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\title{A formal introduction of Feedback gain in non-linear models
\author{\sc  La secte ZOOM\\
{Laboratoire de M\'et\'eorologie Dynamique et d'autres}\\
\it{CNRS - Institut Pierre Simon Laplace - Paris - France}
% \date{Manuscript received ???, in final form ???}
}
}
%
%
\maketitle
%
\begin{tabular}{l}
\hline
{\it Corresponding author's address}:\\
Alain Lahellec, \\
Laboratoire de M\'et\'eorologie Dynamique  - UMR 8539\\
Case courrier 99
Tour 25 - 4, place Jussieu\\ 
75252 Paris Cedex 05 \\ France
E-mail: alain@lmd.jussieu.fr  fax: +33 01 44 27 62 72
\end{tabular}
\newpage
%
%% **** A B S T R A C T ***** and keywords
\begin{abstract}
L'intelligibilité des systèmes numériques complexes requiert la construction
d'outils conceptuels pour interpréter les phénomènes d'ensemble qu'ils manifestent.
Le concept de feedback est aujourdhui largement utilise
en climatologie sans qu'une définition rigoureuse en ai été donnée. Cette
lettre a pour but d'introduire formellement ce concept en illustrant les
possibilites d'application. Un exemple simple est donne sur le modele
de Stommel.
\end{abstract}

\vspace{1cm}%

%%%$\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
%
%
\section{The original concept of feedback gain} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
\label{histo}
Hansen {\it et al.} (1987) quote Bode (1945) as originally introducing the Feedback (FB)
concept in the fields of electric circuitry. It is a concept widely used
also in control theory (e.g. Luenberger 79), as representing a cannonical element
of fully controllable, or observable systems. Basically, it is a composition
of two transfer functions in a loop as in Fig~\ref{fbbd}. The bloc diagram
represents a single input -- single output linear system. Given a function
of time as entry $e$, the output signal $s$ is defined by the FB transfer
function given in Fig.~\ref{fbbd}. The diagram is classically viewed in
a symbolic space, where $p$ is the symbolic variable of a transformation:
Laplace or Borel, Fourier, for continuous time systems, or $z$-transformation
for discrete time systems. Hence, the FB gain is fundamentally a dynamic
view of a particular system. The gain $A$ is the transfer function when the FB
loop is cut ($G$ is set to 0). We define as $s_{ins}$ as the open loop response,
i.e. {\it insensitive} to feedback.
When the loop is closed, the FB factor may increase the
insensitive response (when $AG$ is positive) or decrease the original gain
(when the FB factor AG is negative). The FB sign is thus important when analysing
the stability of a system.
% figure 1 bloc diagram
\begin{figure}[h]
\centerline{\psfig{figure=Cambridge/cambdrig2e.eps,%
        bbllx=1pt,bblly=2pt,bburx=776pt,bbury=280pt,%
        width=15.5cm,clip=}}
\caption{Bloc diagram of Feedback loop}
\label{fbbd}
\end{figure}
Using the relation between the output $s(p)$ and, the output when the system is{\emph insensitive}
to the feedback mecanism: $s(p)_{ins}=A(p)e(p)$, one can define the gain factor as
\begin{equation}
AG(p) = \frac{s(p)-s(p)_{ins}}{s(p)}
\label{agbode}\end{equation}
In control theory, G is sought as the FB solution of a control problem. When
a system is fully controllable, the general placement theorem states that
there is always a $G$ function that can be determined such as the effective
gain has a given  characteristic polynomial. This means that any part of
the output state space can be reached by a proper choice of $G$.

The above short introduction of FB gain explains why FB loop represents
more than one among possible structures of bloc diagrams, and is more
like a paradigm of the behaviour of systems, in spite of the strong
limitation of linearity.

As for a definition of the g eff, one can remarq that g(p) is also the response to a step entry
(citation Bode si on trouve ce f.. article, ou un autre).
%
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
\section{How feedback gain is used in climatology} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
\label{clim}
% figure 2 clim bloc diagram
In climatology, the FB gain is a tentative approach to characterise the
behaviour of a perturbed system. Peixoto (1982) for instance, analyses
the effect of perturbing the extra-atmospheric visible radiation income
on the terrestrial mean surface temperature (Fig~\ref{fbclim}).
In such an analysis, the whole climate systems is analysed in terms
of thought leading processes extracted from the global system. Considering
the effect of the surface albedo for instance, one thinks of opening the
corresponding loop. A decrease of $F_{TA}$ leads to a decrease of $T_{SFC}$.
Closing the loop by restoring the albedo effect, the decreasing temperature
has the consequence of extending sea--ice areas on the long term, as well as continental
ice--shells and also snow cover. These new white surfaces will reflect
more of the visible incoming radiation (increase of the mean terrestrial albedo),
and the insensitive decrease of temperature will be enhanced. One concludes
that the albedo FB loop is associated with a positive feedback. The same
is found for water-vapour and high--cloud cover feedbacks, while low clouds
will have a negative feedback on the temperature.
\begin{figure}[h]
\centerline{\psfig{figure=Cambridge/cambdrig3e.eps,%
        bbllx=1pt,bblly=2pt,bburx=791pt,bbury=360pt,%
        width=15.5cm,clip=}}
\caption{The climate system (Peixoto 1982)}
\label{fbclim}
\end{figure}
%

Contrary to the original concept, these views of the climate system are 
purely conceptual, and does not correspond to well defined ``components''
of the climate. The one input one output diagram is obtained by considering
space and time averages of the geophysical fields to begin with. Moreover,
the climate system is by no means considered as a linear system. An implicit
hypothesis underlying the FB view of the climate system is that when
considering perturbations instead of state variables, the long--term
behaviour of the system responds linearly to perturbation, provided its
amplitude stays within some limits (no climate surprise). The main difference
compared to the original FB concept is the loss of the dynamic aspect of the
feedback function. 
Concerning the linear hypothesis, one more step is applied when featuring
an explanation of the climate system. The leading processes are supposed
well separated, i.e. non internally interacting, so that the overall effective transfer
function at equilibrium is composed of a sum of individual FB factors
(Fig.~\ref{fbclim}). This separation of different process is difficult to
attain when using General Climate Models (GCM) to determine a set of
FB factors (Schneider and Thompson, 1981).
%
Coakley 77 and Wetherald and Manabe 88  give nevertheless a formal justification
of feedback
as follows. Suppose that the climate system perturbed by $\Delta F$ relaxes
to a new equilibrium. Thus $\Delta R$, the modified Output Longwave Radiation (OLR)
of planet
to space equals $\Delta F$ when the surface temperature change $\Delta T_s$ comes
to the new equilibrium. One may write
%
\begin{equation*}
\Delta F = \Delta R = \left( \frac{\partial R}{\partial T_s} + \sum_{i}\frac{\partial R}{\partial x_i}
  \frac{dx_i}{dT_s} \right) \Delta T_s
\label{eq_pdr}
\end{equation*}
%
In this equation, the rhs first partial derivative is said to be the direct effect
of the perturbation of surface temperature on the OLR. The added terms stand for indirect effects,
or ``going through $x_i$'' feedback effects. The separation is implicitly adressed when writing total derivatives
$\frac{dx_i}{dT_s}$. This implies in fact a  limitation of the method: each feedback
loop must be dependant on the surface temperature only. In other words, the $\Delta F$
perturbation must impact directly on $T_s$, which in turn is perturbing processes that
will contribute to the new equilibrium. Again, this equation is purely conceptual and
has no clear counterpoint in the models. It merely is an equivalent  of Fig.~\ref{fbclim}
and a justification of the practical determination described now.
% figure 3 determination pratique
\begin{figure}[h]
\centerline{\psfig{figure=Cambridge/cambdrig4e.eps,%
        bbllx=1pt,bblly=2pt,bburx=750pt,bbury=316pt,%
        width=15.5cm,clip=}}
\caption{The three trajectories FB determination: change in terrestrial
surface temperature resulting from a perturbation of the atmospheric
$CO_2$ content}
\label{fbthree}
\end{figure}
%

How are FB factors determined from GCMs ?
The common practice uses thee trajectories: an open loop perturbed
trajectory is run with a modified version of a GCM, in which some
process is frozen: for instance, impeding a change in sea-ice surfaces.
One obtains in the long term, after eventual chaotic transients generated
by the perturbation have died, the insensitive trajectory of Fig.~\ref{fbthree}.
This determines $\Delta T_{ins}$ as the difference between the insensitive
trajectory and the reference one. The third trajectory is determined with
the full degree of freedom of the GCM restored (closed--loop).
The feedback gain factor is determined as
\begin{equation}
AG(t) = \frac{\Delta T(t)-\Delta T(t)_{ins}}{\Delta T(t)}
\label{aggcm}\end{equation}
in analogy to equation~(\ref{agbode}), with the essential difference that here,
real trajectories are considered. The linearity
is checked by considering the ratio of the two $\Delta T$ differences
variation in time: if it is slowly varying, one considers that the long term
climate model responds linearly to the perturbation and the FB factor
is valuable. This practice is applied in numerous tentatives to sketch
the climate as an intelligible system (see Carney 77, Hansen et al. 84,
Schlesinger et al. 87,  Ramanathan 89, Cess 89, Lindzen 93,
 Dufresne et al. 2003 ...).
It is however a conceptual view projected on models, which means that
the models themselves are inadequate to provide the values. Moreover,
the difficulty of properly separating different processes, and freezing
some of them without violating physical laws render questionable such
pratice, as claimed by Rossow et al. (2002). These authors concluded that
the FB gain concept should be abandoned as lacking of rigorousness.

In summary, it appears that the climate researchers are desperately
looking for some tools that can restore some intelligibility on the
general behaviour of such complex numerical models as the
climate system. The Feedback loop is considered as a paradigm of
such a conceptual tool, but remains out of reach of scientific
rigorousness. It is thus a challenge to build such indispensable tools
while keeping a scientific objectivity. We show that it is possible
in the next sections.

\begin{verbatim}
-> conclusions : 1/ pb de structure
                 2/ linearisation
                 3/ aspect dynamiques important
                    (convolution -> diagnostic de rechauffement cliomatique)
                 3/ definition generale de gain effectif
                    g_eff = sensibilite a un step
\end{verbatim}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Feedback and structure of the model} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
\label{struct}
Si on veut pouvoir étudier les inter-influences entre deux systèmes, il
est nécessaire de définir ces deux systèmes comme,  la fois, séparés, et
couples. Considérons un modèle couple ocean-atmosphere; on a deux systèmes
arbitrairement séparés (l'océan d'une part, l'atmosphere de l'autre). Chacun
d'eux est bien défini des qu'on lui adjoint des conditions limites, et le
modèle mathématique associe doit alors conduire a un problème bien pose.

On prend le cas d'un couplage simplifie où l'océan est spécifié par un flux
provenant de l'atmosphere ($\varphi$), et celle-ci par la SST ($\theta$).
(ref : on pourra citer les papiers de XXX sur la coherence math du couplage). 
On examine d'abord le cas particulier du couplage unicolonne.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
begin{verbatim}
Formulation par le TEF, avec explicitation d'un gain
en fonction des 4 matrices.

Ensuite, montrer qu'on deux gains equivalents scalaires
et deux "matrices de gains" // Fig 4

Dire qu'on peut definir l'aspect stabilisant ou non du couplage
par <Delta2 Phi>/<DeltaPhi> => determinant de [G].
(où Delta2 Phi = Delta Phi - Delta - Phi ins)
Hors, ce determinant est caracteristique de la boucle. En deduire la definition
du gain de la boucle et l'interet de le determiner sous forme scalaire.

on place la remarque sur la possibilite d'aller plus loin en analysant
les structures singulieres de G et donc des champs associes a des gains.
end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Le système forme une boucle de rétroaction parce que, symboliquement :
%
\begin{equation}
\frac{d \eta}{d \zeta} = \frac{\partial \eta}{\partial \theta}\frac{\partial \theta}{\partial \zeta} ;\qquad
 \frac {\partial \eta}{\partial \varphi} = 0
\end{equation}
et que
%
\begin{equation}
\frac{d \zeta}{d \eta} = \frac{\partial \zeta}{\partial \varphi}\frac{\partial \varphi}{\partial \eta} ;\qquad
 \frac {\partial \zeta}{\partial \theta} = 0
\end{equation}
Autrement dit, la structure du système partage les {\it sensibilités} des sous-systèmes
selon les variables d'interface.
% figure systeme Ocean-atm et deux gains
\begin{figure}[h]
\centerline{\psfig{figure=PS/ocatm.eps,%
        bbllx=0pt,bblly=0pt,bburx=626pt,bbury=218pt,%
        width=15.5cm,clip=}}
\caption{A Feedback loop between atmosphere and ocean and two feedback gains}
\label{ocatm}
\end{figure}
%
suite a refaire, mais je laisse parce qu'ilre ste ce que l'on dit a propos du dt !

Ces contraintes peuvent être prises dans leur sens le plus général pour définir
la structure de boucle de rétroaction. On voit cependant que l'utilisation de symboles
de dérivation qui expriment les interdépendances et leur média (variables d'interface) nécessitent
une explicitation : quel est le rôle du temps, que représentent les variables etc.
Ces points seront formellement élucidés dans la section~\ref{gateau}. Néanmoins, la boucle de rétroaction
de la figure~\ref{ocatm} définit deux gains de rétroaction, suivant que l'on s'intéresse
à l'effet de l'océan sur l'atmosphere ou réciproquement. Dans le premier cas par exemple,
on considère une perturbation (un forçage) appliquée à la SST ($\Delta \Pi_{\theta}$). Cette perturbation
modifie l'état de l'atmosphère par changement des conditions limites du système d'équations
associées : c'est l'effet direct. De plus, les flux $\varphi$ subiront les conséquences de ce
changement, perturbant l'océan, de telle manière que la SST en sera affectée.
Autrement dit
la perturbation initiale $\Delta\Pi_{\theta}$ de $\theta$ sera in fine modifiée par la réponse complète de la
boucle de rétroaction ($\Delta\theta$). Pour rester dans le cadre de la définition de gain comme fonction scalaire,
on suppose que dans l'exemple du couplage océan-atmosphère, on ne considère qu'une colonne
verticale du modèle maillé, pour laquelle $\theta$ et $\varphi$ sont deux scalaires.
On peut alors caractériser un gain effectif de rétroaction entre ces deux variables, et écrire
symboliquement :
%
\begin{equation}
\Delta \varphi = \frac{\frac{\partial \varphi}{\partial \eta}\frac{\partial \eta}{\partial \theta}}
{1-\frac{\partial \varphi}{\partial \eta}\frac{d\eta}{d\zeta}\frac{\partial \zeta}{\partial \varphi}}\Delta\Pi_{\theta}
\label{dfi}\end{equation}
%
The converse feedback effective gain would appear as ... {\bf si on a deja dit que meme determinant,
on conclue que c'est le meme - ou dans l'ordre inverse, a voir.}
%
\begin{equation}
\Delta \theta = \frac{\frac{\partial \theta}{\partial \zeta}\frac{\partial \zeta}{\partial \varphi}}
{1-\frac{\partial \theta}{\partial \zeta}\frac{d\zeta}{d\eta}\frac{\partial \eta}{\partial \theta}}\Delta\Pi_{\varphi}
\end{equation}
and would give the response of the SST to a perturbation externally applied to the fluxes
entering the ocean ($\Delta\Pi_{\varphi}$). 

Donc, en gros, on est debarrases du pb de structure et on passe au reste.

Mais il faut bien aborder ici l'aspect dynamique, et prendre l'exemple de l'equilibre avec Borel
avant de sortir ?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A formal introduction of feedback gain in numerical models} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
\label{gateau}
On a definit le gain effectif (1/1-g ou a/1-ag) par une analyse de sensibilite associee
a une structure de boucle. Hors, Cacuci etc, que l'on rappelle.

After twenty years of nuclear--reactor modelling at Oakridge, Cacuci
provided what he called a general sensitivity theory~({\it in J. of
Math. Physics}, Dec. 1981). We
briefly recall this formal theory before calling for the FB concept.
%
\subsection*{The general sensitivity theory of Cacuci} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Consider the following system of equations where small letters symbols
are vectors:

{\bf je verrai mieux s'en tenir au meme modele Oc-Atm et simplifeir les notations au max.}

% Gateau pour TLS
\begin{equation}
\left\{
\begin{array}{cc}
\mathcal{N}(u(x),\alpha(x),x)\quad=&\mathcal{Q}(\alpha(x),x) \,; \,\,x\in\Omega\\
| \mathcal{A}(u,\alpha)- \mathcal{B}(\alpha) \,|_{\partial \Omega}=&0\\
\end{array}
\right.
\label{gat}
\end{equation}
$\mathcal{N}$ stands for a non-linear operator operating on state vectors
$u$ (say in a square integrable Hilbert space). $x$ are space-time coordinates,
$\mathcal{Q}$ represents inhomogeneous sources, and the second equation
is a general form for initial conditions and boundary limit conditions.
$\alpha$ are model parameters.

The general sensitivity theory considers a functional response
$\mathcal{R}(u,\alpha)$ whose variation -- e.g. to a change in parameters $\alpha$
-- is analysed. Cacuci based his definition of sensitivity on the Gâteaux-differential
(G-differential) defined as 
%
\begin{equation*}
\mathcal{V}\mathcal{R}(e) = \lim_{\epsilon\rightarrow 0}\\
{\frac{\mathcal{R}(e +\epsilon \Delta e)-\mathcal{R}(e)}{\epsilon}}
\end{equation*}
%
for a given reference $e$, that may be defined as the reference trajectory.
$\epsilon$ is a scalar and $\Delta e$ is any perturbation vector of $e$ (i.e.
corresponding to a perturbed trajectory).
A variation of $\mathcal{R}$ is related to the G-differential by : 
%
\begin{equation*}
\mathcal{R}(e +\Delta e)-\mathcal{R}(e)= \mathcal{V}\mathcal{R}(e;\Delta e) +\Upsilon(\Delta e)
\end{equation*}
%
where
%
\begin{equation*}
\lim_{\epsilon\rightarrow 0}{\frac{\Upsilon(\epsilon \Delta e)}{\epsilon}} = 0.
\end{equation*}
%
The use of the G-differential has the definite advantage that
it does not require continuity in the vector variable $e$. The result of this operation
on a system neither needs to be linear in $\Delta e$.
For clarity, we will nevertheless consider only systems
that are linear in $\Delta e$, which requires necessaires et suffisantes conditions (see cacuci):
\begin{itemize}
\item a weak Lipschitz condition, that insure numerical stability on a perturbed trajectory
\item and the linearizability condition
%
\begin{equation}
\mathcal{R}(e+\epsilon \Delta e_1 + \epsilon \Delta e_2) -\mathcal{R}(e+\epsilon \Delta e_1 )\\
-\mathcal{R}(e+\epsilon \Delta e_2) +\mathcal{R}(e) = o(\epsilon)
\label{lin}
\end{equation}
\end{itemize}
in that case the G-variation is called a G\^ateaux derivative (G-derivative).
Applying the D-derivative to $\mathcal{R}(u,\alpha)$ yields
\begin{equation}
\mathcal{V}\mathcal{R} = \mathcal{R' }_1 \Delta u + \mathcal{R' }_2 \Delta \alpha
\label{dresp}
\end{equation}
%
Two terms appear in the above equation, the second rhs term giving the
direct contribution of the perturbed parameters, while the first is
depending on the modified trajectory. As the sensitivity of
$\mathcal{R}$ is looked for as a (sensitivity) gradient to perturbation , one looks for
methods to eliminate $\Delta u$, that is the trajectory dependence of
$\mathcal{V}\mathcal{R}$. This requires to forward integrate the system along the reference trajectory.
One might apply the G-differential to~(\ref{gat}) to get
the Linear Tangent System (TLS) related to the reference trajectory
% Gateaux pour TLS
\begin{equation}
\left\{
\begin{array}{cc}
\mathcal{N}(u(x),\alpha(x),x)\quad=&\mathcal{Q}(\alpha(x),x) \\
| \mathcal{A}(u,\alpha)- \mathcal{B}(\alpha)\, |_{\partial \Omega}=&0\\
\end{array}
\right.
\quad\stackrel{TLS}{\longrightarrow}\quad
\left\{
\begin{array}{cc}
\mathcal{N}' _1 \Delta u + \mathcal{N}' _2 \Delta \alpha\qquad=&\mathcal{V}\mathcal{Q}(\alpha;\Delta \alpha) \\
| \mathcal{V}\mathcal{A}(u,\alpha)- \mathcal{V}\mathcal{B}(\alpha)\, |_{\partial \Omega}=&0\\
\end{array}
\right.\label{gat2}
\end{equation}
%
A step by step forward integration of the system is used to
replace the $\Delta u(t)$ vector in~(\ref{dresp}) by algebraic
manipulations. This will provide the variation of $\mathcal{R}$ in a form explicitly
dependent on the perturbed variable. 
Alternatively one can use the Adjoint System of the TLS
(see Cacuci 82,83,84), and backward integrate. The adjoint variables are then used to
eliminate the trajectory dependance on $\mathcal{V}\mathcal{R}$.
We will apply the direct method to
reach the definition of feedback function.
%
\subsection*{A formal definition of feedback} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
On se contenterait alors de montrer comment on utilise l'adjoint pour que la reponse
ne soit plus associee qu'aux perturbations.

Et alors, le gain eff = terme $<d w-_dj>$ dans la reponse $(R=\int_0 ^T <d.u>r(t)dt)$.

retour sur Oc-Atm $<d| = 1/n-mailles-interface <1,1,1,...1|$ donne le gain de la moyenne des u
(temperature ou flux).

On a donc etendu la notion auxx systemes non-L, mais alors le gain eff dyn est lie a un instant
diagnoctic (t=c) : g(tau,c) avec t=c-tau.


\section{Application to the Stommel model of thermohaline circulation} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
on n'a plus qu'a donner un exemple avec des commentaires rassurants sur l'analyse
du gain en faisant varier c.
\section{Conclusive discussion} 
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
\begin{verbatim}
a/   definition formelle de gain dynamique
    -> necessite de calculer les Jacobiennes du systeme  ou de definir le TLS le long...
       ce qui permet le contexte lineaire
    -> obtenu par Borel dans le cas lineaire
    -> extension du sens des tempos caracteristiques obtenus par Borel
       au cas non-L // Cacuci en adjoint
    -> application pour systemes non-L avec instant fixé de diagnostic
       par sensibilite d'une fonctionnelle de reponse.

b/   il est possible d'utiliser le concep de gain dynamique de retro
pour les systemes complexes a condition d'utiliser l'adjoint.

c/   Conclusions generale sur les methodes pragamatiques de la litterature.
et sur la possibilite d'appplication numerique par Gateaux fini sans adjoint.
donc sur l'importance de fondements rigoureux pour prolonger et verifier
les conditions d'applications pratiques des gains.
Enfin prolongements prevus pour exploiter l'analyse de structures multi-dim
associees a des gains // EOF.
\end{verbatim}

\end{document}

%\bibliography{Biblio}
\end
