The Team aims at designing and developing constructive methods in modeling, identification and control of dynamical, resonant and diffusive systems.
Function theory and approximation theory in the complex domain, with applications to frequency identification of linear systems and inverse boundary problems for the Laplace and Beltrami operators:
System and circuit theory with applications to the modeling of analog microwave devices. Development of dedicated software for the synthesis of such devices.
Inverse potential problems in 2D and 3D and harmonic analysis with applications to nondestructive control (from magneto/electroencephalography in medical engineering or plasma confinement in tokamaks for nuclear fusion).
Control and structure analysis of nonlinear systems with applications to orbit transfer of satellites.
Collaboration under contract with Thales Alenia Space (Toulouse, Cannes, and Paris), CNES (Toulouse), XLim (Limoges), CEAIRFM (Cadarache).
Exchanges with UST (Villeneuve d'Asq), University BordeauxI (Talence), University of Orléans (MAPMO), University of Pau (EPI Inria Magique3D), University MarseilleI (CMI), CWI (the Netherlands), SISSA (Italy), the Universities of Illinois (UrbanaChampaign USA), California at San Diego and Santa Barbara (USA), Michigan at EastLansing (USA), Vanderbilt University (Nashville USA), Texas A&M (College Station USA), ISIB (CNR Padova, Italy), Beer Sheva (Israel), RMC (Kingston, Canada), University of Erlangen (Germany), Leeds (UK), Maastricht University (The Netherlands), Cork University (Ireland), Vrije Universiteit Brussel (Belgium), TUWien (Austria), TFHBerlin (Germany), CINVESTAV (Mexico), ENIT (Tunis), KTH (Stockholm).
The project is involved in a EMS21RTG NSF program (with Vanderbilt University), in the Inria Team Enée associated with LAMSINENIT (including the EPI Anubis and Poems), in an EPSRC Grant with Leeds University (UK), in the ANR projects AHPI (Math., coordinator) and FILIPIX (Telecom.).
Identification typically consists in approximating experimental data by the prediction of a model belonging to some model class. It consists therefore of two steps, namely the choice of a suitable model class and the determination of a model in the class that fits best with the data. The ability to solve this approximation problem, often nontrivial and illposed, impinges on the effectiveness of a method.
Particular attention is payed within the team to the class of stable linear timeinvariant systems, in particular resonant ones, and in isotropically diffusive systems, with techniques that dwell on functional and harmonic analysis. In fact one often restricts to a smaller class— e.g.rational models of suitable degree (resonant systems, see section ) or other structural constraints—and this leads us to split the identification problem in two consecutive steps:
Seek a stable but infinite (numerically: high) dimensional model to fit the data. Mathematically speaking, this step consists in reconstructing a function analytic in the right halfplane or in the unit disk (the transfer function), from its values on an interval of the imaginary axis or of the unit circle (the bandwidth). We will embed this classical illposed issue ( i.e.the inverse Cauchy problem for the Laplace equation) into a family of wellposed extremal problems, that may be viewed as a regularization scheme of Tikhonovtype. These problems are infinitedimensional but convex (see section ).
Approximate the above model by a lower order one reflecting further known properties of the physical system. This step aims at reducing the complexity while bringing physical significance to the design parameters. It typically consists of a rational or meromorphic approximation procedure with prescribed number of poles in certain classes of analytic functions. Rational approximation in the complex domain is a classical but difficult nonconvex problem, for which few effective methods exist. In relation to system theory, two specific difficulties superimpose on the classical situation, namely one must control the region where the poles of the approximants lie in order to ensure the stability of the model, and one has to handle matrixvalued functions when the system has several inputs and outputs, in which case the number of poles must be replaced by the McMillan degree (see section ).
When identifying elliptic (Laplace, Beltrami) partial differential equations from boundary data, point 1. above can be recast as an inverse boundaryvalue problem with (overdetermined DirichletNeumann) data on part of the boundary of a plane domain (recover a function, analytic in a domain, from incomplete boundary data). As such, it arises naturally in higher dimensions when analytic functions get replaced by gradients of harmonic functions (see section ). Motivated by free boundary problems in plasma control and questions of source recovery arising in magneto/electroencephalography, we aim at generalizing this approach to the real Beltrami equation in dimension 2 (section ) and to the Laplace equation in dimension 3 (section ).
Step 2. above— i.e.meromorphic approximation with prescribed number of poles—is used to approach other inverse problems beyond harmonic identification. In fact, the way the singularities of the approximant ( i.e.its poles) relate to the singularities of the approximated function is an allpervasive theme in approximation theory: for appropriate classes of functions, the location of the poles of the approximant can be used as an estimator of the singularities of the approximated function (see section ).
We provide further details on the two steps mentioned above in the subparagraphs to come.
Given a planar domain
D, the problem is to recover an analytic function from its values on a subset of the boundary of
D. It is convenient to normalize
Dand apply in each particular case a conformal transformation to meet a “normalized” domain. In the simply connected case, which is that of the halfplane, we fix
Dto be the unit disk, so that its boundary is the unit circle
T. We denote by
H^{p}the Hardy space of exponent
pwhich is the closure of polynomials in the
L^{p}norm on the circle if
1
p<
and the space of bounded holomorphic functions in
Dif
p=
. Functions in
H^{p}have welldefined boundary values in
L^{p}(
T), which make it possible to speak of (traces of) analytic functions on the boundary.
A standard extremal problem on the disk is :
(
P_{0}) Let
1
p
and
fL^{p}(
T); find a function
gH^{p}such that
g
fis of minimal norm in
L^{p}(
T).
When seeking an analytic function in
Dwhich approximately matches some measured values
fon a subarc
Kof
T, the following generalization of (
P_{0}) naturally arises:
(
P) Let
1
p
,
Ka subarc of
T,
fL^{p}(
K),
and
M>0; find a function
gH^{p}such that
and
g
fis of minimal norm in
L^{p}(
K)under this constraint.
Here
is a reference behaviour capsulizing the expected behaviour of the model off
K, while
Mis the admissible error with respect to this expectation. The value of
preflects the type of stability which is sought and how much one wants to smoothen the data.
To fix terminology we generically refer to (
P) as a
bounded extremal problem. The solution to this convex infinitedimensional optimization problem can be obtained upon iteratively solving spectral equations for appropriate Hankel and
Toelitz operators, that involve a Lagrange parameter, and whose right handside is given by the solution to (
P_{0}) for some weighted concatenation of
fand
. Constructive aspects are described in
,
,
, for
p= 2,
p=
, and
1<
p<
, while the situation
p= 1is essentially open.
Various modifications of
(
P)have been studied in order to meet specific needs. For instance when dealing with lossless transfer functions (see section
), one may want to express the constraint on
in a pointwise manner:

g

Ma.e. on
, see
for
p= 2and
= 0.
Another variation of (
P), aiming at solving inverse boundaryvalue problems from mixed DirichletNeumann data (see sections
and
) is to impose as a constraint that
while looking for
to be of minimal norm in
L^{p}(
K)under this constraint. This gives rise to a new identification technique of Robin coefficients, which is interesting for instance in corrosion control, see
.
The abovementioned problems can be stated on an annular geometry rather than a disk. For
p= 2the solution proceeds much along the same lines
,
. When
Kis the outer boundary, (
P) regularizes a classical inverse problem occurring in nondestructive control, namely to recover a harmonic function on the inner boundary from overdetermined DirichletNeumann data on
the outer boundary (see section
). Interestingly perhaps, it becomes a tool to approach Bernoulli type problems for the Laplacian, where overdetermined
observations are made on the outer boundary and we
seek the inner boundaryknowing it is a level curve of the flux (see section
). Here, the Lagrange parameter indicates which deformation should be applied on the inner contour in order to improve the
fit to the data.
Continuing effort is currently payed by the team to carry over bounded extremal problems and their solution to more general settings.
Such generalizations are twofold: on the one hand Apics considers 2D diffusion equations with variable conductivity, on the other hand it investigates the ordinary Laplacian in
R^{n}. The targeted applications are the determination of free boundaries in plasma control and source detection in electro/magnetoencephalography, respectively.
An isotropic diffusion equation in dimension 2 can be recast as a socalled real Beltrami equation
. This way analytic functions get replaced by “generalized” ones in problems (
P_{0}) and (
P). Hardy spaces of solutions, which are more general than Sobolev ones and allow one to handle
L^{p}boundary conditions, have been introduced when
1<
p<
. The expansions of solutions needed to constructively handle such problems, are now under study
. The goal is to solve the analog of (
P) in this context to approach Bernoullitype problems (see section
).
At present, bounded extremal problems for the
nD Laplacian are considered on halfspaces or balls. Following
, Hardy spaces are defined as gradients of harmonic functions satisfying
L^{p}growth conditions on inner hyperplanes or spheres. From the constructive viewpoint, when
p= 2, spherical harmonics offer a reasonable substitute to Fourier expansions
. Only very recently were we able to define operators of Hankel type whose singular values connect to the
solution of (
P_{0}) in BMO norms. The
L^{p}problem also makes contact with some nonlinear PDE's, namely to the
pLaplacian. The goal is here to solve the analog of (
P) on spherical shells to approach inverse diffusion problems across a conductor layer.
Let as before
Ddesignate the unit disk,
Tthe unit circle. We further put
R_{N}for the set of rational functions with at most
Npoles in
D, which allows us to define the meromorphic functions in
L^{p}(
T)as the traces of functions in
H^{p}+
R_{N}.
A natural generalization of problem (
P_{0}) is
(
P_{N}) Let
1
p
,
N0an integer, and
fL^{p}(
T); find a function
g_{N}H^{p}+
R_{N}such that
g_{N}
fis of minimal norm in
L^{p}(
T).
Problem (
P_{N}) aims, on the one hand, at solving inverse potential problems from overdetermined DirichletNeumann data, namely to recover approximate solutions of the inhomogeneous Laplace equation
u=
, with
some (unknown) distribution, which will be discretized by the process as a linear combination of
NDirac masses. On the other hand, it is used to perform the second step of the identification scheme described in section
, namely rational approximation with a prescribed number of poles to a function analytic in the right halfplane, when one
maps the latter conformally to the complement of
Dand solve (
P_{N}) for the transformed function on
T.
Only for
p=
and continuous
fis it known how to solve (
P_{N}) in closed form. The unique solution is given by the AAK theory, that allows one to express
g_{N}in terms of the singular vectors of the Hankel operator with symbol
f. The continuity of
g_{N}as a function of
fonly holds for stronger norms than uniform,
.
The case
p= 2is of special importance. In particular when
, the Hardy space of exponent 2 of the
complementof
Din the complex plane (by definition,
h(
z)belongs to
if, and only if
h(1/
z)belongs to
H^{p}), then (
P_{N}) reduces to rational approximation. Moreover, it turns out that the associated solution
g_{N}R_{N}has no pole outside
D, hence it is a
stablerational approximant to
f. However, in contrast with the situation when
p=
, this approximant may
notbe unique.
The former Miaou project (predecessor of Apics) has designed an adapted steepestdescent algorithm for the case
p= 2whose convergence to a
local minimumis guaranteed; it seems today the only procedure meeting this property. Roughly speaking, it is a gradient algorithm that proceeds recursively with respect to the order
Nof the approximant, in a compact region of the parameter space
. Although it has proved rather effective in all applications carried out so far (see sections
,
), it is not known whether the absolute
minimumcan always be obtained by choosing initial conditions corresponding to
critical pointsof lower degree (as done by the Endymion software section
and RARL2 software, section
).
In order to establish convergence results of the algorithm to the global minimum, Apics has undergone a longhaul study of the number and nature of critical points, in which tools from
differential topology and operator theory team up with classical approximation theory. The main discovery is that the nature of the critical points (
e.g.
local minima, saddles...) depends on the decrease of the interpolation error to
fas
Nincreases
. Based on this, sufficient conditions have been developed for a local
minimumto be unique. This technique requires strong error estimates that are often difficult to obtain, and most of the time only hold for
Nlarge. Examples where uniqueness or asymptotic uniqueness has been proved this way include transfer functions of relaxation systems (
i.e., Markov functions)
, the exponential function, and meromorphic functions
. The case where
fis the Cauchy integral on a hyperbolic geodesic arc of a Dinicontinuous function which does not vanish “too much” has been recently answered in the positive, see section
.
An analog to AAK theory has been carried out for
2
p<
. Although not computationally as powerful, it has better continuity properties and stresses a continuous link
between rational approximation in
H^{2}and meromorphic approximation in the uniform norm, allowing one to use, in either context, techniques available from the other
p<2, problem (
P_{N}) is still fairly open.
A common feature to all these problems is that critical point equations express nonHermitian orthogonality relations for the denominator of the approximant. This is used in an essential manner to assess the behavior of the poles of the approximants to functions with branched singularities ( cf. sections , ), which is of particular interest for inverse source problems.
In higher dimensions, the analog of problem (
P_{N}) is the approximation of a vector field with gradients of potentials generated by
Npoint masses instead of meromorphic functions. The issue is by no means understood at present, and is a major endeavour of future research problems.
Certain constrained rational approximation problems, of special interest in identification and design of passive systems, arise when putting additional requirements on the approximant, for
instance that it should be smaller than 1 in modulus. Such questions became over years an increasingly significant part of the team's activity (see sections
,
,
, and
). When translated over to the circle, a prototypical formulation consists in approximating the modulus of a given function
by the modulus of a rational function of degree
n. When
p= 2this problem can be reduced to a series of standard rational approximation problems, but usually one needs to solve it for
p=
. The case where

fis a piecewise constant function with values 0 and 1 can also be approached
viaclassical Zolotarev problems
, that can be solved more or less explicitly when the passband consists of a single arc. A constructive solution
in the case where

fis a piecewise constant function with values 0 and 1 on several arcs (multiband filters) is one recent achievement of the team. Though the modulus of the response
is the first concern in filter design, the variation of the phase must nevertheless remain under control to avoid unacceptable distortion of the signal. This is an important issue, currently
under investigation within the team under contract with the CNES, see section
.
From the point of view of design, rational approximants are indeed useful only if they can be translated into physical parameter values for the device to be built. This is where system theory enters the scene, as the correspondence between the frequency response ( i.e.the transferfunction) and the linear differential equations that generate this response ( i.e.the statespace representation), which is the object of the socalled realizationprocess. Since filters have to be considered as dual modes cavities, the realization issue must indeed be tackled in a 2×2matrixvalued context that adds to the complexity. A fair share of the team's research in this direction is concerned with finding realizations meeting certain constraints (imposed by the technology in use) for a transferfunction that was obtained with the abovedescribed techniques (see section ).
We refer here to the behavior of the poles of best meromorphic approximants, in the
L^{p}sense on a closed curve, to functions
fdefined as Cauchy integrals of complex measures whose support lies inside the curve. If one normalizes the contour to be the unit circle
T, we are back to the framework of section
and to problem (
P_{N}); the invariance of the problem under conformal mapping was established in
. The research so far has focused on functions whose singular set inside the contour is zero or
onedimensional.
Generally speaking, the behavior of poles is particularly important in meromorphic approximation to obtain error rates as the degree goes large and also to tackle constructive issues like uniqueness. However, the original motivation of Apics is to consider this issue in connection with the approximation of the solution to a DirichletNeumann problem, so as to extract information on the singularities. The general theme is thus how do the singularities of the approximant reflect those of the approximated function? The approach to inverse problem for the 2D Laplacian that we outline here is attractive when the singularities are zero or onedimensional (see section ). It can be used as a computationally cheap preliminary step to obtain the initial guess of a more precise but heavier numerical optimization. For sufficiently smooth crack, or pointwise sources recovery, the approach in question is in fact equivalent to the meromorphic approximation of a function with branch points, and we were able to prove , that the poles of the approximants accumulate in a neighborhood of the geodesic hyperbolic arc that links the endpoints of the crack, or the sources . Moreover the asymptotic density of the poles turns out to be the equilibrium distribution on the geodesic arc of the Green potential and it charges the end points, that are thus well localized if one is able to compute sufficiently many zeros (this is where the method could fail). The case of more general cracks, as well as situations with three or more sources, requires the analysis of the situation where the number of branch points is finite but arbitrary, see section ). This are outstanding open questions for applications to inverse problems (see section ), as also the problem of a general singularity, that may be two dimensional.
Results of this type open new perspectives in nondestructive control, in that they link issues of current interest in approximation theory (the behavior of zeroes of nonHermitian orthogonal polynomials) to some classical inverse problems for which a dual approach is thereby proposed: to approximate the boundary conditions by true solutions of the equations, rather than the equation itself (by discretization).
Let us point out that the problem of approximating, by a rational or meromorphic function, in the
L^{p}sense on the boundary of a domain, the Cauchy transform of a real measure, localized inside the domain, can be viewed as an optimal discretization problem for a logarithmic potential
according to a criterion involving a Sobolev norm. This formulation can be generalized to higher dimensions, even if the computational power of complex analysis is then no longer available,
and this makes for a longterm research project with a wide range of applications. It is interesting to mention that the case of sources in dimension three in a spherical or ellipsoidal
geometry, can be attacked with the above 2D techniques as applied to planar sections (see section
).
Matrixvalued approximation is necessary for handling systems with several inputs and outputs, and it generates substantial additional difficulties with respect to scalar approximation, theoretically as well as algorithmically. In the matrix case, the McMillan degree ( i.e., the degree of a minimal realization in the SystemTheoretic sense) generalizes the degree.
The problem we want to consider reads:
Let
and
nan integer; find a rational matrix of size
m×
lwithout poles in the unit disk and of McMillan degree at most
nwhich is nearest possible to
in
(
H
^{2})
^{m×
l}.Here the
L^{2}norm of a matrix is the square root of the sum of the squares of the norms of its entries.
The approximation algorithm designed in the scalar case generalizes to the matrixvalued situation
. The first difficulty consists here in the parametrization of transfer matrices of given McMillan degree
n, and the inner matrices (
i.e., matrixvalued functions that are analytic in the unit disk and unitary on the circle) of degree
nenter the picture in an essential manner: they play the role of the denominator in a fractional representation of transfer matrices (using the socalled DouglasShapiroShields
factorization).
The set of inner matrices of given degree has the structure of a smooth manifold that allows one to use differential tools as in the scalar case. In practice, one has to produce an atlas of charts (parametrization valid in a neighborhood of a point), and we must handle changes of charts in the course of the algorithm. Such parametrization can be obtained from interpolation theory and Schur type algorithms, the parameters being interpolation vectors or matrices , , . Some of these parametrizations have a particular interest for computation of realizations , , involved in the estimation of physical quantities for the synthesis of resonant filters. Two rational approximation codes (see sections and ) have been developed in the team.
Problems relative to multiple local minima naturally arise in the matrixvalued case as well, but deriving criteria that guarantee uniqueness is even more difficult than in the scalar case. The already investigated case of rational functions of the sought degree (the consistency problem) was solved using rather heavy machinery , and that of matrixvalued Markov functions, that are the first example beyond rational function has made progress only recently . Let us stress that the algorithms mentioned above are first to handle rational approximation in the matrix case in a way that converges to local minima, while meeting stability constraints on the approximant.
Stabilization by continuous state feedback consists in designing a control law which is a smooth (at least continuous) function of the state making a given point (or trajectory) asymptotically stable for the closed–loop system. One can consider this as a weak version of the optimal control problem which is to find a control that minimizes a given criterion (for instance the time to reach a prescribed state). Optimal control generally leads to a rather irregular dependence on the initial state; in contrast, stabilization is a qualitativeobjective ( i.e., to reach a given state asymptotically) which is more flexible and allows one to impose a lot more regularity.
Lyapunov functions are a wellknown tool to study the stability of noncontrolled dynamic systems. For a control system, a Control Lyapunov Functionis a Lyapunov function for the closedloop system where the feedback is chosen appropriately. It can be expressed by a differential inequality called the “Artstein (in)equation” , reminiscent of the HamiltonJacobiBellmann equation but largely underdetermined. One can easily deduce a continuous stabilizing feedback control from the knowledge of a control Lyapunov function; also, even when such a control is known beforehand, obtaining a control Lyapunov function can still be very useful to deal with robustness issues. Moreover, if one has to deal with a problem where it is important to optimize a criterion, and if the optimal solution is hard to compute, one can look for a control Lyapunov function which comes “close” (in the sense of the criterion) to the solution of the optimization problem but leads to a control which is easier to work with.
These constructions were exploited in a joint collaborative research conducted with Thales Alenia Space (Cannes), where minimizing a certain cost is very important (fuel consumption / transfer time) while at the same time a feedback law is preferred because of robustness and ease of implementation (see section ).
Here we study certain transformations of models of control systems, or more accurately of equivalence classes modulo such transformations. The interest is twofold:
From the point of view of control, a command satisfying specific objectives on the transformed system can be used to control the original system including the transformation in the controller. This is relevant when the transformed system has a structure that can easily be exploited, e.g.linear controllable.
From the point of view of identification and modeling, the interest is either to derive qualitative invariants to support the choice of a nonlinear model given the observations, or to contribute to a classification of nonlinear models which is missing sorely today. Indeed, the success of the linear model in control and identification is due to the deep understanding one has of it; a more complete knowledge of invariants of nonlinear models under basic transformations is a prerequisite for a more general theory of nonlinear identification.
Concerning the classes of transformations, a static feedbacktransformation is a (nonsingular) reparametrization of the control depending on the state, together with a change of coordinates in the state space. A dynamic feedbacktransformation consists of a dynamic extension (adding new states, and assigning them a new dynamics) followed by a state feedback on the augmented system. Let us now stress two specific problems that we are tackling.
Dynamic Equivalence.This equivalence is more general than static equivalence and therefore more interesting for a classification. Very few invariants are known. Any insight on this problem is relevant to the above questions. See some result in and section .
A special equivalence class is the one containing linear controllable systems. It turns out that a system is in this class— i.e.is dynamic linearizable—if and only if there is a formula that gives the general solution by applying a nonlinear differential operator to a certain number of arbitrary functions of time; such a formula is often called a (Monge) parametrizationand the order of the differential operator the order of the parametrization. Existence of such a parametrization has been emphasized over the last years as very important and useful in control, see ; this property (with additional requirements on the parametrization) is also called flatness.
An important question remains open: how can one algorithmically decide whether a given system has this property or not, i.e., is dynamic linearizable or not? The mathematical difficulty is that no a priori bound is known on the order of the above mentioned differential operator giving the parametrization. Within the team, results on low dimensional systems have been obtained , see also ; the above mentioned difficulty is not solved for these systems but results are given with prioriprescribed bounds on this order.
From the differential algebraic point of view, the module of differentials of a controllable system is free and finitely generated over the ring of differential polynomials in
d/
dtwith coefficients in the ring of functions on the system's trajectories; the above question is the one of finding out whether there exists a basis consisting of
closed differential forms. Expressed in this way, it looks like an extension of the classical Frobenius integrability theorem to the case where coefficients are differential operators. Of
course, some non classical conditions have to be added to the classical stability by exterior differentiation, and the problem is open. In
, a partial answer to this problem was given, but in a framework where infinitely many variables are allowed and
a finiteness criterion is still missing. The goal is to obtain a formal and implementable algorithm to decide whether or not a given system is flat around a regular point.
Topological Equivalence.Compared to static equivalence, dynamic equivalence is more general, hence might offer some more robust “qualitative” invariants; another way to enlarge equivalence classes is to look for equivalence modulo possibly nondifferentiable transformations.
In the case of dynamical systems without control, the HartmanGrobman theorem states that every system is locally equivalent via a transformation that is solely bicontinuous, to a linear system in a neighborhood of a nondegenerate equilibrium. A HartmanGrobman theorem for control systems would say, typically, that outside a “meager” class of models (for instance, those whose linear approximation is noncontrollable), and locally around nominal values of the state and the control, no qualitative phenomenon can distinguish a nonlinear system from a linear one, all nonlinear phenomena being thus either of global nature or singularities. Such a statement is wrong, at least away from singularities: if a system is locally equivalent to a controllable linear system via a bicontinuous transformation –a local homeomorphism in the statecontrol space– it is alsoequivalent to this same controllable linear system via a transformation that is as smooth as the system itself . A contrario, under weak regularity conditions, linearization can be done by noncausal transformations (see ) whose structure remains unclear, but acquires a concrete meaning when the entries are themselves generated by a finitedimensional dynamics.
The above considerations call for the following question, which is important for modeling control systems: are there local “qualitative” differences between the behavior of a nonlinear system and that of its linear approximation when the latter is controllable?
Many systems coming from mathematical physics, applied mathematics and engineering sciences can be described by means of systems of ordinary or partial differential equations, difference equations, differential timedelay equations... In the case of linear systems, these systems can be defined by means of matrices with entries in noncommutative algebras of functional operators such as differential operators, shift operators, timedelay operators, difference operators...
The methods of
algebraic analysis
The research of Apics on such topics combines Gröbner bases techniques over some noncommutative polynomial rings with the development of new algorithms of algebraic
analysis in order to effectively check classical properties of module theory (e.g., existence of a nontrivial torsion submodule or
rpure torsion submodules, torsionfreeness, reflexiveness, projectiveness, stably freeness, freeness, simple or decomposable modules), give their systemtheoretical interpretations
(existence of autonomous elements or successive parametrizations, existence of minimal/injective parametrizations or Bézout equations) and compute important tools of homological algebra (e.g.,
(minimal) free resolutions, split long exact sequences, extension and torsion functors, projective and Krull dimensions, Hilbert power series). The developed algorithms are implemented in
various symbolic packages that are used to apply our results to systems theory (e.g., parameterizability, flatness, autonomous elements, equivalences of systems, factoring and decomposing
linear functional systems) and to mathematical physics (e.g., research of potentials, computations of the field equations and the conservation laws).
The bottom line of the team's activity is twofold, namely function theory and optimization in the frequency domain on the one hand, and the control of certain systems governed by differential equations on the other hand. Therefore one can distinguish between two main families of applications: one dealing with the design and identification of diffusive and resonant systems (these are inverse problems), and one dealing with the control of certain mechanical systems. For applications of the first type, approximation techniques as described in section allow one to deconvolve linear equations, analyticity being the result of either the use of Fourier transforms or the harmonic character of the equation itself. Applications of the second type mostly concern the control of systems that are “poorly” controllable, for instance low thrust satellites or optical regenerators. We describe all these below in more detail.
We are mainly concerned with classical inverse problems like the one of localizing defaults (as cracks, pointwise sources or occlusions) in a two or three dimensional domain from boundary data (which may correspond to thermal, electrical, or magnetic measurements), of a solution to Laplace or to some conductivity equation in the domain. These defaults can be expressed as a lack of analyticity of the solution of the associated DirichletNeumann problem that may be approached, in balls, using techniques of best rational or meromorphic approximation on the boundary of the object (see sections , ).
Indeed, it turns out that traces of the boundary data on 2D cross sections (disks) coincide with analytic functions in the slicing plane, that has branched singularities inside the disk . These singularities are related to the actual location of the sources (namely, they reach in turn a maximum in modulus when the plane contains one of the sources). Hence, we are back to the 2D framework where approximately recovering these singularities can be performed using best rational approximation.
In this connection, the realistic case where data are available on part of the boundary only offers a typical opportunity to apply the analytic extension techniques (see section ) to Cauchy type issues, a somewhat different kind of inverse problems in which the team is strongly interested.
The approach proposed here consists in recovering, from measured data on a subset
Kof the boundary
Dof a domain
Dof
R^{2}or
R^{3}, say the values
F_{K}on
Kof some function
F, the subset
of its singularities (typically, a crack or a discrete set of pointwise sources), provided that
Fis an analytic function in
.
The analytic approximation techniques (section
) first allow us to extend
Ffrom the given data
F_{K}to all of
D, if
KD, which is a Cauchy type issue for which our algorithms provide robust solutions, in plane domains (see
for the case of the 2D disk or simply connected domains,
,
,
,
for 2D annular domains, and
for 3D spherical situations, also discussed in section
). Note that identification schemes for an unknown Robin coefficient together with stability properties have been obtained
in the same way
.
From these extended data on the whole boundary, one can obtain information on the presence and the location of , using rational or meromorphic approximation on the boundary (sections , ). This may be viewed as a discretization of by the poles of the approximants . This is the case in dimension 2, using classical links between analyticity and harmonicity , but also in dimension 3, at least in spherical or ellipsoidal domains, working on 2D plane sections, , ).
The two above steps are shown in to provide a robust way of locating sources from incomplete boundary data in a 2D situation with several annular layers. Numerical experiments have already yielded excellent results in 3D situations and we are now on the way to process real experimental magnetoencephalographic data, see also sections , . The doctoral work of A.M. Nicu and M. Zghal are concerned with these applications, in collaboration with the Odyssée team of Inria SAM, and with neurosciences teams in partnerhospitals (Timone, Marseille, and Salpêtrière, Paris).
Such methods are currently being generalized to problems with variable conductivity governed by a 2D Beltrami equation, see . The application we have in mind is to plasma confinement for thermonuclear fusion in a Tokamak, more precisely with the extrapolation of magnetic data on the boundary of the chamber from the outer boundary of the plasma, which is a level curve for the poloidal flux solving the original divgrad equation. Solving this inverse problem of Bernoulli type is of importance to determine the appropriate boundary conditions to be applied to the chamber in order to shape the plasma . These issues are the topics of the thesis of S. Chaabi and Y. Fischer, and of a joint collaboration with the CEAIRFM (Cadarache), the Laboratoire J.A. Dieudonné at the Univ. of NiceSA, and the CMILATP at the Univ. of Marseille I (see section ).
One of the best training grounds for the research of the team in function theory is the identification and design of physical systems for which the linearity assumption works well in the considered range of frequency, and whose specifications are made in the frequency domain. Resonant systems, either acoustic or electromagnetic based, are prototypical devices of common use in telecommunications.
In the domain of space telecommunications (satellite transmissions), constraints specific to onboard technology lead to the use of filters with resonant cavities in the microwave range. These filters serve multiplexing purposes (before or after amplification), and consist of a sequence of cylindrical hollow bodies, magnetically coupled by irises (orthogonal double slits). The electromagnetic wave that traverses the cavities satisfies the Maxwell equations, forcing the tangent electrical field along the body of the cavity to be zero. A deeper study (of the Helmholtz equation) states that essentially only a discrete set of wave vectors is selected. In the considered range of frequency, the electrical field in each cavity can be seen as being decomposed along two orthogonal modes, perpendicular to the axis of the cavity (other modes are far off in the frequency domain, and their influence can be neglected).
Near the resonance frequency, a good approximation of the Maxwell equations is given by the solution of a second order differential equation. One obtains thus an electrical model for our filter as a sequence of electricallycoupled resonant circuits, and each circuit will be modeled by two resonators, one per mode, whose resonance frequency represents the frequency of a mode, and whose resistance represent the electric losses (current on the surface).
In this way, the filter can be seen as a quadripole, with two ports, when plugged on a resistor at one end and fed with some potential at the other end. We are then interested in the power
which is transmitted and reflected. This leads to defining a scattering matrix
S, that can be considered as the transfer function of a stable causal linear dynamical system, with two inputs and two outputs. Its diagonal terms
S_{1, 1},
S_{2, 2}correspond to reflections at each port, while
S_{1, 2},
S_{2, 1}correspond to transmission. These functions can be measured at certain frequencies (on the imaginary axis). The filter is rational of order 4 times the number of cavities (that is 16 in
the example), and the key step consists in expressing the components of the equivalent electrical circuit as a function of the
S_{ij}(since there are no formulas expressing the lengths of the screws in terms of parameters of this electrical model). This representation is also useful to analyze the numerical
simulations of the Maxwell equations, and to check the design, particularly the absence of higher resonant modes.
In fact, resonance is not studied via the electrical model, but via a lowpass equivalent circuit obtained upon linearizing near the central frequency, which is no longer conjugate symmetric ( i.e., the underlying system may not have real coefficients) but whose degree is divided by 2 (8 in the example).
In short, the identification strategy is as follows:
measuring the scattering matrix of the filter near the optimal frequency over twice the pass band (which is 80Mhz in the example).
solving bounded extremal problems for the transmission and the reflection (the modulus of he response being respectively close to 0 and 1 outside the interval measurement, cf.section ). This provides us with a scattering matrix of order roughly 1/4 of the number of data points.
Approximating this scattering matrix by a rational transferfunction of fixed degree (8 in this example) via the Endymion or RARL2 software ( cf.section ).
A realization of the transfer function is thus obtained, and some additional symmetry constraints are imposed.
Finally one builds a realization of the approximant and looks for a change of variables that eliminates nonphysical couplings. This is obtained by using algebraicsolvers and continuation algorithms on the group of orthogonal complex matrices (symmetry forces this type of transformation).
The final approximation is of high quality. This can be interpreted as a validation of the linearity hypothesis for the system: the relative
L^{2}error is less than
10
^{3}. This is illustrated by a reflection diagram (Figure
). Nonphysical couplings are less than
10
^{2}.
The above considerations are valid for a large class of filters. These developments have also been used for the design of nonsymmetric filters, useful for the synthesis of repeating devices.
The team investigates today the design of output multiplexors (OMUX) where several filters of the previous type are coupled on a common guide. In fact, it has undergone a rather general analysis of the question “How does an OMUX work?” With the help of numerical simulations and Schur analysis, general principles are being worked out to take into account:
the coupling between each channel and the “Tee” that connects it to the manifold,
the coupling between two consecutive channels.
The model is obtained upon chaining the corresponding scattering matrices, and mixes up rational elements and complex exponentials (because of the delays) hence constitutes an extension of the previous framework. Its study is being conducted under contract with Thales Alenia Space (Toulouse) (see sections and ).
Generally speaking, aerospace engineering requires sophisticated control techniques for which optimization is often crucial, due to the extreme functioning conditions. The use of satellites in telecommunication networks motivates a lot of research in the area of signal and image processing; see for instance section for an illustration. Of course, this requires that satellites be adequately controlled, both in position and orientation (attitude). This problem and similar ones continue to motivate research in control. The team has been working for six years on control problems in orbital transfer with lowthrust engines, including four years under contract with Thales Alenia Space (formerly Alcatel Space) in Cannes.
Technically, the reason for using these (ionic) low thrust engines, rather than chemical engines that deliver a much higher thrust, is that they require much less “fuel”; this is decisive because the total mass is limited by the capacity of the launchers: less fuel means more payload, while fuel represents today an impressive part of the total mass.
From the control point of view, the low thrust makes the transfer problem delicate. In principle of course, the control law leading to the right orbit in minimum time exists, but it is quite
heavy to obtain numerically and the computation is nonrobust against many unmodelled phenomena. Considerable progress on the approximation of such a law by a feedback has been carried out
using
ad hocLyapunov functions.These approximate surprisingly well timeoptimal trajectories. The easy implementation of such control laws makes them attractive as compared to genuine optimal
control. Here the
n1first integrals are an easy means to build control Lyapunov functions since any function of these first integrals can be made monotone decreasing by a suitable
control. See
and the references therein.
The development of the
RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see section
)
http://
This software takes as input a stable transfer function of a discrete time system represented by
either its internal realization,
or its first
NFourier coefficients,
or discretized values on the circle.
It computes a local best approximant which is
stable, of prescribed McMillan degree, in the
L^{2}norm.
It is akin to the arl2 function of Endymion from which it differs mainly in the way systems are represented: a polynomial representation is used in Endymion, while RARL2 uses realizations, this being very interesting in certain cases. It is implemented in Matlab. This software handles multivariablesystems (with several inputs and several outputs), and uses a parametrization that has the following advantages
it incorporates the stability requirement in a builtin manner,
it allows the use of differential tools,
it is wellconditioned, and computationally cheap.
An iterative research strategy on the degree of the local minima, similar in principle to that of arl2, increases the chance of obtaining the absolute minimum (see section ) by generating, in a structured manner, several initial conditions.
RARL2 performs the rational approximation step in our applications to filter identification (section ) as well as sources or cracks recovery (section ). It was released to the universities of Delft, Maastricht, Cork and Brussels. The parametrization embodied in RARL2 was recently used for a multiobjective control synthesis problem provided by ESTECESA, The Netherlands (section ). An extension of the software to the case of triple poles approximants is now available. It gives nice results in the source recovery problem (section ).
The identification of filters modeled by an electrical circuit that was developed by the team (see section
) led us to compute the electrical parameters of the underlying filter. This means finding a particular realization
(
A,
B,
C,
D)of the model given by the rational approximation step. This 4tuple must satisfy constraints that come from the geometry of the equivalent electrical network and
translate into some of the coefficients in
(
A,
B,
C,
D)being zero. Among the different geometries of coupling, there is one called “the arrow form”
which is of particular interest since it is unique for a given transfer function and also easily computed. The
computation of this realization is the first step of RGC. Subsequently, if the target realization is not in arrow form, one can nevertheless show that it can be deduced from the arrowform by a
complex orthogonal change of basis. In this case, RGC starts a local optimization procedure that reduces the distance between the arrow form and the target, using successive orthogonal
transformations. This optimization problem on the group of orthogonal matrices is nonconvex and has a lot of local and global minima. In fact, there is not always uniqueness of the realization
of the filter in the given geometry. Moreover, it is often interesting to know all the solutions of the problem, because the designer cannot be sure, in many cases, which one is being handled,
and also because the assumptions on the reciprocal influence of the resonant modes may not be equally well satisfied for all such solutions, hence some of them should be preferred for the
design. Today, apart from the particular case where the arrow form is the desired form (this happens frequently up to degree 6) the RGC software gives no guarantee to obtain a single
realization that satisfies the prescribed constraints. The software DedaleHF (see
), which is the successor of RGC, solves in a guaranteed manner this constraint realization problem.
PRESTOHF: a toolbox dedicated to lowpass parameter identification for microwave filters http://wwwsop.inria.fr/apics/personnel/Fabien.Seyfert/Presto_web_page/presto_pres.html. In order to allow the industrial transfer of our methods, a Matlabbased toolbox has been developed, dedicated to the problem of identification of lowpass microwave filter parameters. It allows one to run the following algorithmic steps, either individually or in a single shot:
determination of delay components, that are caused by the access devices (automatic reference plane adjustment);
automatic determination of an analytic completion, bounded in modulus for each channel,
rational approximation of fixed McMillan degree;
determination of a constrained realization.
For the matrixvalued rational approximation step, PrestoHF relies either on hyperion (Unix or Linux only) or RARL2 (platform independent), both rational approximation engines were developed within the team. Constrained realizations are computed by the RGC software. As a toolbox, PrestoHF has a modular structure, which allows one for example to include some building blocks in an already existing software.
The delay compensation algorithm is based on the following strong assumption: far off the passband, one can reasonably expect a good approximation of the rational components of
S_{11}and
S_{22}by the first few terms of their Taylor expansion at infinity, a small degree polynomial in
1/
s. Using this idea, a sequence of quadratic convex optimization problems are solved, in order to obtain appropriate compensations. In order to check the previous
assumption, one has to measure the filter on a larger band, typically three times the pass band.
This toolbox is currently used by Thales Alenia Space in Toulouse and a license agreement has been recently negotiated with Thales airborne systems. XLim (University of Limoges) is a heavy user of PrestoHF among the academic filtering community and some free license agreements are currently being considered with the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingston, Canada).
The development of Endymion, http://wwwsop.inria.fr/apics/endymion/index.htmlhas been pursued. It is a software licensed under the CeCILL license version two, see http://www.cecill.info. It was developed on Linux, but works as well on MacOS (PowerPC and Intel processors). The core of the system is formed by a library that handles numbers (short integers, arbitrary size rational numbers, floating point numbers, quadruple and octuple precision floating point numbers, arbitrary precision real numbers, complex numbers), polynomials, matrices, etc. Specific data structures for the rational approximation algorithm arl2and the bounded extremal problem bepare also available. One can mention for instance splines, Fourier series, Schur matrices, etc. These data structures are manipulated by dedicated algorithms (matrix inversion, roots of polynomials, a gradientbased algorithm for minimising , Newton method for finding a critical point of , etc), and inputouput functions that allow one to save data on disk, restore them, plot them, etc.
The software is interactive. Basically there is a Lisp interpreter: one says
(getcoef P 2)if one wants the coefficient of
z^{2}in P. A symbolic interpreter is based on top of the Lisp interpreter, for instance one can say
P[2]++; this increments the quantity
P_{2}, which is the coefficient of
z^{2}in
Pif
Pis a polynomial, or the third element if
Pis a list, or whatever, etc. There is also a compiler, that transforms a file containing symbolic instructions into a file containing Lisp instructions.
Lets us discuss one algorithm:
arl2with poles of multiplicity
k. The problem is to find
pand
qsuch that
f
p/
qis of minimal norm, where
fis given. One can easily eliminate
p, so that the problem consists in finding
qminimising
_{f}(
q). The same problem holds in the nonscalar case, where the DouglasShapiroShields factorization is used (see section
). In the scalar case,
qis a polynomial defined by its coefficients, in the nonscalar case,
is a function of
Q, a matrix determined by its Schur parameters. One can add the constraint that all poles have the same multiplicity
k. This means that
q=
r^{k}for some polynomial
r. The algorithm works in the case where the denominator is a scalar (in the multiinput multioutput case, this means that there is a single input or a single output), irrespective of
whether
qis given by its coefficients or Schur parameters. Note that the program computes the derivatives of
qfrom the derivatives of
rusing automatic differentiation rules in direct and reverse mode. Clearly
q^{'}=
kr^{'}r^{k1}, but in reverse mode, things are less easy, and we compute
s=
r^{2}then either
q=
s,
q=
rsor
q=
s^{2}. This means that the algorithm is only implemented for
k4.
DedaleHF is a software meant to solve exhaustively the coupling matrix synthesis problem in reasonable time for the users of the filtering community. For a given coupling topology the coupling matrix synthesis problem (C.M. problem for short) consists in finding all possible electromagnetic coupling values between resonators that yield a realization of a given filter characteristics (see section ). Solving the latter problem is crucial during the design step of a filter in order to derive its physical dimensions as well as during the tuning process where coupling values need to be extracted from frequency measurements (see Figure ).
DedaleHf consists in two parts: a database of coupling topologies as well as a dedicated predictorcorrector code. Roughly speaking each reference file of the database contains, for a given coupling topology, the complete solution to the C.M. problem associated to a particular filtering characteristics. The latter is then used as a starting point for a predictorcorrector integration method that computes the solution to the C.M. problem of the user, i.e.the one corresponding to a userspecified filter characteristics. The reference files are computed off line using Groebner basis techniques or numerical techniques based on the exploration of a monodromy group. The use of such a continuation technique combined with an efficient implementation of the integrator produces a drastic reduction of the computational time, say, by a factor of 20.
Access to the database and integrator code is done via the web on http://wwwsop.inria.fr/apics/Dedale/WebPages. The software is free of charge for academic research purposes: a registration is however needed in order to access full functionality. Up to now 90 users have registered among the world (mainly: Europe, U.S.A, Canada and China) and 4000 reference files have been downloaded.
As mentioned in an extension of this software that handles symmetrical networks is under construction.
The OreModuleslibrary of Ore _{}algebra ( Ore _{}algebrais a part of the commercial release of Maple) is dedicated to the study of linear functional systems defined over certain Ore algebras of functional operators and their applications in mathematical systems theory and mathematical physics.
The main novelty of OreModulesis to combine the recent developments of the Gröbner bases over some noncommutative polynomial rings with new algorithms of algebraic analysis in order to effectively check classical properties of module theory (e.g., existence of a nontrivial torsion submodule, torsionfreeness, reflexiveness, projectiveness, stably freeness, freeness), give their systemtheoretical interpretations (existence of autonomous elements or successive parametrizations, existence of minimal/injective parametrizations or Bézout equations) and compute important tools of homological algebra (e.g., (minimal) free resolutions, split exact sequences, extension functors, projective or Krull dimensions, Hilbert power series).
The abstract language of homological algebra used in the algebraic analysis approach carries over to the implementations in OreModules: up to the choice of the domain of functional operators which occurs in a given system, all algorithms are stated and implemented in sufficient generality such that linear systems defined over the Ore algebras developed in the Maple package of Ore _{}algebraare covered at the same time. Applications of the OreModulespackage to mathematical systems theory are illustrated in a large library of examples.
The
Staffordpackage of
OreModulescontains an implementation of constructive versions of J. T. Stafford's famous but difficult theorem stating that every ideal over
the Weyl algebras
A_{n}(
k)and
B_{n}(
k)(
kis a field of characteristic 0) can be generated by two generators. Based on this implementation and on algorithmic results recently obtained by the authors of this package, two
algorithms have been implemented which compute bases of free modules over the Weyl algebras
and
.
The forthcoming QuillenSuslinpackage of OreModules, developed by A. Fabiańska (University of Aachen) with the help of A. Quadrat, contains an implementation of the famous QuillenSuslin theorem. In particular, this implementation allows us to compute bases of free modules over a commutative polynomial ring with coefficients in the field and in the principal ideal domain .
The OreMorphismspackage of OreModuleswas developed by T. Cluzeau (ENSIL, Limoges) and A. Quadrat in order to handle some homological tools such as computations of some morphisms between two finitely presented modules over Ore algebras, compute kernel, coimage, image and cokernel of such morphisms and projectors. Using the packages Staffordand QuillenSuslin, these results allow us to compute factorizations as well as finding some decompositions of linear systems over Ore algebras. In terms of module theory, the OreMorphismspackage gives some methods to test if two modules are isomorphic, if a given module contains a submodule (reducible modules) or if it can be written as the direct sum of two submodules. Applications of the OreMorphismspackage to mathematical physics and mathematical systems theory are illustrated in a library of examples.
FindSources3D is a software, being implemented (see
http://
The major use of Tralics remains the production of the RAWEB (Scientific Annex to the Annual Activity Report of Inria), as explained schematically on figure
. The input is a
Other applications of Tralics consist in putting scientific papers on the Web; for instance Cedram (
http://
The main philosophy of Tralics is to have the same parser as
Two major versions have been released this year, namely 2.12 in April and 2.13 in October. The documentation consists in some technical reports and , , , they are regularly updated, especially the HTML version (produced by Tralics). Some new packages were added to the system (graphicx, xkeyval, color), and for efficiency reasons, part of the code is implemented in the C++ kernel. The referencing system was completely rewritten, so that for instance the XML document contains the same equation numbers as the PostScript version (in the Raweb case, equation numbers are computed by the XMLtoHTML style sheet). The raweb preprocessor was removed: all commands specific to the Activity Report are now defined in package files, the kernel containing some primitives that can check the validity of some arguments versus a keyword list defined in the configuration file.
Solving overdetermined Cauchy problems for the Laplace equation on a spherical layer (in 3D) in order to treat incomplete experimental data is a necessary ingredient of the team's approach to inverse source problems, in particular for applications to EEG since the latter involves propagating the initial conditions from the boundary to the center of the domain where the singularities ( i.e.the sources) are sought. Here, the domain is typically made of several homogeneous layers of different conductivities.
Such problems offer an opportunity to state and solve extremal problems for harmonic fields for which an analog of the Toeplitz operator approach to bounded extremal problems
has been obtained. Still, a best approximation on the subset of a general vector field by a harmonic gradient
under a
L^{2}norm constraint on the complementary subset can be computed by an inverse spectral equation for some Toeplitz operator. Constructive and numerical aspects of the procedure (harmonic
3D projection, Kelvin and Riesz transformation, spherical harmonics) and encouraging results have been obtained on numerically simulated data
.
Issues of robust interpolation on the sphere from incomplete pointwise data are also under study (splines, spherical harmonics, spherical wavelets, spherical Laplace operator, ...), in order to improve numerical accuracy of our reconstruction schemes.
The analogous problem in
L^{p},
p2, is considerably more difficult. A collaborative work with A. Bonami and S.
Grellier (université d'Orléans) in the framework of the ANR project AHPI is going on, aiming mainly at the case
p=
. It was obtained that the BMO distance between a bounded vector field on the
sphere and a bounded harmonic gradient is within a constant of the norm of a Hankellike operator, acting on
L^{2}divergencefree vector fields with values in
L^{2}gradients. Estimating the constant requires solving further extremal problems in
L^{1}on the best approximation of a gradient by a divergence free vector field. This issue is currently being studied in
L^{p}where it leads to analyze particular solutions to the the
pLaplacian on the sphere.
The problem of sources recovery can be handled in 3D balls by using best rational approximation on 2D cross sections (disks) from traces of the boundary data on the corresponding circles (see section ).
The team started to consider more realistic geometries for the 3D domain under consideration. A possibility is to parametrize it in such a way that its planar crosssections are quadrature domains or Rdomains. In this framework, best rational approximation can still be performed in order to recover the singularities of solutions to Laplace equations, but complexity issues are delicate. The preliminary case of an ellipsoid, which requires the preliminary computation of an explicit basis of ellipsoidal harmonics, has been studied in and is one of the topics of the PhD thesis of M. Zghal.
In 3D, epileptic regions in the cortex are often represented by pointwise sources that have to be localized from measurements on the scalp of a potential satisfying a Laplace equation (EEG, electroencephalography). A breakthrough was made which makes it possible now to proceed via best rational approximation on a sequence of 2D disks along the inner sphere . A dedicated numerical software “FindSources3D” (see section ) is being developed (R. Bassila), in collaboration with the team Odyssée.
Also, magnetic data from MEG (magnetoencephalography) will soon become available, which should enhance sources recovery. Indeed, the radial component of the magnetic field on the boundary also reflects the presence of current dipoles in the domain. This is the topic of A.M. Nicu's PhD thesis. Magnetic sources localization by analytic and rational approximation on plane sections is currently analyzed from experimental SQUID data, from Vanderbilt University Physics Dept. (also within M. Zghal's thesis).
Further, it appears that in the rational approximation step of these schemes, multiplepoles possess a nice behaviour with respect to the branched singularities (see figure ). This is due to the very basic physical assumptions on the model (for EEG data, one should consider triplepoles). Though numerically observed, there is no mathematical justification why these multiple poles have such strong accumulation properties, which remains an intriguing observation.
In collaboration with the CMILATP (University Marseille I) and in the framework of the ANR AHPI, the team considers 2D diffusion processes with variable conductivity. In particular its complexified version, the socalled real Beltrami equation, was investigated. In the case of a smooth domain, and for a smooth conductivity, we analyzed the Dirichlet problem for solutions in Sobolev and then in Hardy classes .
Their traces merely lie in
L^{p}(
1<
p<
) of the boundary, a space which is suitable for identification from pointwise
measurements. Again these traces turn out to be dense on strict subsets of the boundary. This allows us to state Cauchy problems as bounded extremal issues in
L^{p}classes of generalized analytic functions, in a reminiscent manner of what was done for analytic functions as discussed in section
. Recently, dual formulations were obtained and some multiplicative (fibered) structure for the solution was obtained based
on old work by Bers and Nirenberg on pseudoanalytic functions. An article is being written on these topics.
The case of a conductivity that is merely in
, which is important for inverse conductivity problems, is under examination (PhD thesis of S. Chaabi). There, it is still unknown whether solutions exist for all
p.
The application that initially motivated this work comes from free boundary problems in plasma confinement (in tokamaks) for thermonuclear fusion. This work was started in collaboration with the Laboratoire J. Dieudonné (University of Nice) and is now the topic of a collaboration with two teams of physicists from the CEAIRFM (Cadarache).
In the transversal section of a tokamak (which is a disk if the vessel is idealized into a torus), the socalled poloidal flux is subject to some conductivity outside the plasma volume for some simple explicit smooth conductivity function, while the boundary of the plasma (in the Tore Supra Tokamak) is a level line of this flux . Related magnetic measurements are available on the chamber, which furnish incomplete boundary data from which one wants to recover the inner (plasma) boundary. This free boundary problem (of Bernoulli type) can be handled through the solutions of a family of bounded extremal problems in generalized Hardy classes of solutions to real Beltrami equations, in the annular framework. Such approximation problems also allow us to approach a somewhat dual extrapolation issue, raised by colleagues from the CEA for the purpose of numerical simulation. It consists in recovering magnetic quantities on the outer boundary (the chamber) from an initial guess of what the inner boundary (plasma) is.
In the particular case at hand, it seems possible to explicitly compute a basis of solutions (Bessel functions) that should greatly help the computations, see . This is the topic of the PhD thesis of Y. Fischer.
In the most recent tokamaks, like Jet or ITER, an interesting feature of the level curves of the poloidal flux is the occurrence of a cusp (a saddle point of the poloidal flux, called an X point), and it is desirable to shape the plasma according to a level line passing through this X point for physical reasons relating to the efficiency of the energy transfer. This will be the topic of future studies.
We mentioned in section the role of parameters defining an atlas of charts in rational matrix approximation. Charts for the class of lossless systems were obtained from interpolation theory , which lead to a simple and robust computation of balanced realizations and form the basis of RARL2 (see section ).
This year, a particular attention was paid to some (local) canonical forms presenting a particular structure, the subdiagonal pivot structure. These forms generalize the wellknown Hessenberg form in discretetime and SchwarzOber form in continuoustime, which are involved in the estimation of physical quantities in many applications. A very flexible, straightforward algorithm to put any system into these canonical forms under state isometries was obtained. These results have been presented at the ERNSI meeting and at the regular seminar of the Cambridge University Engineering Department. For the class of lossless discretetime systems, these forms are precisely those computed from a backward Schur algorithm in which the interpolation points are at zero and the interpolation directions standard basis vectors. In continuoustime, the relevant interpolation problem in connection with these forms happens to be a boundary interpolation problem: the interpolation points are still at zero, which is no longer in the analyticity domain but on its boundary. In the SISO case, the parameters in the OberSchwarz canonical form can be interpreted as boundary interpolation values in a recursive Schur type algorithm . To our knowledge, boundary interpolation has never been used in the description or in the parametrization of stable systems and this approach seems to be promising.
Passive devices play an important role in a lot of application areas: telecommunication, chemical process control, economy, biomedical processes. Network simulation software packages (as ADS
or SPICE) require passive models for their components. However, identifying a passive model from band limited frequency data is still an open and challenging problem. Schur rational
approximation is a new way to approach this problem and was the subject of the Phd Thesis of Vincent Lunot
. In this work, a parametrization of all strictly Schur rational functions of degree
nis constructed from a multipoint Schur algorithm, the parameters being both the interpolation values and interpolation points. Examples are computed by an
L^{2}norm optimization process and the results are validated by comparison with the unconstrained
L^{2}rational approximation. Last year, the results of
on the hyperbolic convergence of the classical Schur algorithm were generalized to the case of the multipoint Schur
algorithm. Orthogonal rational functions and a recent generalization of Geronimus theorem were used
. This year, an analog of the Szegö theorem was obtained where the interpolation points tend to the boundary,
provided the approximated function is continuous and less than 1 in a neighborhood of the accumulation set of the interpolation points. It generalizes the results in
and is of novel type since the
nth orthogonal rational function inverts the Szegö function
modulothe Poisson kernel. This provides one with a rather unexpected theorem on the behaviour of certain orthogonal polynomials with varying weight. This research is partly contained in
V. Lunot's doctoral dissertation, and an article is being written on the results.
Yet the case where the function attains 1 in modulus is very important since it justifies much of the use of the Schur algorithm. Indeed, it is difficult to obtain Schur approximants to a function which is Schur but not of modulus strictly less than 1, because approximants tend to wind around the function thereby producing overshoot. This situation is now our next goal. We proved already that if the value 1 is matched slowly enough at an isolated place, the result continues to hold. It depends on LittlewoodPaley type estimates of the error.
The results of
and
were extensively used over the last years to prove the convergence in capacity of
L^{p}best meromorphic approximants on the circle (
i.e.solutions to problem (
P_{N}) of section
) when
p2, for those functions
fthat can be written as Cauchy transforms of complex measures supported on a hyperbolic geodesic arc
,
,
. A rational function can also be added to
fwithout modifying the results, which is useful for applications to inverse sources problems. Some mild conditions (bounded variation of the argument and powerthickness of the total
variation) were required on the measure. Here, we recall that convergence in capacity means that the (logarithmic) capacity of the set where the error is greater than
goes to 0 for each fixed
>0. This convergence can be quantified, namely it is geometric with
pointwise rate
exp{1/
C
G}where
Cis the capacity of the condensor
and
Gthe Green potential of the equilibrium measure. The results can be adapted to somewhat general interpolation schemes
,
. From this work it follows that the counting measures of the poles of the approximants converge, in the weak*
sense, to the Green equilibrium distribution on
. In particular the poles cluster to the endpoints of the arc, which is of fundamental use in the team's approach to source detection (see section
).
The technique we just described only yields convergence in capacity and
nth root asymptotics. To obtain strong asymptotics, additional assumptions must usually be made on the approximated function. This year, we proved strong asymptotics of appropriate
interpolants to Cauchy integrals
over arbitrary analytic arcs, when the density of the measure with respect to a positive power of the equilibrium distribution on the arc is Dinismooth. Moreover, if the density is
Höldercontinuous, the result holds without restrictions, that is, the power of the equilibrium distribution to which we compare the density needs no longer be positive (in the language of
orthogonal polynomials, this means we can handle arbitrary Jacobi weights). In addition, the density
may in fact vanishin finitely many points like a small fractional power of the distance to these point
. This result is a significant achievement in the theory of Padé approximation, in that it is the first asserting
that uniform convergence holds, for a quite large class of functions, when the interpolation points are chosen in some appropriate manner (symmetric with respect to the weighted equilibrium
potential on the contour). Moreover, the polar singularities of the function, if any, are asymptotically reproduced by the approximants with their multiplicities. This is important for inverse
problem of mixed type, like those mentioned in section
, where monopolar and dipolar sources are handled simultaneously. Some convergence even holds on the support of the measure.
Moreover, this yields bounds on the multiplicity of the singular values of the underlying Hankel operators
.
The method is to translate the critical point equation into a RiemannHilbert problem on an analytic curve where harmonic analysis techniques can be used. Analyzing the solution in terms of Hankel and Toeplitz spectral equations, the conclusion ultimately follows from estimates on the essential spectrum of Hankel operators. In the case of nonpositive Jacobi powers, estimates and Muckenhoupt weights are also needed.
In another collection, the results of
have been carried over for analytic approximation to the matrix case in
. The surprising fact was that not every matrix valued function generates a vectorial Hankel operator meeting the
AAK theorem when
p<
. This led us to the generalization of the latter based on Hankel operators with
matrix argument.
Specifically, under weak regularity conditions on the measure, the counting measure of the poles converges weakstar to the equilibrium distribution of the condenser
(
T,
)in the case of meromorphic approximants, and of the condenser
(
,
)where
is the support of the asymptotic distribution of the interpolation points if one deals with convergent Padé approximants
,
,
. The more general situation where
is a socalled “minimal contour” for the Green potential (of which a geodesic arc is the simplest example) has been settled with the same conclusion. This technical result is still under
writing. Below, we illustrate numerically these facts which are of particular significance to locate several 2D sources or piecewise analytic cracks from overdetermined boundary data (see
sections
and
).
Figure shows the location of the poles of various approximants to the function:
Groomed by industrial users like Thales AleniaSpace, we made some progress in the analysis of the realizations of 2x2 lossless scattering systems whose scattering response
(
S
_{i,
j})satisfies the socalled
autoreciprocalcondition
S_{1, 1}=
S_{2, 2}. It was shown that autoreciprocal inner responses admit a canonical circuit realisation of the form of Fig.
. The length difference (
m
l) of the two antennas of Fig.
is equal to the Cauchy index on the imaginary axes of the filter function to be realised. Surprisingly enough this form
appears to be central in the new modal framework S.Amari is currently developing on dual mode filters (
). It was shown that the classical folded form can be advantageously replaced by the latter yielding a design
procedure with nearly no tuning required (all the physical dimensions of the filter can be computed exactly from the circuit parameters): a paper has been submitted on this topic and is
currently being reviewed. In future work, we will focus on the practical implementation of this analysis within the software DedaleHf
.
We also made some progress on the problem of circuit realisations with mixed type (inductive or capacitive) coupling elements. An algebraic formulation of the synthesis problem of circuits with mixed type elements has been obtained which relies on a set of two matricial equations. As opposed to the classical low pass case with frequency independent couplings the unknown is no longer a similarity transform but a general nonsingular matrix acting on two coupling matrices: the capacitive and the inductive one. Special structures of the underlying equations are being studied and approaches relying on the efficient use of Groebner basis and continuation techniques will be investigated.
Finally a general survey on applications of our work to microwave filters synthesis has been published as well as an application to insertion loss minimization .
This is part of V. Lunot's doctoral work. The theoretical developments took place over the last two years, while deepenings of the numerical aspects were carried out in 2007. This study was conducted under contract with the CNES and ThalèsAleniaSpace (Toulouse). The problem goes as follows. On introducing the ratio of the transmission and reflexion entries of a scattering matrix, the design of a multiband filter response (see section ) reduces to the following optimization problem of Zolotarev type :
where
(resp.
) is a finite union of compact intervals
I_{i}of the real line corresponding to the passbands (resp. stopbands), and
P_{m}(
K)stands for the set of polynomials of degree less than
mwith coefficients in the field
K. Depending on the physical symmetry of the filter, it is interesting to solve problem (
) either for
(“real” problem) or
(“mixed” problem), or else
(“complex” problem). The “real” Zolotarev problem can be decomposed into a sequence of concave maximization problems, whose solution we were able to characterize in terms of an
alternation property. Based on this, a Remezlike algorithm has been derived in the polynomial case (
i.e.when the denominator
qof the scattering matrix is fixed), which allows for the computation of a dualband response (see Figure
) according to the frequency specifications (see Figure
for an example from the spacecraft SPOT5 (CNES)). We have designed an algorithm in the rational case which, unlike linear
programming, avoids sampling over all frequencies. This raises the issue of the “generic normality” (
i.e.the maximum degree) of the approximant with respect to the geometry of the intervals. This question remains open. The design of efficient procedures to tackle the “mixed” and
“complex” cases remains a challenge. A preliminary version of
easyFF, the software by V. Lunot to treat the complex case has been released this year to our academic partners: Xlim and the Royal Military College of Canada. Applications of the Remez
algorithm to filter synthesis are described in
,
. An article on the general approach based on linear programming has been published
.
Some important results have been obtained in order to handle tuning and synthesis of broad band filters. One of the major problems when dealing with wide band filters is the break down of
the classical low pass model which relies on a narrow band assumption. We showed however that there exists a unifying “low pass formalism” which is valid in the narrow band as well as in the
wide band situations. The latter relies on the following remark. Let
Sbe any inner, real, symmetric (
S^{t}=
S), rational matrix, which is identity at infinity and has MacMillan degree
n. Then the rational matrix
S_{r}defined by:
is again an inner, complex, symmetric matrix, which is identity at infinity, and has MacMillan degree
n. It can be shown that
Sis entirely characterised by the knowledge of its reduced “complex” version
S_{r}. Measurements of
Son two conjugate frequency bands are mapped to measurements of
S_{r}on a single band, which up to the use of a linear frequency transformation can be cast to the normalized band
[1, 1]. Usual techniques used to recover rational models from low pass responses measured on a single frequency interval can therefore be used to recover
high pass responses via the use of the generalized reduced response
S_{r}. Implementation attempts of the latter in the PrestoHf software were started and encouraging results where obtained for the tuning of an ultrawide band filter realized with suspended
strip lines. Figure
shows data and their rational approximation of this 10
^{th}order filter (reduced order 5) with a bandwidth ratio of approximately
10%(in collaboration with the university of Ulm, Germany).
Concerning the synthesis of the response of such filters we had already shown that the latter amounts to a Zolotarev problem with a nonpolynomial weight (with a square root singularity).
For fixed transmission zeros we were however able to derive explicit formulas for the optimal (in the Chebychev sense) filter function
F_{n}:
where
is a suitable parabolic frequency transformation and the
z_{k}^{'}sare prescribed transmission zeros. Recurrence formulas for the practical computation of
F_{N}have been derived and implemented as part of the DedaleHf software package. We also made progress about the realizability of such responses.
In collaboration with the RMC and possibly with XLIM and STMicroelectronics (Tours) our goal is now to test the validity of our unified approach on real examples. Data collection campaigns obtained during tuning phases are scheduled. Joint publications about the topic are also in progress.
An OMUX (Output MUltipleXor) can be modeled in the frequency domain through scattering matrices of filters, like those described in section , connected in parallel onto a common guide. The problem of designing an OMUX with specified performance in a given frequency range naturally translates into a set of constraints on the values of the scattering matrices and of the phase shift introduced by the guide in the considered bandwidth.
An OMUX simulator on a Matlab platform was designed last year and checked against a number of designs proposed by Thalès Alenia Space. Under the terms of a contract with Thales Alenia Space (see section ), it has been used to design a dedicated software to optimize OMUXes whose second release to AAS has taken place this year..
The software proceeds by adding channels recursively, applying to the new channel the above shortcircuit and reflectioninthe bandwidth rules. This yields an initial guess for the global “optimizer” which seems to regularly outperform those currently used by AAS. More extensive tests are being conducted. A natural sequel should consist of the study of the socalled “manifoldpeaks” that may impede a design based on ideal assumptions of losslesness.
As mentioned in section , dynamic equivalence is more general than static. Also, no necessary conditions for equivalence are known, so that it is in general difficult to prove that two systems are notdynamic equivalent.
If two control systems on manifolds of the same dimension are dynamic equivalent, we prove in that either they are static equivalent – i.e.equivalent via a classical diffeomorphism– or they are both ruled; for systems of different dimensions, the one of higher dimension must be ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a LieBäcklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems.
It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is “trivial” ( i.e.linear controllable).
In the terminology of
,
, a
Kepler control systemis a system in dimension
nwhose drift has
n1first integral and compact trajectories and where the control is “small” in the sense that we are interested in asymptotic properties as the bound on the control
tends to zero. It is the case in low thrust orbital transfer, see section
, for negative energy,
i.e.in the socalled elliptic domain.
For this class of systems, a notion of average control systemis introduced in , . Using averaging techniques in this context is rather natural, since the free system produces a fast periodic motion and the smallcontrol a slow one; averaging is a widespread tool in perturbations of integrable Hamiltonian systems, and the small control is in some sense a “perturbation”. In some recent literature, one proceeds as follows: the control is preassigned, for instance to time optimal control via Pontryagin's Maximum Principle or else to some feedback designed beforehand. Then, averaging is performed on the resulting ordinary differential equation, whose limit behavior is analyzed when the control magnitude tends to zero.
The novelty of (see also ) is to average beforeassigning the control, hence getting a control systemthat describes the limit behavior better. For that reason, the average control system is a convenient tool when comparing different control strategies.
It allowed us to answer an open question stated in on the minimum transfertime between two elliptic orbit when the thrust magnitude tends to zero, see .
Under some controllability conditions that are trivially satisfied in the case at hand, we proved that the average system is one where the velocity set has nonempty interior, i.e. all velocity directions are allowed at any point, and the constraint is convex; mathematically this yields a Finsler structure (in the same way as a controllable system without drift with a quadratic constraint on the control yields a subRiemannian structure). An article is in progress, reporting on these results .
The problems of factoring, reducing and decomposing a linear system of ordinary differential (ODEs) or difference equations with varying coefficients have long been studied in mathematical literature. In , , we recast these problems within the algebraic analysis framework developed in and extend them to general functional linear systems (e.g., determined/overdetermined/underdetermined linear systems of ODEs, PDEs, difference equations, differential timedelay equations). Based on endomorphism computations of a finitely presented left module over a noncommutative polynomial ring of functional operators (e.g., differential, shift, timedelay, difference operators) associated with the linear functional system, general conditions for the existence of nontrivial factorizations, reductions and decompositions are obtained in , . Moreover, under certain conditions of freeness, the linear functional system is then equivalent to a block triangular or a block diagonal system. We prove that endomorphisms of the module define Galoislike transformations of the corresponding linear functional system and show how to compute quadratic first integrals of motion/conservation laws of linear differential systems. The different algorithms are implemented in the package OreMorphisms( ). Using the packages Stafford( ) and QuillenSuslin( ), we can then constructively study the factorization, reduction and decomposition problems for general linear functional systems. Explicit examples coming from engineering sciences, control theory and mathematical physics illustrate OreMorphisms. Finally, a very efficient method for reducing linear functional systems is developed in and the corresponding algorithms will be soon implemented in a package called Serre.
Using algebraic analysis and the socalled Baer extensions, we constructively solve in
the following open problem in mathematical systems theory: given two linear systems
S_{1}and
S_{2}, parametrize all the linear functional systems
Swhich contain
S_{1}as a subsystem and satisfy that the quotient
S/
S_{1}is isomorphic to
S_{2}. In particular, these results are applied to parametrize all the equivalence classes of linear systems
Swhich admit a fixed parametrizable subsystem
S_{1}and satisfy that
S/
S_{1}is isomorphic to a fixed autonomous system
S_{2}. The different algorithms are implemented in the package
OreMorphisms(
) and illustrated on different classical examples appearing in the literature of differential timedelay systems.
Finally, these results have other interesting applications we are now developing.
Finally, the writing of a book untitled “Systems and Structures: An algebraic analysis approach to mathematical systems theory” by A. Quadrat has recently been finished and some parts have been taught in a series of lectures at the Korea Institute for Advanced Studies (KIAS) and presented at a semiplenary talk at the Eighteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), Virginia Tech (USA) ( ).
Contract (reference Inria: 2470, CNES: 60465/00) involving CNES, XLIM and Inria, whose objective is to work out a software package for identification and design of microwave devices. The work at Inria concerns the design of multiband filters with constraints on the group delay. The problem is to control the logarithmic derivative of the modulus of a rational function, while meeting specifications on its modulus.
A contract (reference Inria: 1931, AAS: B00375) has been signed between Inria and Thales Alenia Space (branch of Toulouse), in which Inria will design and provide a software for OMUX simulation with efficient initial condition for an optimisation algorithm based on recursive tuning of the channels.
L. Baratchart is a member of the editorial board of Computational Methods and Function Theoryand Complex Analysis and Operator Theory.
J. Leblond was a member of the scientific committee of the conference PICOF'08 (4th International Conference on Inverse Problems, Control and Shape Optimization, Marrakech, april 2008).
A. Quadrat is an associate editor of the journal Multidimensional Systems and Signal Processing(Springer).
AHPI (Analyse Harmonique et Problèmes Inverses), is a “Projet blanc” in Mathematics involving InriaSophia (L. Baratchart coordinator), the Université de Provence (LATP), the Université Bordeaux I (LATN), the Université d'Orléans (MAPMO), InriaBordeaux (Magique 3D), the Université de Pau. It aims at developing Harmonic Analysis techniques to approach inverse problems in seismology, Electroencephalography, tomography and nondestructive control.
FILIPIX (FILtering for Innovative Payload with Improved fleXibility) is a “Projet Thématique en Télécommunications”, involving InriaSophia (Apics), XLIM, Thales Alenia Space (Centre de Toulouse, coordinator).
EPSRCresearch grant EP/F020341/1 (Operator theory in function spaces on finitelyconnected domains), with Leeds University (UK) and the University Lyon I, 20072009.
Équipe associée Énéelinks the LAMSINENIT (Tunis) to three INRIA teams: Anubis, Poems and Apics.
NSF EMS21RTG is a students exchange program with Vanderbilt University.
Apics is linked with the CEAIRFM (Cadarache), through the Région PACA, for the thesis of Y. Fischer.
Apics is part of the regional working group SBPI (Signaux, bruits, problèmes inverses) (Signal, Noise, Inverse Problems) whose aim is to find methods for reducing noise in Virgo http://wwwsop.inria.fr/apics/sbpi.
The following scientists gave a talk at the seminar:
Laurent Baratchart, Extremal problems for the Beltrami equation.
Frédéric Boyer (IRCCyN), De la dynamique de Lagrange à celle de Poincaré.
Yves Dermenjian (CMILATP), Une application d'une inégalité de type Carleman au contrôle dans un milieu stratifié.
Bruno Fabre, Transformations de Radon complexes et réelles.
Sandrine Grellier (Département de Mathématiques, Université d'Orléans), Opérateurs de Hankel et équation de Schrödinger non linéaire.
Bernard Hanzon (University College Cork), Dynamic StateSpace Models with Cointegration.
Johan Karlsson (KTH, Stockholm), The Inverse Problem and Weight Selection for DegreeConstrained Rational Analytic Interpolation.
Stéphanie Nivoche (UNSA), Généralisation du Théorème de Hilbert sur les Lemniscates
Yannick Privat (Institut Elie Cartan, Nancy), Quelle est la forme optimale d'un tuyau ?.
Eva Sincich (RICAM), Stabilité et reconstruction pour certains problèmes liés à la corrosion et à la diffusion inverse.
L. Baratchart, DEA Géométrie et Analyse, LATPCMI, University Marseille I.
J. Leblond, Centre Montessori, collège, MouansSartoux.
M. Olivi, Mathématiques pour l'ingénieur (Fourier analysis and integration), section Mathématiques Appliquées et Modélisation, 3ème année, École Polytechnique NiceSophia Antipolis.
A. Quadrat, ISIA, Master affiliated to the École des Mines de Paris (computer algebra).
Slah Chaabi, « Problèmes extrémaux pour l´équation de Beltrami réelle 2dimensionnelle et application à la détermination de frontières libres », since October.
Ahed Hindawi « Transport optimal en contrôle », started in October.
Yannick Fischer, « Problèmes inverses pour l'équation de Beltrami et extrapolation de quantités magnétiques dans un Tokamak », since October (région PACAInria).
Vincent Lunot, « Techniques d'approximation rationnelle en synthèse fréquentielle : problème de Zolotarov et algorithme de Schur », defended in May, .
Moncef Mahjoub, “Approximation harmonique dans une couronne et application à la résolution numérique de quelques problèmes inverses”, defended in February, .
AnaMaria Nicu, « Inverse potential problems for MEG/EEG », since November (MEN).
Meriem Zghal, « Constructive aspects of some inverse problems (Cauchy, sources) for Laplace equation in ellipsoidal domains », cotutelle with LamsinENIT (Tunis, ImageenErasmus Mundus).
L. Baratchart was a member of the HDR defense committee of S. Kupin, Univ. de Provence, Marseille.
J. Leblond was a member of the HDR defense committee of P. Gaïtan, Univ. Méditerranée, Marseille.
L. Baratchart is Inria's representative at the « conseil scientifique » of the Université de Provence.
J. Grimm is a representative at the « comité de centre ».
J. Leblond is a member of the « Commission d'évaluation » (CE) of Inria. She is a member of the « Commission d'Animation Scientifique » (CAS) of the Research Center, and participates to the working group « Méditerranée 3+3 ».
M. Olivi is a member of the CSD (Comité de Suivi Doctoral) of the Research Centre of Sophia Antipolis.
J.B. Pomet is a representative at the « comité technique paritaire » (CTP).
F. Seyfert is a member of the CDL (Comité de Développement Logiciel) of the Research Centre of Sophia Antipolis.
L. Baratchart, J. Leblond and M. Yattselev presented communications and a poster at the workshop Approximation, Modélisation Géométrique et Applications, CIRM (Luminy, Marseille, France), Nov.
L. Baratchart and J. Leblond respectively presented a communication and attended the conference PICOF'08 (Marrakech, April). They participated to the meeting of the ANR project AHPI (Orleans, Sept.). L. Baratchart and M. Yattselev delivered talks at the International Workshop on Orthogonal Polynomials and Approximation Theory (Madrid, Spain, Sept.).
L. Baratchart was an invited speaker at the conference ”Hilbert spaces of entire functions”, CRM Montreal, Dec., at the colloquium of Newcastle University, Apr., and at the « journées d'approximation », Lille, May. He also delivered a talk at the conference “Foundations of Constructive Mathematics”, HongKong, June.
J. Grimm gave a talk at the Journée GUTenberg 2008 (Paris) .
J. Leblond gave presentations at the working group « gtsignalmeeg » (organized by the Odyssée team), March, at the seminar EDP of IECN (Institut Elie Cartan de Nancy), April, for the Comité directeur de la fédération sur la fusion magnétique (UNSA, labo. JAD), and at CEAIRFM (Cadarache), Nov.
M. Olivi gave a talk at CDC 2008, Cancun, Mexico.
M. Olivi and J. B. Pomet attended the 2008 ERNSI Meeting in Sigtuna (Sweeden).
J.B. Pomet gave a talk at the seminar in Université de Mulhouse, Mathematics Department.
A. Quadrat gave a presentation at the Groupe de Travail EDP of GDR MACS (Lyon, France). He was invited to the Premier colloque FrancoMaghrébin de Calcul Formel (Iles de Kerkennah, Tunisa), where he gave a talk. He was semiplenary speaker at the 18th International symposium on Mathematical Theory of Networks and Systems (MTNS) (Virginia Tech, USA), where he also presented two articles and was invited to the Conference “Mathematics, Algorithms and Proofs (MAP)”, The Abdus Salam International Centre for Theoretical Physics (Trieste, Italy) and at “A workshop on linear systems theory: model reduction” (Sde Boker, Israel) where he presented his recent works. Finally, he was invited at Korea Institute for Advanced Study (KIAS) at Seoul (South Korea) to give four lectures and develop a collaboration.
F. Seyfert coorganized with P. Machiarella (Politecnico, Milano) the workshop “Multiband Filters: Design and Application” at the IEEE International Microwave Symposium in Atlanta, USA, June. He also gave a talk at this conference on the synthesis of filter responses with global optimality conditions.
M. Yattselev attended the conference “Théorie spectrale des opérateurs et applications”, CIRM (Luminy, Marseille, France) and he gave talks at the Departamento de Matemáticas Colloquium, U. Carlos III (Madrid, Spain, Oct.) together with the following seminars: Séminaire Analyse et Géométrie, U. Provence, Marseille, France, Nov., Seminaro de Matemática Aplicada, U. Almería, Spain, Oct., Computational Analysis Seminar, Vanderbilt U., USA, Apr., Seminaro de Matemática Aplicada, U. Almería, Spain, March.
M. Zghal presented a poster at the 10th International Workshop on Optimization and Inverse Problems in Electromagnetism (OIPE 2008), Ilmenau, Germany, September.