Reminders
=====


Optimization problem associated with an energy functional
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This operator solves the following problem:

Find the triple of eligible fields :math:`(u,v,w)\text{}` that minimize functionality:

:math:`{e}_{\omega }^{2}(u,v,w)=\frac{\gamma }{2}{(u-v)}^{T}[K](u-v)+\frac{1-\gamma }{2}{(u-w)}^{T}{\omega }^{2}[M](u-w)+\frac{1-\alpha }{\alpha }{(\mathit{Hu}-\widehat{u})}^{T}\text{}[\mathit{Gr}](\mathit{Hu}-\widehat{u})`

under duress:

:math:`\left[K\right]v-{\omega }^{2}\left[M\right]\text{w=}0`

Where:

:math:`K` represents a real stiffness matrix

:math:`M` represents a mass matrix

:math:`H` represents an observation matrix

:math:`\mathit{Gr}` represents a symmetric positive definite matrix serving as the norm of errors in the observation space

:math:`\omega =2\pi f`: pulsation of excitement

:math:`\widehat{u}` observation of movements at the pulse :math:`\omega`

:math:`\gamma` parameter for weighting errors :math:`(u-v)` and :math:`(u-w)`

:math:`\alpha` functional weighting parameter similar to a regularization term


Problem solving equations
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Obtaining the triplet :math:`(u,v,w)\text{}` associated with the problem under stress above leads, at each frequency considered, to the resolution of the following system of linear equations:

:math:`Al=b`

with, for each natural pulsation :math:`{\omega }_{i}`:

:math:`{A}_{i}=\left(\begin{array}{cc}\gamma \left(\text{K+γ}/\left(1-\gamma \right){\omega }_{i}^{2}M\right)& -\gamma \left(K-{\omega }_{i}^{2}M\right)\\ -\gamma \left(K-{\omega }_{i}^{2}M\right)& \left(-\text{2α}/\left(1-\alpha \right)\right){H}^{T}{G}_{r}H\end{array}\right)`; and :math:`{b}_{i}=\left(\begin{array}{c}{0}_{n}\\ \left(-\text{2α}/\left(1-\alpha \right)\right){H}^{T}{G}_{r}{\widehat{u}}_{i}\end{array}\right)`

For more details on the wording, refer to document [:external:ref:`R4.10.07 <R4.10.07>`].