.. _strong_maximum_test:
Strong Maximum Test
-------------------
| The Strong Maximum Test is used under the following context:
:math:`\vect{X}` denotes a random input vector, representing the
sources of uncertainties, :math:`\pdf` its joint density probability,
:math:`\vect{d}` a deterministic vector, representing the fixed
variables :math:`g(\vect{X}\,,\,\vect{d})` the limit state function of
the model,
:math:`\cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\}`
the event considered here and :math:`{g(\vect{X}\,,\,\vect{d}) = 0}`
its boundary (also called limit state surface).
| The probability content of the event :math:`\cD_f`:
.. math::
:label: PfX12
\begin{aligned}
P_f &=& \int_{{g(\vect{X}\,,\,\vect{d}) \le 0}} \pdf\, d\vect{x}.
\end{aligned}
may be evaluated with the FORM or SORM method.
| In order to evaluate an approximation of :math:`P_f`, these analytical
methods uses the Nataf isoprobabilistic transformation which maps the
probabilistic model in terms of :math:`\vect{X}` onto an equivalent
model in terms of :math:`n` independent standard normal random
:math:`\vect{U}` (refer to to have details on the transformation). In
that new :math:`\vect{u}`-space, the event has the new expression
defined from the transformed limit state function of the model
:math:`G` :
:math:`\cD_f = \{\vect{U} \in \Rset^n \, / \, G(\vect{U}\,,\,\vect{d}) \le 0\}`
and its boundary:
:math:`\{\vect{U} \in \Rset^n \, / \,G(\vect{U}\,,\,\vect{d}) = 0\}`.
| These analytical methods rely on the assumption that most of the
contribution to :math:`P_f` comes from points located in the vicinity
of a particular point :math:`P^*`, the design point, defined in the
:math:`\vect{u}`-space as the point located on the limit state surface
and of maximal likelihood. Given the probabilistic characteristics of
the :math:`\vect{u}`-space, :math:`P^*` has a geometrical
interpretation : it is the point located on the event boundary and at
minimal distance from the center of the :math:`\vect{u}`-space. Thus,
the design point :math:`P^*` is the result of a constrained
optimization problem.
| The FORM/SORM methods suppose that :math:`P^*` is unique.
| One important difficulty comes from the fact that numerical method
involved in the determination of :math:`P^*` gives no guaranty of a
global optimum : the point to which it converges might be a local
optimum only. In that case, the contribution of the points in the
vicinity of the real design point is not taken into account, and this
contribution is the most important one.
| Furthermore, even in the case where the global optimum has really been
found, there may exist another local optimum :math:`\tilde{P}^*` which
likelihood is slightly inferior to the design point one, which means
its distance from the center of the :math:`\vect{u}`-space is slightly
superior to the design point one. Thus, points in the vicinity of
:math:`\tilde{P}^*` may contribute significantly to the probability
:math:`P_f` and are not taken into account in the FORM and SORM
approximations.
| In these both cases, the FORM and SORM approximations are of bad
quality because they neglect important contributions to :math:`P_f` .
| The Strong Maximum Test helps to evaluate the quality of the design
point resulting from the optimization algorithm. It checks whether the
design point computed is:
- the *true* design point, which means a global maximum point,
- a *strong* design point, which means that there is no other local
maximum located on the event boundary and which likelihood is
slightly inferior to the design point one.
| This verification is very important in order to give sense to the FORM
and SORM approximations.
| The principle of the Strong Maximum Test relies on the geometrical definition
of the design point.
| The objective is to detect all the points :math:`\tilde{P}^*` in the
ball of radius :math:`R_{\varepsilon} = \beta(1+\delta_{\varepsilon})`
which are potentially the real design point (case of
:math:`\tilde{P}_2^*`) or which contribution to :math:`P_f` is not
negligible as regards the approximations Form and SORM (case of
:math:`\tilde{P}_1^*`). The contribution of a point is considered as
negligible when its likelihood in the :math:`\vect{u}`-space is more
than :math:`\varepsilon`-times lesser than the design point one. The
radius :math:`R_{\varepsilon}` is the distance to the
:math:`\vect{u}`-space center upon which points are considered as
negligible in the evaluation of :math:`P_f`.
| In order to catch the potential points located on the sphere of radius
:math:`R_{\varepsilon}` (frontier of the zone of prospection), it is
necessary to go a little further more : thatâ€™s why the test samples
the sphere of radius :math:`R = \beta(1+\tau \delta_{\varepsilon})`,
with :math:`\tau >0`.
| Points on the sampled sphere which are in the vicinity of the design
point :math:`P^*` are less interesting than those verifying the event
and located *far* from the design point : these last ones might reveal
a potential :math:`\tilde{P}^*` which contribution to :math:`P_f` has
to be taken into account. The vicinity of the design point is defined
with the angular parameter :math:`\alpha` as the cone centered on
:math:`P^*` and of half-angle :math:`\alpha`.
| The number :math:`N` of the simulations sampling the sphere of radius
:math:`R` is determined to ensure that the test detect with a
probability greater than :math:`(1 - q)` any point verifying the event
and outside the design point vicinity.
.. image:: FigureStrongMaxTest.svg
:align: center
The vicinity of the Design Point is the arc of the sampled sphere which
is inside the half space which frontier is the linearized limit state
function at the Design Point: the vicinity is
the arc included in the half space :math:`D_1`.
.. image:: StrongMaxTest_vicinity.png
:align: center
:scale: 50
The Strong Maximum Test proceeds as follows. The User selects the
parameters:
- the importance level :math:`\varepsilon`, where
:math:`0 < \varepsilon < 1`,
- the accuracy level :math:`\tau`, where :math:`\tau >0`,
- the confidence level :math:`(1 - q)` where :math:`0