99` are also implemented, but are not active).
Combining the ``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 ``refinement`` specification allows the seletion of basic
importance sampling ( ``import``), adaptive importance sampling (
``adapt_import``), or multimodal adaptive importance sampling (
``mm_adapt_import``), along with the specification of number of samples
( ``samples``) and random seed ( ``seed``). Additional details
on these methods are available in :cite:p:`Eld04`
and :cite:p:`Eld06a` and in the main :ref:`Uncertainty Quantification Capabilities section <uq>`.


