\documentclass[a4paper,12pt]{article}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amsfonts}
%pour les figures
\usepackage{eepic,psfig,pstricks,multirow}
\usepackage{floatflt}
\usepackage{fancyheadings}                                
\usepackage{footnpag}        
\usepackage{rotating}
\usepackage{euscript}
\usepackage{tabularx}
\usepackage[frenchb]{babel}
\usepackage[latin1]{inputenc}
\addtolength{\textheight}{2cm}
%
\def\dt{{\delta t}}
\def\edtaux{{exp (-{\delta t  \over \taux } ) }}
\def\deta{{\delta \eta } }
\def\Deta{{\Delta \eta } }
\def\Dt{{\delta t } }
\def\dfi{{\delta \varphi } }
\def\Dfi{{\Delta \varphi } }
\def\dts{{\delta t\over 2 } }
\def\dtaf{{\overline {\overline {\partial _\eta f }}} }
\def\d(#1)G{{\overline {\overline {\partial_#1 G}}} }
\def\VD{{\mathcal V \mathcal D}}
%à ù
\newcommand{\cad}{c'est-\`a-dire }
\newcommand{\dgdeta}{\overline{ \overline{ \frac{\partial G}{\partial \eta}}}}
\newcommand{\dgdphi}{\overline{ \overline{ \frac{\partial G}{\partial \varphi}}}}
\newcommand{\dgdt}{\frac{\partial \vec{G}}{\partial t}}
\newcommand{\dfdeta}{\overline{ \overline{ \frac{\partial f}{\partial \eta}}}}
\newcommand{\dfdphi}{\overline{ \overline{ \frac{\partial f}{\partial \varphi}}}}
\newcommand{\dfdt}{\vec{ \frac{\partial f}{\partial t}}}
\newcommand{\vecdeltaeta}{\overrightarrow{ \delta \eta}}
\newcommand{\vecdeltaetadec}{\overrightarrow{ \delta \eta}_{dec}}
\newcommand{\vecdeltaphi}{\overrightarrow{ \delta \varphi}} 
\newcommand{\vecdeltaphiins}{\overrightarrow{ \delta \varphi}_{ins}}
\newcommand{\vecdeltaz}{\overrightarrow{ \delta z}} 
\newcommand{\vecdeltazins}{\overrightarrow{ \delta z_{ins}}} 
\newcommand{\vecdeltazinsst}{\overrightarrow{ \delta z^*_{ins}}} 
\newcommand{\vecdeltax}{\overrightarrow{ \delta x}}  
\newcommand{\vecdeltaxins}{\overrightarrow{ \delta x_{ins}}} 
\newcommand{\matra}{\overline{ \overline{ A}}} 
\newcommand{\matramun}{\overline{ \overline{ A^{-1}}}}
\newcommand{\matrb}{\overline{ \overline{ B}}} 
\newcommand{\matrcplus}{\overline{ \overline{ C^{+}}}} 
\newcommand{\matrcbardt}{\overline{ \overline{\mathbb{C}_{(\delta t)}}}}
\newcommand{\matrcbardtst}{\overline{ \overline{\mathbb{C}^*_{(\delta t)}}}}
\newcommand{\matrd}{\overline{ \overline{ D}}} 
\newcommand{\matrf}{\overline{ \overline{ F}}}
\newcommand{\matri}{\mathbb{I}}

\newcommand{\B}{\mathcal{B}}
\newcommand{\C}{\underline{\underline{\mathcal{C}}}}
\newcommand{\eC}{\mathcal{C}}
\newcommand{\rdelta}{\mathring{\delta}}

\textwidth 15cm
                                                                     
\begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{The Transfer Evolution Formalism: \\
                an introduction to model partitioning and coupling,
and feedback analysis
\author {The TEF-ZOOM Collaboration\footnote{Paper originated by Venance Journé.
Final redaction: Jean-Yves Grandpeix, Stéphane Hallegatte, Alain Lahellec and
Venance Journé. Reviewers: Stéphane Blanco,
Patrice Dumas, Jean-Louis Joly and Vincent Platel.}
\\(CIRED\footnote{Centre International de Recherche sur l'Environnement et
le Développement , Paris} ,
LE\footnote{Laboratoire d'Énergétique, Université Paul Sabatier, Toulouse and its
Tarbes antenna} , LMD\footnote{Laboratoire de Météorologie Dynamique, Paris} )}
%\nodate 
} 
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\vspace{2cm}
%\begin{center}
%{\bf VERY PRELIMINARY VERSION}
%\end{center}
%
\vspace{2cm}
The purpose of the Transfer Evolution Formalism, from its genesis, is to allow
the modelling of complex systems in which
various involved processes are more or less strongly coupled. The analysis of the coupling
mechanisms which are
influencing the behaviour of the different parts of the model is fundamental, as
it allows some understanding of the links between elementary processes
that are to be 
modelled. The interaction between sub-models which may be ``black boxes'' needs specific 
methods to drive the analysis of the imbedded couplings and feedbacks.

The modelling procedure should meet several requirements:
%
\begin{itemize}
\item modularity to allow the implementation, in one single model, of various 
sub-models suited to each process or each part of the system,
\item able to provide information on the couplings between the various evolutions of different parts 
of the system,
\item it must allow to study the non-stationary dynamics as well as the search of stationary 
solutions.
\end{itemize}
%
The description of a system using a modular model will proceed in two 
complementary steps: partitioning of the global system in different subsystems
(corresponding to different phenomena), and connecting of all these parts to 
recover the global system. To acomplish that feature, One replaces the global system by a set of
partial models
and  (eventually) of additional variables and relations, such that the new set of equations be
mathematicaly equivalent to the global model. 
\newpage
For example, in the field of environmental integrated assessment\footnote{TEF-ZOOM has been
developped in the fields of Engineering Sciences. This text was written for
a European Program towards integrated assesment, which explains the choice
of some illustrative examples.}, there exist
a number of detailed disciplinary models. It is important to be able to study
the interfaces between these 
models and such a formalism as the TEF is well suited to this task. One reason
is that the resolution of the system of equations proceeds by the elimination
of all variables except the ones describing the interfaces to be analysed. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Partitioning and re-coupling a system
                              using the TEF: cells and transfers.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Modelling of a system through partitioning and re-coupling is involved in
system analysis, when asking questions as: what cross-influence two subsystems have
on one another. To answer such question, each sub-model must be well defined, and an interface
between these objects generally needs to be conceived. This is what is called partitioning here.

 A TEF partitioning is applied in a recursive manner,
each subsystem of a partition being submit to partitioning again.
The final elements of the partitioning should represent a satisfying
level of analysis of the original system.
A good practice considers an elementary object as eventually submit to an
experimental procedure, hence quasi isolated from the environment,
and linked to it through well identified physical quantities (interface conditions).

Let start with a system defined from its state equations:
%
\begin{equation}
\frac{d \vec{\eta} }{dt}  = \vec{ G} ( \vec{\eta},\vec{\psi}, t )
\label{deta}
\end{equation}
%
Given initial conditions $\vec{\eta} (t_0)$, and boundary conditions, and
forcing (independent sources) $\vec{\psi} (t)$, a system trajectory
is hence determined\footnote{between eventual singularities}: one
says that the mathematical system is ``well posed''.

The {\bf partitioning} is applied to the vector space $\{\vec{\eta}\}$,
leading to a state equation for each sub-vector space $\{\vec{\eta_{\alpha}}\}$:
%
\begin{equation}
\frac{d \vec{\eta}_{\alpha} }{dt}  = \vec{ G} ( \vec{\eta}_{\alpha},\vec{\varphi},\vec{\psi},t )
\label{deta2}
\end{equation}
%
where some new boundary conditions $\vec{\varphi}$ had to be added as needed by
the TEF to have well posed sub-systems. Two consequences are drawn:
\begin{enumerate}
\item each element, called ``a cell'', should not explicitly depend on
another cell state variables;
\item the interface that had to be introduced represents {\bf dependant}
boundary conditions, which means they depend on cells state.
\end{enumerate}
%
Suppose these interfacing conditions are chosen, hence each cell will follow
a trajectory that we call a ``decoupled trajectory''. For these trajectories to
become the true system trajectory, one needs to specify a supplemental equation for
each interface variable which has to explicit how the variable is constrained
by the state space environment. The TEF calls ``transfers'' these interfacing
objects:
%
\begin{equation}
\vec{\varphi}  = \vec{ f} ( \vec{\eta} , \vec{\varphi}, t )
\label{vecphi}
\end{equation}
%
Equation~(\ref{vecphi}) expresses that transfer variables are statically
determined by state variables, and hence do not introduce new initial conditions.
{\bf Coupling} is the operation that defines all transfers between cells.
They usually correspond to mathematical constraints (continuity) or physical
quantities exchanged between cells (as fluxes).

On a mathematical point of view, systems~($\ref{deta2},\ref{vecphi}$)
and~(\ref{deta}) have the same solution.
To summarise, partitioning and coupling with the TEF uses two classes of
objects, and the two classes of associated variables.
{\bf Cells} are mathematically represented by state variables and
state equations (equations of evolution using partial time derivative), and
{\bf transfers} as static constraint equations.

Cells models are very similar to boundary condition problems.
The difference lies in the introduction of a dependence of these
conditions as being function of the rest of the system. One can infer from
the TEF structure of a system that the analysis elements of inter-influence
are set: a cell has to go through transfers to influence the rest of
the system. Reciprocally, the reacting system retro-influences each cell
through the set of transfers connecting the cell to the system.

To fully explicit the mutual influences, the TEF modelling of a cell
explicits the sensitivity of the state variables to changing interface
conditions. This is mathematically applied through the linearisation
of the state space equations, as will now be described.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linearisation, analysis of the  TEF \\
operators, and resolution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
The relation between sub-systems is excessively difficult to
exhibit when having to cope with non-linear systems. To meat that purpose,
the TEF uses
the Tangent Linear System (TLS) along a trajectory.
The continuous time linearised equations are built firstly, and a time
scheme is applied in a second phase.

After these two steps, one gets each time step, an algebraic system
equivalent to the full system.
The linearised system allows analyses in terms of
sensitivity to perturbation, coupling matrices or feedback factors.
The TEF uses the TLS both for simulation (system trajectory), and
analysis.
%
%--------------------------------------------------
\subsection{Linearisation}
%--------------------------------------------------

One considers the non-linear system $(\ref{deta2},\ref{vecphi})$, over
a small portion along its trajectory (say between $t_0$ and $t_0+\delta t$).
The variation $\vecdeltaeta$ of $\vec{\eta}$, and $\vecdeltaphi$ of
$\vec{\varphi}$ is obtained through Taylor expansion:
%
\begin{equation}
\frac{\partial \vecdeltaeta }{\partial t}(\tau)  = \left. \vec{G} \right|_{t_0} +
                                 \left. \dgdeta \right|_{t_0}  \vecdeltaeta (\tau)+
                                 \left. \dgdphi \right|_{t_0}  \vecdeltaphi (\tau)+
                                 \left. \dgdt \right|_{t_0}  \tau
\label{pardeltaeta}
\end{equation}               

\begin{equation}
\vecdeltaphi (\tau) =                  \left. \dfdeta \right | _{t_0} \vecdeltaeta (\tau)+
                                 \left. \dfdphi \right | _{t_0} \vecdeltaphi (\tau)+
                                 \left. \dfdt \right | _{t_0} \tau 
\label{deltaphi}
\end{equation}
%
One obtains a partial difference equation system called the Tangent Linear System
to the trajectory, with $\tau \in [t,t+\delta t [$ parameterizing the trajectory.

%--------------------------------------------------
\subsection{Time scheme}
%--------------------------------------------------
\label{discrtemp}

Solving a linear system is classical and may be done using either a time scheme or
a linear transformation (Laplace, Fourier, Borel,...).
We use here a time scheme to obtain the lumped trajectory. The application
of a linear transformation will be used later to introduce the system analysis.

A Crank-Nicolson scheme is applied to system~(\ref{pardeltaeta},\ref{deltaphi})
to obtain:
%                          
\begin{equation}
\left\lbrace 
\begin{array}{l}
\vspace{0.5cm}
\matra \vecdeltaeta  + \matrb \vecdeltaphi = \vec{\Gamma} \delta t  \\
\matrcplus \vecdeltaeta - (\matri  +\matrd) \vecdeltaphi = \vec{\Omega} \delta t
\end{array} 
\right.
\label{eqtef}
\end{equation}
%
where the introduced symbols of matrices and vectors are defined by:                                                  
% 
\begin{equation}
\left\lbrace 
\begin{array}{l}
\matra = \matri - \frac{\delta t}{2}  \left. \dgdeta \right|_{t_0}  \\
\matrb =  - \frac{\delta t}{2}  \left. \dgdphi \right|_{t_0} \\
\vec{\Gamma} = \left. \vec{G} \right|_{t_0}
                   + \frac{\delta t}{2} \left. \dgdt \right|_{_t0} \\
\matrcplus =  \left. \dfdeta \right | _{t_0}   \\
\matrd = - \left. \dfdphi \right | _{t_0}  \\
\vec{\Omega} = - \left. \dfdt \right | _{t_0}  
\end{array}              
\right.
\label{coefmat}
\end{equation}
%
The matrix $(\matri  + \matrd)$ has been introduced because the overall set
of transfer equations
is also partitioned, so that subsets of transfers can be linked between
each-other, contrary to cells.

The preceding formulation is responding to our programme as we now will show.

%--------------------------------------------------
\subsection{Interpreting the elements of the TEF}
%--------------------------------------------------
An ultimate rewriting of system~(\ref{eqtef}) is used to open to an
interpretation of the set of mathematical elements introduced by the TEF in
modelling the original system.
Let eliminate $\vecdeltaeta$ from the second equation of~(\ref{eqtef}):
%
\begin{equation}
\left\lbrace
\begin{array} {l}
\vecdeltaeta = \vecdeltaetadec + \matrf \vecdeltaphi \\    
\left( \matri + \matrcbardt \right) \vecdeltaphi = \vecdeltaphiins  
\end{array}
\right.
\label{eqtef2}
\end{equation}
%
The Jacobian matrices in cells are defining $\matrf = -\matramun \matrb$.
This matrix expresses the sensitivity of a cell to change in its interfacing
conditions $\vec{\varphi}$. When this matrix is a null matrix, or when
$\vecdeltaphi = 0$, one gets $\vecdeltaeta = \vecdeltaetadec$, which is the
cell evolution when the interface is ``frozen''. It is called the ``decoupled
evolution'' of the cell (i.e. along the local TLS associated to a time-step
\footnote{more rigorously, in the local differential manifold}):
%
\begin{equation*}
\vecdeltaetadec = \matramun \, \vec{\Gamma} \, \delta t
\end{equation*}
%
In an analog way , $\matrcplus$ from~(\ref{eqtef}) determines the evolution
of the transfers during the time-step while cells are evolving.
When $\vecdeltaeta$ is eliminated from the transfer equation, which is
always possible\footnote{being a well-posed problem, $\matra$ is regular},
one obtains the second line of~(\ref{eqtef2}), with:
%
\begin{equation*}                          
\matrcbardt = \matrd + \matrcplus \, \matramun \, \matrb
\end{equation*}
%
and
%
\begin{equation*}
\vecdeltaphiins = \matrcplus \, \vecdeltaetadec - \vec{\Omega} \delta t
\end{equation*}
%
$\matrcbardt$ is called a ``coupling matrix'' between transfers.
Under this form, it is depending on $\delta t$. We shall review later on
how this matrix is important to system analysis.

$\vecdeltaphiins$ is the transfers evolution when $\matrb = 0$.
$\vecdeltaphiins$ is called the ``insensitive transfer evolution'' because
cells are following their decoupled trajectory. In that case, one notices that
$\matrcbardt = \matrd$, which means that cells are not coupling any more
transfers among themselves.
\bigskip

To summarise, using the local TLS gives access to a mathematical formulation that provides
the full dynamics of a system, allowing to explicit the operators (matrices) which determine
the inter-influence between elementary objects. It also allows to define
a portion of the trajectory, using the solution of a matrix algebraic system.
%
%--------------------------------------------------
\subsection{Matricial structure of the system and solution}
%--------------------------------------------------
The matrix structure of system~(\ref{eqtef}) is the following:
%
\begin{equation*}
\left[
\begin{array}{cc}
\matra & \matrb  \\
-\matrcplus & \matri +\matrd
\end{array}              
\right]
\left[
\begin{array}{c}
\vecdeltaeta  \\
\vecdeltaphi
\end{array}
\right]
=
\left[
\begin{array}{c}
\Gamma \delta t  \\
-\Omega \delta t
\end{array}
\right]
\label{matef}
\end{equation*}
%
The system is solved in a two-step process, eliminating firstly the cell variables
using the same procedure as for~(\ref{eqtef2}):
%
\begin{equation*}
\vecdeltaeta = -\matramun \, \matrb \, \vecdeltaphi + \matramun \,  \vec{\Gamma} \, \delta t 
\end{equation*} 
%
secondly, $\vecdeltaeta$ is introduced in the second equation:
%
\begin{equation*}        
\matrcplus \left( -\matramun \, \matrb \, \vecdeltaphi 
+ \matramun \vec{\Gamma} \delta t \right)
- \left( \matri + \matrd \right) \vecdeltaphi = \vec{\Omega} \, \delta t
\end{equation*}       
%
\begin{equation*}
\left(\matri + \matrd + \matrcplus \, \matramun \, \matrb \right) \vecdeltaphi =
\matrcplus \, \matramun \, \vec{\Gamma} \delta t - \vec{\Omega} \, \delta t
\end{equation*}
%
This linear system is classically solved to give $\vecdeltaphi$. Each cell-equation
can be solved, as all terms except $\vecdeltaeta$ are known.
%
%--------------------------------------------------
\subsection{Numerical process}
%--------------------------------------------------
Once given the general procedure of solving a TEF system, a few difficulties may be described.
The manipulation of the Jacobian matrices can lead to problems.
Let explicit an arbitrary system of three cells
labelled $(\alpha, \beta, \gamma)$ and the set of all transfers connecting them,
as a unique vector~$\vec{\varphi_u}$:
%
\begin{equation*}
\begin{array}{c}
\alpha  \\
\beta  \\
\gamma \\
u 
\end{array}
\left(
\begin{array}{cccc}
\matra_{\alpha\alpha}  & 0 & 0 & \matrb_{\alpha u}  \\
0 & \matra_{\beta\beta} & 0  & \matrb_{\beta u}  \\
0 & 0 & \matra_{\gamma\gamma} & \matrb_{\gamma u} \\
-\matrcplus_{u \alpha}  & -\matrcplus_{u \beta} & -\matrcplus_{u \gamma} & \matri +\matrd
\end{array}              
\right)
\label{matroiscel}                  
\end{equation*}
%
It is conceivable that for systems of high dimension, the full matrix be huge
and hollow. An appropriate solving process should make use of block-matrices and
of the tree-structure defining their imbedding in the global matrix.

The chosen structure applied to a TEF system is closely following the partitioning
procedure.
``Families'' of cells are defined, gathered themselves in super-families, following
the recursive partitioning of the system. The cells are the leaves of a tree
corresponding to the partition in families. Subsets of transfers linking the cells
can be associated with each family in an automated mode.
It can be shown that their exists an isomorphism between that tree-structure and
the imbedding
of the block-matrices in the overall system matrix. Hence, that tree can be used to link to
each block-matrix and solve partial algebraic systems recursively until all variables
are computed. This process refers to a so-called ``hyper multi-frontal method
with super-relaxed nodes'', see sources at the end of the article.

Before proceeding to the system analysis using the TEF, let comie back to an alternative
solution of the TLS using a linear transformation as stated in~\ref{discrtemp}.
It will be the occasion to show the equivalence
between the two solutions (time scheme and symbolic transformation).
%
%------------------------------------------------------------
\subsection{Borel transformation on the Tangent Linear System}
%------------------------------------------------------------
%
We previously mentioned that the TEF elements (vectors and matrices) are function of
the time step $\delta t$. In order to show how the TEF elements exhibit the whole
system dynamics, we introduce a TLS symbolic transformation. In lieu of the classical
Laplace transformation, whose definition involves the complex space, we use the
Borel transformation, defined as:
%
\begin{equation} 
f{(t)}\stackrel{\cal B}{\rightarrow }\B[{f}]{(\tau )}=\frac{1}{\tau }% 
\int_{0}^{\infty }e^{-t/\tau }f{(t)}dt=\frac{1}{\tau }\widetilde{f}{\left(\frac{% 
1}{\tau }\right)} 
\end{equation} 
% 
where $\widetilde{f}{(p)}$ is the Laplace transformation of $f(t)$. It is seen
that the Borel variable $\tau$ is real and homogeneous to time.
Applying this definition to system~(\ref{pardeltaeta},\ref{deltaphi}), one has
% 
\begin{center} 
\begin{equation} 
\label{slt-KN}
\left\{  
\begin{array}{c} 
\B[\boldsymbol{\vecdeltaeta }_{\alpha }]=\overbrace{\left[ 1-\tau \left. \overline{% 
\overline{\frac{\partial G_{\alpha}}{\partial \eta _{\alpha }}}}\right| 
_{t_{n}}\right] ^{-1}\tau\ \mathbf{G_{\alpha}}|_{t_{n}}}^{\B[\boldsymbol{\vecdeltaeta }% 
_{\alpha ,dec}]}+\overbrace{\tau \left[ 
1-\tau\left. \overline{\overline{\frac{\partial G_{\alpha}}{\partial 
\eta _{\alpha }}}}\right| _{t_{n}}\right] ^{-1}\left. \overline{\overline{% 
\frac{\partial G_{\alpha}}{\partial \varphi }}}\right| _{t_{n}}}^{\overline{\overline{\mathcal{F}}}}
\B[\boldsymbol{\vecdeltaphi  }] \\  
\B[\boldsymbol{\vecdeltaphi } ]=\sum_{\beta} \left. \overline{\overline{\frac{% 
\partial f}{\partial \eta _{\beta }}}}\right| _{t_{n}}\B[\boldsymbol{\vecdeltaeta }% 
_{\beta }]+\left. \overline{\overline{\frac{\partial f}{\partial \varphi }}}% 
\right| _{t_{n}}\B[\boldsymbol{\vecdeltaphi }] 
\end{array} 
\right.  
\end{equation} 
\end{center} 
%
this is obtained because $\B[\partial f/\partial t]=(1/\tau)\B[f]$.
After eliminating cell variables from the first equation into the second equation,
the following equivalent system is obtained:
%
\begin{equation} \label{eqn:c}
\left\{  
\begin{array}{c} 
\B[\boldsymbol{\vecdeltaeta }]=\B[\boldsymbol{\vecdeltaeta }_{dec}]+% 
\overline{\overline{\mathcal{F}}} \, \B[\boldsymbol{\vecdeltaphi }] \\ 
\left[ 1+\overline{\overline{\mathcal{C}}}\right]
\B[\boldsymbol{\vecdeltaphi }]=% 
\B[\boldsymbol{\vecdeltaphi }_{ins}] 
\end{array} 
\right.
\end{equation} 
% 
this system is analog to (\ref{eqtef2}), but here $\tau$ is the dynamic variable
instead of $\delta t$. This is the main property that explains the link between
the forward time resolution and the dynamical characteristics of the system.
This link between system (\ref{eqtef2}) and (\ref{eqn:c}) is obtained thanks to
a property of the Crank-Nicolson time scheme, that is true for small  $t$:
%
\begin{equation}
\label{corres}
  \delta \eta(t) \sim 2 \cdot \B(\delta \eta)\left(\frac{t}{2}\right)
\end{equation}
%
As a result, when the TLS is solved for finite increments of $\delta t$, it can
be formally
identified with the Borel transformation of the TLS. Said differently, the TEF
elements, as functions of $\tau$, are solutions of the TLS. The trajectory is just
defined for small values of the Borel variable.
In the sequel, we shall make no more distinction between the two equivalent forms.
We now examine how the TEF structure of the system can bring some information
related to the intelligibility of systems behaviour.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{TEF and system analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The problem of obtaining some intelligibility from numerical models
involves a lot of effort in numerical fluid dynamics or climate modelling
in particular. The post-treatment of simulation can reach same complexity as
the simulation itself. Leaving aside the problem of analysing the spatial
structures of simulated fields of parameters, we shall take as an illustration
a classical type of analysis on how a system is regulating, dumping or
diverging under perturbation. The concept of {\bf feedback gain}
is currently applied to explain the response of climate to forcing.
Such tentative is still controversial today, because it addresses a theoretical
problem that was not in the original concept definition.
%_______________________________________
\begin{figure}[h]
\begin{minipage}[h]{.4\linewidth}
\psfig{figure=gainctrl.eps,%
bbllx=1pt,bblly=1pt,bburx=309pt,bbury=230pt,width=6.5cm,clip=}
\end{minipage}\hfill
\begin{minipage}[h]{.6\linewidth}
\begin{equation}
s(p) = H(p) e(p) = \frac{A(p)}{1-A(p)G(p)} e(p)
\end{equation}
\end{minipage}\hfill
\caption{Elementary feedback loop and associated transfer function.}
\label{fbbd}
\end{figure}
%_______________________________________
%
The concept of feedback originates in Electric Circuit Theory (Bode 1945). The feedback gain
is defined as a function of a symbolic variable (Laplace or Fourier transformation).
Considering the input--output circuit of Fig.~\ref{fbbd}, the derivation through $G$ takes
the output signal $s$ as amplified by $A$ from entry, apply $G$, and is
added to the entry $e$. The transfer function $H$ directly gives the output signal
when the input is given. Hence for system $A$, the so called feedback gain $G$
allows to shape up signals, reduce noise, etc. Such a feedback loop is though as
an active circuit built up from well known linear components.
More complex circuits can be analysed in terms of arrangement of the elementary
feedback-loop of Fig.~\ref{fbbd}.

The same concept is nowadays used in a very different theoretical frame, and appears more
as a conceptual analysis applied to systems that are difficult to analyse in terms
of elementary components, and usually non-linear. The tentative extension of the
original feedback concept emphasises on the need for building analysis tools devoted to
complex systems. We show in the following that the rigorousness of
the original concept can be restored for non-linear systems and in the very general
frame of numerical modelling.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Dynamic feedback gain using the TEF}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To be illustrative, let consider a macro-eco\-no\-mic model, the one of\\
 IMACLIM-PROJ
(cf sources). Without having to enter into details, the model is a national compatibility
input-output table reduced to six sectors
(oil, transportation, energy, residential housing, commercial sector and a composite sector).
The model gives the response of the economic system to an exogenous increase of
the oil price. It was built to reproduce global equilibrium between householder
consumption, industrial production, and employment. The system dynamics is driven by the level
of consumption
allowed by the prices and the wages, the repartition of consumption, and intermediate
production as given by a constant Leontiev matrix. For instance, whenever the price of
a product is increasing, it can be substituted by a less expensive one to optimise
economic efficiency.

A more general macro-economic model may quickly reach a hight level of complexity, as the number
of branches and products is more than a hundred for each country. Numerous parts
of the model are also non-linear. As a result, the obtained trajectory of internal variables
can be used to check for the level of realism reached by the model. It does not
inform on how and why it is behaving as it does, particularly when submitted to a shock (as oil-crises,
tsunamis, hurricanes).
Let see how the feedback gain can help in understanding model behaviour. Take one variable
of the system, which is equivalent to isolating a process associated to this variable.
This operation has to be associated to a feedback loop, because the feedback is
measuring the effect of opening that loop when analysing its response to a perturbation,
as shown in the figure:
%_______________________________________
\begin{center}
\begin{figure}[ht]
\begin{minipage}[h]{.4\linewidth}
\psfig{figure=PS/gainsltc_E.eps,%
bbllx=7pt,bblly=9pt,bburx=320pt,bbury=310pt,width=6.5cm,clip=}
\end{minipage}\hfill
\begin{minipage}[h]{.6\linewidth}
\begin{equation}
\left\lbrace 
\begin{array}{l}
\delta p + \mathbb{C}_{px} \vecdeltax
                                         = \delta p_{ins}  \\             
\
\mathbb{C}_{xp} \delta p + \left( 1 + \mathbb{C}_{xx} \right) \vecdeltax 
                                        = \vecdeltaxins  
\end{array}
\right.
\label{prixt}
\end{equation}
\end{minipage}\hfill
\caption{closed loop: only one variable ($p$) of the system is remaining after
the elimination process.}
\end{figure}
\end{center}
%_______________________________________
%
The figure explicits how is the system acting on the price of transportation ($p$).
The corresponding system of equations is~(\ref{prixt}). In this system, $\overrightarrow{x}$
represents the set of all system variables but $p$.
Matrix $\mathbb{C}_{px}$ is a row vector and $\mathbb{C}_{xp}$ is a column vector.
One may notice that the first equation is the transfer equation giving the price when
$\overrightarrow{x}$ is representing all other variables.
Let us explicit what is the effect of opening the loop:
%
\begin{center}
[Price of transportation $p$] $\rightarrow$ [Rest of the system] $\rightarrow$ [$p$]
\end{center}
%
\bigskip
%
%
One eliminates the vector of variables $\vecdeltax$ in the price-equation $p$:
%      
\begin{equation}
\left( 1 - \mathbb{C}_{px}(1 + \mathbb{C}_{xx})^{-1}\mathbb{C}_{xp} \right) \delta p
                   = \delta p_{ins} - \mathbb{C}_{px}(1 + \mathbb{C}_{xx})^{-1}\vecdeltaxins  
\label{deltap}
\end{equation}
%
\underline{open loop:} the loop is open after the transfer $p$, which hence cannot influence
the rest of the system. This is formally done by zeroing the raw vector-matrix
$\mathbb{C}_{xp}$ (the system becomes insensitive to perturbations from $p$). 
%
%_______________________________________
\begin{center}
\begin{figure}[h]
\psfig{figure=PS/gainscoup_E.eps,%
bbllx=8pt,bblly=11pt,bburx=322pt,bbury=310pt,width=6.5cm,clip=}
\caption{The open-loop system is obtained zeroing all matrices that allowed the rest
of the system to react to a perturbation in $p$.
}
\end{figure}
\end{center}
%_______________________________________
%
notice that the term $\mathbb{C}_{px}(1 + \mathbb{C}_{xx})^{-1}\vecdeltaxins$
at the right of (\ref{deltap}) is unchanged. When the loop is closed again,
the original equation is restored:
%
\begin{equation*}
\boxed{
(1-g) \delta p = \delta \acute{p}_{ins}
}
\end{equation*}
%
where $g =\mathbb{C}_{px}(1 + \mathbb{C}_{xx})^{-1}\mathbb{C}_{xp}$, and
$\delta \acute{p}_{ins} = \delta p_{ins} - \mathbb{C}_{px}(1 + \mathbb{C}_{xx})^{-1}\vecdeltaxins$.
It can be shown that this function is independent of the algebraic manipulation used to
eliminate all other variables than $p$ (the ZOOM tree-structure).

The term {\bf $g$} is called the ``feedback gain'' of the system for the price of transportation.
It is a function of $\tau$ (or equivalently of $\delta t$) which gives the linear response
of the system when the price is perturbed (at time $t$ on the trajectory).
In our model, we found an annual gain ($\tau=1$ year) of 15\%, which has the meaning that,
the system responds with an amplification of the initial perturbation.

Another result with the same model shows that for the wages of the composite sector,
the system feedback leads to a gain of 50\% per year. Suppose that one injects
an increase of one unit to the wages in that sector, the wages will undergo an
extra of 1.5 units by the end of the year after (looks like a Keynesian model).
So clearly, a procedure such as the TEF allowing to explicit feedback gains do
provide important information about system behaviour.
One might go further, and ask why is the system reacting in such manner. One can for
example ``freeze'' a bunch of cells or transfers and compare the two gains, to
look for the elements of the system that contribute principally to the feedback.
Another possibility would involve coupling matrix analysis that consists in
isolating two variables instead of one for feedback analysis.

\vspace{0.7cm}
As conclusion of this section, when comparing the proposed TEF method to the original one of
electric circuit theory
in looking for feedback gain, we obtained here a method to evaluate feedback gain
numerically along the trajectory of non-linear systems. Whatever the complexity of a model,
it is easy to analyse feedback loops.
%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Computational implementation \\ The ZOOM software}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%------------------------------------------------
\subsection*{Generalities}
%------------------------------------------------
ZOOM (Zone Organised Optimal Modelling) is the software implementation
of the TEF: it computes time dependent or stationary simulations
for the models which have been partitioned into cells and transfers, and
provides as well, Borel transform functions devoted to system analysis
\footnote{A simplified version that can handle a few hundreds of equations, using the same TEF principles
is also available, which includes automatic sensitivity analysis, adjoint modelling etc.
see text: \it{ Mini$\_$Ker et Adjoint: ''un résumé'', and the English version of the Mini-ker manuals.}}.
Programming a model proceeds in two steps: 

({\bf 1}) The first step is the description of the different objects in which the model is 
partitioned. To each class of cells and transfers corresponds one model (a set of equations)
which is programmed in 
''a processor''. Processors are written in extended Fortran, so that the 
connections can be managed. Same processors can be used in different kinds of models
so that in general, the writing of new models 
does not generally need more than the reprogramming of a few processors. Therefore, one can 
expect that if Zoom is used in economics, the accumulation of a number of processors will 
greatly ease the writing of new models. 

({\bf 2}) The second step is the the description of the global structure of the system,
that is how cells are gathered into families, what transfers are connecting cells etc.
Here also, a pseudo-Fortran programming language is used, with some facilities as:

- (i) object creation instructions ;
 
- (ii) connecting instructions ; 

- (iii) instructions allowing the grouping of cells in families and the arrangement of families in 
a tree structure ; 

In addition, as for any model, one has to specify initial state conditions (cells only), give value
to processors parameters, choose among particular graphic output standards.
By default, all variables are available as output (see ZOOM manuals).

When ZOOM is run, the first task is to connect the objects and reserve dynamic
memory to store all matrices and vectors as prescribed by the TEF.
Each processor is then called. The cell and transfer processors provide the necessary
elements for the computation of the TEF system, so that the system can be solved each
time step.
Graphic output are stored in the same tree-structure as defined in the second step of programming.
The graphic processing (called VIZU) is an extension of the CERN PAW code.
Finally, VIZU allows for some post-processing to compute coupling matrices and feedback
gains.
%------------------------------------------------
\subsection*{Processors}
%------------------------------------------------

Processors are programmed to provide the TEF vectors and matrices needed by the TEF.


\vspace{0.5cm}
\underline{Cells processor:}

\vspace{0.3cm}
The state evolution of a cell is given on one time step by:
%
\begin{equation*}
\vecdeltaeta = \vecdeltaetadec + \matrf \vecdeltaphi
\end{equation*}
%
where vector $\vecdeltaphi$ represents a subset of the system transfers - the ones that
are connected to each particular cell ($\alpha$). The information of transfer variables
comes with a connecting object. The cell processor can thus compute
$\vecdeltaetadec$, and $\matrf$. 
%
\begin{itemize}
\item On entry, one has: 
\begin{itemize}
\item the time step value $\delta t$, 
\item initial state variable values of the cell, the ones of the connected transfers
and eventual outside sources or independent boundary conditions (${\overrightarrow{ \phi_0}}$).
\end{itemize}
\item On output, one gets:
\begin{itemize}
\item the decoupled evolution vector $\overrightarrow{ \delta \eta} _{\alpha dec}$
\item along with matrix $\matrf$ that gives the sensitivity of state variables to
change in the connecting transfers.
\end{itemize}
\end{itemize}

\vspace{0.5cm}
\underline{Transfer processor:}

\vspace{0.3cm}
Transfer variables are associated to static constraint equations that links them to connecting
cells. Contrary to cell processors, transfers are also dependent from other transfer
variables in general. The other transfer has to be connected to the set of cells
that are also connected to the transfer at work.
%
\begin{equation*}
\overrightarrow{\phi} = \overrightarrow{f} ( 
\overrightarrow{\eta_{\alpha 0}} , \overrightarrow{\phi_0} ) 
\end{equation*}           
%
A transfer processor compute matrix $\matrcplus$ which, analog to cells,
give the sensitivity of the transfer vector to a perturbation of the connected
cell variables (for the current time-step).
\begin{itemize}
\item On entry, one has: 
\begin{itemize}
\item the time step value  $\delta t$, 
\item initial values of connected objects: $\overrightarrow{\eta_0}$ for cells,
$\overrightarrow{\phi_0}$ for eventual other transfers.
\end{itemize}
\item On output, the processor provides:
\begin{itemize}
\item  the vector of transfer variables
\item and matrices $\matrcplus$ and $\matrd$.
\end{itemize}
\end{itemize}
the fact that transfers are inter-depending between themselves implies that an implicit
system has to be solved to compute transfers. This is done in ZOOM at the
beginning of each time step.

\newpage
\begin{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{CONCLUSIONS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{center}
The TEF procedure of building models, as the result of partionning and re-coupling a system,
gives rise to analysis of well-defined objects. The structure of the partition
allows to study the inter-influence between parts of a system. Every objetc of the system
has to provide TEF elements that are used to solve the overall system of equations.
The elementary vectors are interpreted in terms of intrinsc  behavior of the objects, while
Jacobian matrices are representing local sensitivity operators.

Hence, one has the general view that explains how our original programm can be
accomplished. The solution of the partictionned system has the consequence of explicitating
the partial derivative chain rules that propagates local sensitivities troughout the
system. Once a system reduced after eliminating a set of variables, the remaining variables
keep track of their coupling through the eliminated variables.
As a consequence, sensitivity to perturbations, error or noise,
propagates through the system and can be analysed. The satisfactry exploitation of such facilities
leads to numerous
difficulties, considering in particular non-linear systems. We have shown how to characterize
the local dynamics of a system using the Borel transform of feedback function.
More generally, the presence of the Jacobian matrices are sufficient to automatically
get the adjoint model, and hence allow to analyse the sensitivity of non-linear
models and to solve optimalisation problems. Forward sensitivity also shares the same ste
of Jacobian matrices, and can be used to study non-linear feedback gain as shown in the Mini-ker
manuals.

Our collaboration is on its way to use and test different techiques to solve
problems in a diversity of scientif fields (Engineering Sciences, Econmy and Climate,
and also Biology modelling). More draft-papers are available on this web site).

\vspace{0.7cm}
\hrule
\vspace{0.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Sources:}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{source}

http://www.lmd.jussieu.fr/zoom/documents.dir

- on Borel: article submit to the Journal of Atmospheric Sciences (2005)

- on the IMACLIM-PROJ model: rapport d'avancement ZOOM-ProjTES (2002) on line

- see also TEF-ZOOM seminar Volume 1, TEF, EDH, L'Agelonde, 21-25 octobre 1991 (soon on line).

- Thèses de Stéphane Blanco et Jean-Louis Joly (on request).

- on ZOOM tree-structure: Liu J.W.H. 1992: The multi-frontal method for sparse matrix
solution: Theory and Practice. {\it Society for Industrial and Applied
Mathematics Review},{\bf 34}, 82-109. Voire aussi manuel EDH (soon on line).
\end{document}
\end

