.. _strong_maximum_test: Strong Maximum Test ------------------- The Strong Maximum Test is used under the following context: let :math:`\vect{X}` be a probabilistic input vector with joint density probability :math:`\pdf`, let :math:`\vect{d}` be a deterministic vector, let :math:`g(\vect{X}\,,\,\vect{d})` be the limit state function of the model and let :math:`\cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\}` be an event whose probability :math:`P_f` is defined as: .. math:: :label: PfX12 P_f = \int_{{g(\vect{X}\,,\,\vect{d}) \le 0}} \pdf\, d\vect{x} The probability :math:`P_f` is evaluated with the :ref:`form_approximation` and :ref:`sorm_approximation` methods. These methods use 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 :ref:`isoprobabilistic_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 \, / \, h(\vect{U}\,,\,\vect{d}) \le 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. Both FORM and SORM methods assume that :math:`P^*` is unique. One important difficulty comes from the fact that numerical methods involved in the determination of :math:`P^*` gives no guaranty of a global optimum: the point to which they converge 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}^*` with likelihood only slightly inferior to the likelihood of the design point one, which means that it is only slightly further apart from the center of the :math:`\vect{u}`-space than the design point. 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 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 with likelihood only slightly inferior to the likelihood of 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 whose 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 less than :math:`\varepsilon` times the likelihood of the design point. The radius :math:`R_{\varepsilon}` is the distance to the :math:`\vect{u}`-space center upon which points are considered as negligible :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 : this is 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 : the latter might reveal a potential :math:`\tilde{P}^*` whose 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 detects 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 whose 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