3.2.1 Elliptic partial differential equations (PDE)

Participants: Paul Asensio, Laurent Baratchart, Sylvain Chevillard, Mubasharah Khalid Omer, Juliette Leblond, Masimba Nemaire, Dmitry Ponomarev, Cristóbal Villalobos Guillén, Anass Yousfi.

By standard properties of conjugate differentials, reconstructing Dirichlet-Neumann boundary conditions for a function harmonic in a plane domain, when these conditions are already known on a subset $E$ of the boundary, is equivalent to recover a holomorphic function in the domain from its boundary values on $E$. This is the problem raised on the half-plane in step 1 of Section 3.1. It makes good sense in holomorphic Hardy spaces where functions are entirely determined by their values on boundary subsets of positive linear measure, which is the framework for Problem $\left(P\right)$ more precisely described in Section 3.3.1.

$\left(P\right)$  Let $1\le p\le \infty$, $K$ a sub-arc of $T$, $f\in L^p\left(K\right)$, $\psi \in L^p(T\setminus K)$ and $M>0$; find a function $g\in H^p$ such that ${\parallel g-\psi \parallel }_{L^p(T\setminus K)}\le M$ and $g-f$ is of minimal norm in $L^p\left(K\right)$ under this constraint.

Such issues naturally arise in nondestructive testing of 2-D (or 3-D cylindrical) materials from partial electrical measurements on the boundary. For instance, the ratio between the tangential and the normal currents (the so-called Robin coefficient) tells one about corrosion of the material. Thus, solving Problem $\left(P\right)$ where $\psi$ is chosen to be the response of some uncorroded piece with identical shape yields non destructive testing of a potentially corroded piece of material, part of which is inaccessible to measurements. This was an initial application of holomorphic extremal problems to non-destructive control  61, 63.

Studying Hardy spaces of conjugate Beltrami equations is another interesting topic. For Sobolev-smooth coefficients of exponent greater than 2, they were investigated in 5,  37. The case of the critical exponent 2 is treated in  33, which apparently provides the first example of well-posed Dirichlet problem in the non-strictly elliptic case: the conductivity may be unbounded or zero on sets of zero capacity and, accordingly, solutions need not be locally bounded. More importantly perhaps, the exponent 2 is also the key to a corresponding theory on very general (still rectifiable) domains in the plane, as coefficients of pseudo-holomorphic functions obtained by conformal transformation onto a disk are merely of $L^2$-class in general, even if the initial problem deals with coefficients of $L^r$-class for some $r>2$. Such generalizations are now under study within the team, in collaboration with E. Pozzi (Saint Louis Univ., Missouri, USA) and E. Russ (Univ. J. Fourier, Grenoble), and fairly deep connections between the regularity of the conformal parametrization of the domain and the range of exponents $p$ for which the Dirichlet problem is solvable in $L^p$ were brought to light.

Generalized Hardy classes as above are used in  34 where we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain of $\Omega \subset {ℝ}^n$, $n\ge 2$, with uniformly bounded Robin coefficient, $L^2$ Neumann data and conductivity of Sobolev class $W^{1,r}\left(\Omega \right)$, $r>n$. We show that uniqueness of the Robin coefficient on a subset of the boundary, given Cauchy data on the complementary part, does hold in dimension $n=2$, thanks to a unique continuation result, but needs not hold in higher dimension. In higher dimension, this raises an open issue on harmonic gradients, namely whether the positivity of the Robin coefficient is compatible with identical vanishing of the boundary gradient on a subset of positive measure.

The 3-D version of step 1 in Section 3.1 is another subject investigated by Factas: to recover a harmonic function (up to an additive constant) in a ball or a half-space from partial knowledge of its gradient. This prototypical inverse problem (i.e., inverse to the Cauchy problem for the Laplace equation) often recurs in electromagnetism. At present, Factas is involved with solving instances of this inverse problem arising in two fields, namely medical imaging, e.g., for electroencephalography (EEG) or magneto-encephalography (MEG), and paleomagnetism (recovery of rocks magnetization) 1,  41, see Section 6.1. The question is considerably more difficult than its 2-D counterpart, due mainly to the lack of multiplicative structure for harmonic gradients. Still, substantial progress has been made over the last years using methods of harmonic analysis and operator theory.

The team is further concerned with 3-D generalizations and applications to non-destructive control of step 2 in Section 3.1. A typical problem is here to localize inhomogeneities or defaults such as cracks, sources or occlusions in a planar or 3-dimensional object, knowing thermal, electrical, or magnetic measurements on the boundary. These defaults can be expressed as a lack of harmonicity of the solution to the associated Dirichlet-Neumann problem, thereby posing an inverse potential problem in order to recover them. In 2-D, finding an optimal discretization of the potential in Sobolev norm amounts to solve a best rational approximation problem, and the question arises as to how the location of the singularities of the approximant (i.e., its poles) reflects the location of the singularities of the potential (i.e., the defaults we seek). This is a fairly deep issue in approximation theory, to which the project Apics (predecessor of Factas2) contributed convergence results for certain classes of fields (expressed as Cauchy integrals over extremal contours for the logarithmic potential 7,  42, 60). Initial schemes to locate cracks or sources via rational approximation on planar domains were obtained this way  45, 49, 61. It is remarkable that finite inverse source problems in 3-D balls, or more general algebraic surfaces, can be approached using these 2-D techniques upon slicing the domain into planar sections 9,  46. More precisely, each section cuts out a planar domain, the boundary of which carries data which can be proved to match an algebraic function. The singularities of this algebraic function are not located at the 3-D sources, but are related to them: the section contains a source if and only if some function of the singularities in that section meets a relative extremum. Using bisection it is thus possible to determine an extremal place along all sections parallel to a given plane direction, up to some threshold which has to be chosen small enough that one does not miss a source. This way, we reduce the original source problem in 3-D to a sequence of inverse poles and branch-points problems in 2-D. This bottom line generates a steady research activity within Factas, and again applications are sought to medical imaging and geosciences, see Sections 4.2, 4.3 and 6.1.

Conjectures may be raised on the behavior of optimal potential discretization in 3-D, but answering them is an ambitious program still in its infancy.

3.2.2 Systems, transfer and scattering

Participants: Laurent Baratchart, Sylvain Chevillard, Adam Cooman, Martine Olivi, Fabien Seyfert.

Through contacts with CNES (French space agency), members of the team became involved in identification and tuning of microwave electromagnetic filters used in space telecommunications. The initial problem was to recover, from band-limited frequency measurements, physical parameters of the device under examination. The latter consists of interconnected dual-mode resonant cavities with negligible loss, hence its scattering matrix is modeled by a $2\times 2$ unitary-valued matrix function on the frequency line, say the imaginary axis to fix ideas. In the bandwidth around the resonant frequency, a modal approximation of the Helmholtz equation in the cavities shows that this matrix is approximately rational, of Mc-Millan degree twice the number of cavities.

This is where system theory comes into play, through the so-called realization process mapping a rational transfer function in the frequency domain to a state-space representation of the underlying system of linear differential equations in the time domain. Specifically, realizing the scattering matrix allows one to construct a virtual electrical network, equivalent to the filter, the parameters of which mediate in between the frequency response and the geometric characteristics of the cavities (i.e., the tuning parameters).

Hardy spaces provide a framework to transform this ill-posed issue into a series of regularized analytic and meromorphic approximation problems. More precisely, the procedure sketched in Section 3.1 goes as follows:

1. infer from the pointwise boundary data in the bandwidth a stable transfer function (i.e., one which is holomorphic in the right half-plane), that may be infinite dimensional (numerically: of high degree). This is done by solving a problem analogous to $\left(P\right)$ in Section 3.2.1, while taking into account prior knowledge on the decay of the response outside the bandwidth, see 12 for details.
2. A stable rational approximation of appropriate degree to the model obtained in the previous step is performed. For this, a descent method on the compact manifold of inner matrices of given size and degree is used, based on an original parametrization of stable transfer functions developed within the team  29, 12.
3. Realizations of this rational approximant are computed. To be useful, they must satisfy certain constraints imposed by the geometry of the device. These constraints typically come from the coupling topology of the equivalent electrical network used to model the filter. This network is composed of resonators, coupled according to some specific graph. This realization step can be recast, under appropriate compatibility conditions  62, as solving a zero-dimensional multivariate polynomial system. To tackle this problem in practice, we use Gröbner basis techniques and continuation methods which team up in the Dedale-HF software (see Section 3.4.2).

Factas also investigates issues pertaining to design rather than identification. Given the topology of the filter, a basic problem in this connection is to find the optimal response subject to specifications that bear on rejection, transmission and group delay of the scattering parameters. Generalizing the classical approach based on Chebyshev polynomials for single band filters, we recast the problem of multi-band response synthesis as a generalization of the classical Zolotarev min-max problem for rational functions  28, 80. Thanks to quasi-convexity, the latter can be solved efficiently using iterative methods relying on linear programming. These were implemented in the software easy-FF. Currently, the team is engaged in the synthesis of more complex microwave devices like multiplexers and routers, which connect several filters through wave guides. Schur analysis plays an important role here, because scattering matrices of passive systems are of Schur type (i.e., contractive in the stability region). The theory originates with the work of I. Schur  84, who devised a recursive test to check for contractivity of a holomorphic function in the disk. The so-called Schur parameters of a function may be viewed as Taylor coefficients for the hyperbolic metric of the disk, and the fact that Schur functions are contractions for that metric lies at the root of Schur's test. Generalizations thereof turn out to be efficient to parametrize solutions to contractive interpolation problems  31. Dwelling on this, Factas contributed differential parametrizations (atlases of charts) of lossless matrix functions  29, 81, 72 which are fundamental to our rational approximation software RARL2 (see Section 3.4.5). Schur analysis is also instrumental to approach de-embedding issues, and provides one with considerable insight into the so-called matching problem. The latter consists in maximizing the power a multiport can pass to a given load, and for reasons of efficiency it is all-pervasive in microwave and electric network design, e.g., of antennas, multiplexers, wifi cards and more. It can be viewed as a rational approximation problem in the hyperbolic metric. Factas made significant contributions to this subject 6, in particular within the framework of the (defense funded) ANR Cocoram.

In recent years, our attention was driven by CNES and UPV (Bilbao) to questions about stability of high-frequency amplifiers. Contrary to previously discussed devices, these are active components. The response of an amplifier can be linearized around a set of primary current and voltages, and then admittances of the corresponding electrical network can be computed at various frequencies, using the so-called harmonic balance method. The initial goal is to check for stability of the linearized model, so as to ascertain existence of a well-defined working state. The network is composed of lumped electrical elements namely inductors, capacitors, negative and positive resistors, transmission lines, and controlled current sources. Our research so far has focused on describing the algebraic structure of admittance functions, so as to set up a function-theoretic framework where the two-steps approach outlined in Section 3.1 can be put to work. The main discovery is that the unstable part of each partial transfer function is rational and can be computed by analytic projection, see  10. We now start investigating the linearized harmonic transfer-function around a periodic cycle, to check for stability under non necessarily small inputs.

3.3.1 Best analytic approximation

In dimension 2, the prototypical problem to be solved in step 1 of Section 3.1 may be described as: given a domain $D\subset {ℝ}^2$, to recover a holomorphic function from its values on a subset $K$ of the boundary of $D$. For the discussion it is convenient to normalize $D$, which can be done by conformal mapping. So, in the simply connected case, we fix $D$ to be the unit disk with boundary unit circle $T$. We denote by $H^p$ the Hardy space of exponent $p$, which is the closure of polynomials in $L^p\left(T\right)$-norm if $1\le p<\infty$ and the space of bounded holomorphic functions in $D$ if $p=\infty$. Functions in $H^p$ have well-defined boundary values in $L^p\left(T\right)$, which makes it possible to speak of (traces of) analytic functions on the boundary.

To find an analytic function $g$ in $D$ matching some measured values $f$ approximately on a sub-arc $K$ of $T$, we formulate the constrained best approximation problem $\left(P\right)$ as in Section 3.2.1.

There, $\psi$ is a reference behavior capturing a priori assumptions on the behavior of the model off $K$, while $M$ is some admissible deviation thereof. The value of $p$ reflects the type of stability which is sought and how much one wants to smooth out the data. The choice of $L^p$ classes is suited to handle pointwise measurements.

To fix terminology, we refer to $\left(P\right)$ as a bounded extremal problem. As shown in 11, 44, 47, 55, the solution to this convex infinite-dimensional optimization problem can be obtained when $p\ne 1$ upon iterating with respect to a Lagrange parameter the solution to spectral equations for appropriate Hankel and Toeplitz operators. These spectral equations involve the solution to the special case $K=T$ of $\left(P\right)$, which is a standard extremal problem  70:

($P_0$)  Let $1\le p\le \infty$ and $\phi \in L^p\left(T\right)$; find a function $g\in H^p$ such that $g-\phi$ is of minimal norm in $L^p\left(T\right)$.

In the case $p=1$, partial results are known but computational issues remain open.

Various modifications of $\left(P\right)$ can be tailored to meet specific needs. For instance when dealing with lossless transfer functions, one may want to express the constraint on $T\setminus K$ in a pointwise manner: $|g-\psi |\le M$ a.e. on $T\setminus K$, see  48. In this form, the problem comes close to (but still is different from) $H^{\infty }$ frequency optimization used in control  73, 83. One can also impose bounds on the real or imaginary part of $g-\psi$ on $T\setminus K$, which is useful when considering Dirichlet-Neumann problems.

In view of our current research on stability of active devices via analyticity of the harmonic transfer function, on inverse magnetization issues, and on inverse scattering via identification of the frequency response, bounded extremal problems for analytic functions are receiving renewed interest by the team. In such issues, a function on an interval of the real line (or an arc of the circle) must be approximated by the trace of a function holomorphic in the half-plane (or the disk), that meets suitable size constraints.

The analog of Problem $\left(P\right)$ on an annulus, $K$ being now the outer boundary, can be seen as a means to regularize a classical inverse problem occurring in nondestructive control, namely to recover a harmonic function on the inner boundary from Dirichlet-Neumann data on the outer boundary (see Sections 3.2.1, 4.2, 6.1.2). It may serve as a tool to approach Bernoulli type problems, where we are given data on the outer boundary and we seek the inner boundary, knowing it is a level curve of the solution. In this case, the Lagrange parameter indicates how to deform the inner contour in order to improve data fitting. Similar topics are discussed in Section 3.2.1 for more general equations than the Laplacian, namely isotropic conductivity equations of the form $\mathrm{div}\left(\sigma \nabla u\right)=0$ where $\sigma$ is no longer constant (i.e., varies in the space). Then, the Hardy spaces in Problem $\left(P\right)$ are those of a so-called conjugate Beltrami equation: $\overline{\partial }f=\nu \overline{\partial f}$  74, with $\nu =(1-\sigma )/(1+\sigma )$, which are studied for $1<p<\infty$ in 5,  33,  37 and  64. Expansions of solutions needed to constructively handle such issues in the specific case of linear fractional conductivities have been expounded in  67.

Though originally considered in dimension 2, Problem $\left(P\right)$ carries over naturally to higher dimensions where analytic functions get replaced by gradients of harmonic functions. Namely, given some open set $\Omega \subset {ℝ}^n$ and some ${ℝ}^n$-valued vector field $V$ on an open subset $O$ of the boundary of $\Omega$, we seek a harmonic function in $\Omega$ whose gradient is close to $V$ on $O$.

When $\Omega$ is a ball or a half-space, a substitute for holomorphic Hardy spaces is provided by the Stein-Weiss Hardy spaces of harmonic gradients  86. Conformal maps are no longer available when $n>2$, so that $\Omega$ can no longer be normalized. More general geometries than spheres and half-spaces have not been much studied so far.

On the ball, the analog of Problem $\left(P\right)$ is

$\left(P_1\right)$  Let $1\le p\le \infty$ and $B\subset {ℝ}^n$ the unit ball. Fix $O$ an open subset of the unit sphere $S\subset {ℝ}^n$. Let further $V\in L^p\left(O\right)$ and $W\in L^p(S\setminus O)$ be ${ℝ}^n$-valued vector fields. Given $M>0$, find a harmonic gradient $G\in H^p\left(B\right)$ such that ${\parallel G-W\parallel }_{L^p(S\setminus O)}\le M$ and $G-V$ is of minimal norm in $L^p\left(O\right)$ under this constraint.

When $p=2$, Problem $\left(P_1\right)$ was solved in 1 as well as its analog on a shell, when the tangent component of $V$ is a gradient (when $O$ is Lipschitz the general case follows easily from this). The solution extends the work in  44 to the 3-D case, using a generalization of Toeplitz operators. The case of the shell was motivated by applications to the processing of EEG data. An important ingredient is a refinement of the Hodge decomposition, that we call the Hardy-Hodge decomposition, allowing us to express a ${ℝ}^n$-valued vector field in $L^p\left(S\right)$, $1<p<\infty$, as the sum of a vector field in $H^p\left(B\right)$, a vector field in $H^p({ℝ}^n\setminus \overline{B})$, and a tangential divergence free vector field on $S$; the space of such divergence-free fields is denoted by $D\left(S\right)$. If $p=1$ or $p=\infty$, $L^p$ must be replaced by the real Hardy space or the space of functions with bounded mean oscillation. More generally this decomposition, which is valid on any sufficiently smooth surface (see Section 6.1), seems to play a fundamental role in inverse potential problems. In fact, it was first introduced formally on the plane to describe silent magnetizations supported in ${ℝ}^2$ (i.e., those generating no field in the upper half space)  41.

Just like solving problem $\left(P\right)$ appeals to the solution of problem $\left(P_0\right)$, our ability to solve problem $\left(P_1\right)$ will depend on the possibility to tackle the special case where $O=S$:

$\left(P_2\right)$  Let $1\le p\le \infty$ and $V\in L^p\left(S\right)$ be a ${ℝ}^n$-valued vector field. Find a harmonic gradient $G\in H^p\left(B\right)$ such that ${\parallel G-V\parallel }_{L^p\left(S\right)}$ is minimum.

Problem $\left(P_2\right)$ is simple when $p=2$ by virtue of the Hardy-Hodge decomposition together with orthogonality of $H^2\left(B\right)$ and $H^2({ℝ}^n\setminus \overline{B})$, which is the reason why we were able to solve $\left(P_1\right)$ in this case. Other values of $p$ cannot be treated as easily and are still under investigation, especially the case $p=\infty$ which is of particular interest and presents itself as a 3-D analog to the Nehari problem  82.

Companion to problem $\left(P_2\right)$ is problem $\left(P_3\right)$ below.

$\left(P_3\right)$  Let $1\le p\le \infty$ and $V\in L^p\left(S\right)$ be a ${ℝ}^n$-valued vector field. Find $G\in H^p\left(B\right)$ and $D\in D\left(S\right)$ such that ${\parallel G+D-V\parallel }_{L^p\left(S\right)}$ is minimum.

Note that $\left(P_2\right)$ and $\left(P_3\right)$ are identical in 2-D, since no non-constant tangential divergence-free vector field exists on $T$. It is no longer so in higher dimension, where both $\left(P_2\right)$ and $\left(P_3\right)$ arise in connection with inverse potential problems in divergence form, like source recovery in electro/magneto encephalography and paleomagnetism, see Sections 3.2.1 and 4.2.

3.3.2 Best meromorphic and rational approximation

The techniques set forth in this section are used to solve step 2 in Section 3.2 and they are instrumental to approach inverse boundary value problems for the Poisson equation $\Delta u=\mu$, where $\mu$ is some (unknown) measure.

3.3.3 Behavior of poles of meromorphic approximants

Participants: Paul Asensio, Laurent Baratchart.

We refer here to the behavior of poles of best meromorphic approximants, in the $L^p$-sense on a closed curve, to functions $f$ defined as Cauchy integrals of complex measures whose support lies inside the curve. Normalizing the contour to be the unit circle $T$, we are back to Problem $\left(P_N\right)$ in Section 3.3.2; invariance of the latter under conformal mapping was established in  49. Research so far has focused on functions whose singular set inside the contour is polar, meaning that the function can be continued analytically (possibly in a multiple-valued manner) except over a set of logarithmic capacity zero.

Generally speaking, in approximation theory, assessing the behavior of poles of rational approximants is essential to obtain error rates as the degree goes large, and to tackle constructive issues like uniqueness. However, as explained in Section 3.2.1, the original twist by Apics, now Factas, is to consider this issue also as a means to extract information on singularities of the solution to a Dirichlet-Neumann problem. The general theme is thus: how do the singularities of the approximant reflect those of the approximated function? This approach to inverse problem for the 2-D Laplacian turns out to be attractive when singularities are zero- or one-dimensional (see Section 4.2). It can be used as a computationally cheap initial condition for more precise but much heavier numerical optimizations which often do not even converge unless properly initialized. As regards crack detection or source recovery, this approach boils down to analyzing the behavior of best meromorphic approximants of given pole cardinality to a function with branch points, which is the prototype of a polar singular set. For piecewise analytic cracks, or in the case of sources, we were able to prove (7,  49, 42), that the poles of the approximants accumulate, when the degree goes large, to some extremal cut of minimum weighted logarithmic capacity connecting the singular points of the crack, or the sources  45. Moreover, the asymptotic density of the poles turns out to be the Green equilibrium distribution on this cut in $D$, therefore it charges the singular points if one is able to approximate in sufficiently high degree (this is where the method could fail, because high-order approximation requires rather precise data).

The case of two-dimensional singularities is still an outstanding open problem.

It is remarkable that inverse source problems inside a sphere or an ellipsoid in 3-D can be approached with such 2-D techniques, as applied to planar sections, see Section 6.1. The technique is implemented in the software FindSources3D, see Section 3.4.3.

Another, extremely classical technique to approximate –more accurately: extrapolate– a function given pointwise values is to compute a rational interpolant of minimal degree to match the values. This method, know as Padé (or multipoint Padé) approximation has been intensively studied for decades 30 but fails to produce pointwise convergence, even if the data are analytic. The best it can give in general is convergence in capacity, at least to functions whose singular set has capacity zero, and this does not prevent spurious poles of the approximant from wandering about the domain of analyticity of the approximated function 79. This phenomenon is standard in numerical practice, and gives rise in physics and engineering circles to a distinction between “mathematical” and “physical” poles; note that this distinction ignores the possibility that the function has other singularities than poles (for example branch-points or essential singularities). A modification of the multipoint Padé technique, where the degree is kept much smaller than the number of data and only approximate interpolation is performed in the least-square sense, has become especially popular over the last decade under the name vector fitting; this is in trend with the soaring development of computational methods in the frequency domain. Although their behavior looks similar to the one of multipoint Padé approximants from a numerical viewpoint, there seems to be no convergence result available for such approximate interpolants so far. Motivated by the behavior of numerical schemes developed at LEAT to recover resonance frequencies of conductors under electromagnetic inverse scattering (see section 4.5), we started investigating the behavior of such least-square rational approximants to functions with polar singular set, see section 6.4.

3.4.1 pisa

• Name:
  pisa
• Keywords:
  Electrical circuit, Stability
• Functional Description:

  To minimise prototyping costs, the design of analog circuits is performed using computer-aided design tools which simulate the circuit's response as accurately as possible.

  Some commonly used simulation tools do not impose stability, which can result in costly errors when the prototype turns out to be unstable. A thorough stability analysis is therefore a very important step in circuit design. This is where pisa is used.

  pisa is a Matlab toolbox that allows designers of analog electronic circuits to determine the stability of their circuits in the simulator. It analyses the impedance presented by a circuit to determine the circuit's stability. When an instability is detected, pisa can estimate location of the unstable poles to help designers fix their stability issue.

• Release Contributions:
  First version
• URL:
  https://project.inria.fr/pisa
• Publications:
  hal-01381731, hal-01098616
• Authors:
  Adam Cooman, David Martinez Martinez, Fabien Seyfert, Martine Olivi
• Contact:
  Fabien Seyfert

3.4.2 DEDALE-HF

• Keyword:
  Microwave filter
• Scientific Description:

  Dedale-HF consists in two parts: a database of coupling topologies as well as a dedicated predictor-corrector code. Roughly speaking each reference file of the database contains, for a given coupling topology, the complete solution to the coupling matrix synthesis problem associated to particular filtering characteristics. The latter is then used as a starting point for a predictor-corrector integration method that computes the solution to the coupling matrix synthesis problem corresponding to the user-specified filter characteristics. The reference files are computed off-line using Gröbner basis techniques or numerical techniques based on the exploration of a monodromy group. The use of such continuation techniques, combined with an efficient implementation of the integrator, drastically reduces the computational time.

  Dedale-HF has been licensed to, and is currently used by TAS-Espana.

• Functional Description:
  Dedale-HF is a software dedicated to solve exhaustively the coupling matrix synthesis problem in reasonable time for the filtering community. Given a coupling topology, the coupling matrix synthesis problem consists in finding all possible electromagnetic coupling values between resonators that yield a realization of given filter characteristics. 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.
• URL:
  http://www-sop.inria.fr/apics/Dedale/
• Contact:
  Fabien Seyfert
• Participant:
  Fabien Seyfert

3.4.3 FindSources3D

• Keywords:
  Health, Neuroimaging, Visualization, Compilers, Medical, Image, Processing
• Scientific Description:
  Though synthetic data could be static, actual signal recordings are dynamical. The time dependency is either neglected and the data processed instant by instant, or separated from the space behavior using a singular value decomposition (SVD). This preliminary step allows to estimate the number of independent activities (uncorrelated sources) and to select the corresponding quantity of principal static components. After a first data transmission (“cortical mapping”) step of the static data, using the harmonicity property of the potential in the outermost layers (solving BEP problems on spherical harmonics bases), FS3D makes use of best rational approximation on families of 2-D planar cross-sections and of the software RARL2 in order to locate singularities and to determine the expected quantity of sources. From those planar singularities, the 3-D sources are finally estimated, together with their moment, in a last clustering step. Through this process, FS3D is able to recover time correlated sources, which is an important advantage. When simultaneously available, EEG and MEG data can now be processed together, and this also improves the recovery performance. In case of dynamical data, a recent additional step is to find the linear combination of the preliminary selected static components (change of basis) that produces source estimates which minimize the error w.r.t. data, an original criterion, which allows to improve the recovery quality.
• Functional Description:
  FindSources3D (FS3D) is a software program written in Matlab dedicated to the resolution of inverse source problems in brain imaging, electroencephalography (EEG) and magnetoencephalography (MEG). From data consisting in pointwise measurements of the electrical potential taken by electrodes on the scalp (EEG), or of a component of the magnetic field taken on a helmet (MEG), FS3D estimates pointwise dipolar current sources within the brain in a spherical layered model. Each layer (brain, skull, scalp) is assumed to have a constant conductivity.
• URL:
  http://www-sop.inria.fr/apics/FindSources3D/en/index.html
• Contact:
  Juliette Leblond
• Participants:
  Jean-Paul Marmorat, Juliette Leblond, Maureen Clerc, Nicolas Schnitzler, Théodore Papadopoulo

3.4.4 PRESTO-HF

• Keywords:
  CAO, Telecommunications, Microwave filter
• Scientific Description:

  For the matrix-valued rational approximation step, Presto-HF relies on RARL2. Constrained realizations are computed using the Dedale-HF software. As a toolbox, Presto-HF 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 assumption: far off the pass-band, one can reasonably expect a good approximation of the rational components of S11 and S22 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 has been licensed to (and is currently used by) Thales Alenia Space in Toulouse and Madrid, Thales airborne systems and Flextronics (two licenses). Xlim (University of Limoges) is a heavy user of Presto-HF among the academic filtering community and some free license agreements have been granted to the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingston, Canada).

• Functional Description:
  Presto-HF is a toolbox dedicated to low-pass parameter identification for microwave filters. In order to allow the industrial transfer of our methods, a Matlab-based toolbox has been developed, dedicated to the problem of identification of low-pass microwave filter parameters. It allows one to run the following algorithmic steps, either individually or in a single stroke:

  • Determination of delay components 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.
• URL:
  https://project.inria.fr/presto-hf/
• Contact:
  Fabien Seyfert
• Participants:
  Fabien Seyfert, Jean-Paul Marmorat, Martine Olivi

3.4.5 RARL2

• Name:
  Réalisation interne et Approximation Rationnelle L2
• Keyword:
  Approximation
• Scientific Description:

  The method is a steepest-descent algorithm. A parametrization of MIMO systems is used, which ensures that the stability constraint on the approximant is met. The implementation, in Matlab, is based on state-space representations.

  RARL2 performs the rational approximation step in the software tools PRESTO-HF and FindSources3D. It is distributed under a particular license, allowing unlimited usage for academic research purposes. It was released to the universities of Delft and Maastricht (the Netherlands), Cork (Ireland), Brussels (Belgium), Macao (China) and BITS-Pilani Hyderabad Campus (India).

• Functional Description:

  RARL2 is a software for rational approximation. It computes a stable rational L2-approximation of specified order to a given L2-stable (L2 on the unit circle, analytic in the complement of the unit disk) matrix-valued function. This can be the transfer function of a multivariable discrete-time stable system. RARL2 takes as input either:

  • its internal realization,
  • its first N Fourier coefficients,
  • discretized (uniformly distributed) values on the circle. In this case, a least-square criterion is used instead of the L2 norm.

  It thus performs model reduction in the first or the second case, and leans on frequency data identification in the third. For band-limited frequency data, it could be necessary to infer the behavior of the system outside the bandwidth before performing rational approximation.

  An appropriate Möbius transformation allows to use the software for continuous-time systems as well.

• URL:
  http://www-sop.inria.fr/apics/RARL2/rarl2.html
• Contact:
  Martine Olivi
• Participants:
  Jean-Paul Marmorat, Martine Olivi

3.4.6 Sollya

• Keywords:
  Computer algebra system (CAS), Supremum norm, Proof synthesis, Code generator, Remez algorithm, Curve plotting, Numerical algorithm
• Functional Description:

  Sollya is an interactive tool where the developers of mathematical floating-point libraries (libm) can experiment before actually developing code. The environment is safe with respect to floating-point errors, i.e., the user precisely knows when rounding errors or approximation errors happen, and rigorous bounds are always provided for these errors.

  Among other features, it offers a fast Remez algorithm for computing polynomial approximations of real functions and also an algorithm for finding good polynomial approximants with floating-point coefficients to any real function. As well, it provides algorithms for the certification of numerical codes, such as Taylor Models, interval arithmetic or certified supremum norms.

  It is available as a free software under the CeCILL-C license.

• URL:
  https://sollya.org/
• Contact:
  Sylvain Chevillard
• Participants:
  Christoph Lauter, Jérôme Benoit, Marc Mezzarobba, Mioara Joldes, Nicolas Jourdan, Sylvain Chevillard
• Partners:
  CNRS, UPMC, ENS Lyon, LIP6, UCBL Lyon 1, Loria

6.1.1 Inverse magnetization issues for planar and volumetric samples

The goal is to invert magnetizations carried by a rock sample from measurements of the magnetic field nearby. A typical application is when the sample is shaped into thin slabs, with measurements taken by a superconducting quantum interference device (SQUID). Figure 2 sketches the corresponding experimental set up, brought up to our knowledge by collaborators from the Earth and Planetary Sciences Laboratory at MIT, in the case when the sample is modeled as a parallelepiped.

Show image description

Schematic view of the experimental setup : the horizontal plane on which the rock sample lies (the sample being a parallelepiped with height $r$ and a square basis of half-size $s$). Its basis is at height 0, and the center of the square basis is at the origin of the system of coordinates). Above, the measurements are performed on a parallel plane at height $z$. Here, for instance, it is a square, whose half-size is $R$ and which is horizontally centered.

When the measurement area is a disk (or even a cylinder, if its thickness is not negligible), the issue of asymptotic estimates has undergone new developments. As part of A. Yousfi's thesis work, we derived a much more straightforward way of establishing the asymptotic estimates of the moments of $B_3$, a key step to derive asymptotic estimates of the net moment of a sample. This work is still ongoing and a report on the subject is being written. Meanwhile, a further simplification was achieved in the computations that we made in 2022 and the previously uploaded preprint 27 was updated accordingly. The work is now submitted to a journal.

Instability of high-order asymptotic formulas for the net moment and the lack of the measurements naturally give rise to two aspects: noise filtering and field prolongation. One way to attempt both issues at once is to use the so-called spectral extrapolation. Such an approach consists in finding a suitable choice of basis functions which are adapted to problem structure and geometry, but at the same time are naturally defined outside of the original region. Using an eigenbasis of the suitable integral operator related to the forward problem, we attempt to construct a field extrapolant which would extend the available measurements in a way respecting the structure of the problem (such as harmonicity, support of the source, qualitative behavior at infinity). Restricting the number of terms in the expansion over this basis also provides a regularization strategy achieving a desired degree of the field denoising.

The work on the field extrapolation initiated at the end of 2022 has been advanced further. In particular, there were several directions taken to develop a different methodology.

First, after realizing that the previously developed spectral extrapolation procedure based on the forward operator featuring only one kernel function (the vertical derivative of the Poisson kernel) was somewhat naive, we have demonstrated that the extrapolation results could be improved by introducing several local extrapolation steps. Namely, by a procedure reminiscent of the chain method for local analytic continuation, we gradually increase the extrapolation area up to some intermediate size after which the previous naive procedure can be applied with a better outcome. The local stepping is computationally expensive and the details of its particular implementation should be investigated more thoroughly for its further improvement.

However, we have delved into developing another approach in a somewhat similar fashion. Namely, it was observed that the spectral extrapolation approach could be made exact if the basis functions are chosen as vector-eigenfunctions of a $2\times 2$ matrix integral operator. This would invoke a preliminary solution of a canonical auxiliary problem whose ill-posedness reflects the same phenomenon of the original problem of the field extrapolation. One would also need to employ this solution to preprocess the initial measurements. While computationally heavy and difficult to implement numerically, this approach was verified and showed much better result compared to its more naive counterpart.

A third approach for the field extrapolation is much more in the spirit of previous works of the team. Here, we use the techniques of constrained approximation in Hilbert spaces. Namely, given a tolerance level for measurement errors, we search for an extrapolant which agrees in a best possible manner with the original field data respecting the harmonicity of the field. Currently, we have done it only for the scalar potential simulated data (a toy problem for the internship work of M. Khalid Omer) rather than the vertical field component data (that would correspond to the physical SQUID setup at EAPS, MIT). The extension of this methodology to the latter case must be straightforward.

Finally, we have also started exploring a possibility of obtaining an extrapolant directly from solution of some integral equation. Even though, we have succeeded obtaining such an equation, no constructive results are yet available here. Any progress in this direction would allow dealing with volumetric magnetization sources.

Besides, we pursued a joint research effort with E. A. Lima and B. P. Weiss (EAPS dept. MIT, USA), C. S. Borlina (EAPS dept. Johns Hopkins Univ. USA) and D. Hardin (Vanderbilt Univ. USA), investigating the question whether is it possible to estimate the moment of an unknown magnetization by the one of a magnetization producing a field close enough to the observations? In other words, is full-inversion a demonstrably good (though admittedly costly) method to estimate the moment? We showed in 2022 that the answer is essentially no. More precisely, the field-to-moment map (that exists because silent magnetizations have zero moment) is discontinuous, even when magnetizations are endowed with a rather weak topology (like the weak-star topology on distributions of fixed order and given support), even if models for the magnetization are restricted to a very small class. In 2023, we started studying the speed of approximation of a given field by the field of a multipole when the degree of the latter goes large. This speed must be compared with the norm of the operator mapping the field (or the observed portion thereof) to the moment within the model class of multipoles, in order to decide if full inversion allows one to efficiently estimate the moment this way. This line of investigation is still in progress, and an article has been published that describes the method 15.

In collaboration with D. Hardin at Vanderbilt University (Nashville, USA), We also made theoretical progress describing the outcome of regularization techniques that penalize the total variation of an unknown measure modeling the magnetization, for inverse magnetization problems on compact samples of zero 3-D Lebesgue measure with connected complement. Namely, we proved that natural discretizations of the regularized criterion have a unique minimizer in this case, which allows us to establish the consistency of such schemes. Also when the sample is volumic (like a ball), we showed that these schemes select a solution on the boundary of the sample (which is the balayage of the “true” magnetization, whatever the latter is).

In another connection, the development of QDM sensors has heightened the need for 3D-inversion of field maps. Indeed, while the QDM can be put very close to the surface of the sample which makes for improved resolution, the thickness of the sample becomes appreciable compared to the distance to the sensor and can no longer be neglected. In his thesis 20, Masimba Nemaire has taken up the study of silent magnetization of $L^p$ class on rather general bounded volumes for $1<p<\infty$, see Section 6.3 and the preprint 26.

6.1.2 Inverse problems in medical imaging

Our software FindSources3D (FS3D, see Sections 3.4.3, 4.2) dedicated to pointwise source estimation in EEG–MEG is now used by some of our collaborators. Together with M. Darbas (LAGA, Univ. Sorbonne Paris Nord) and P.-H. Tournier (labo. JLL, Sorbonne Univ), we considered the EEG inverse problem with a variable conductivity in the intermediate skull layer, in order to model hard / spongy bones, especially for neonates. Coupled with FS3D, the related transmission step is performed using a mixed variational regularization and finite elements (FreeFem++) on tetrahedral meshes, and furnishes very promising results 14.

We studied the uniqueness of a critical point of the quadratic criterion in the electroencephalography (EEG) problem for a single dipole situation (PhD of P. Asensio 19, see also 22). This issue is essential for the use of descent algorithms. This leads to the study of the following criterion:

where $x$ and $x_0$ (actual dipole position) are in a smooth open set $\Omega$ included in ${ℝ}^3$ and $p$, $p_0$ (actual dipole moment) belong to ${ℝ}^3$. In the half-space case where $\Omega =\{(x_1,x_2,x_3)\in {ℝ}^3,x_3<0\}$, we have proven that there is a unique critical point; i.e., a unique location and moment pair $(x,p)$ where the gradient of $J_{\partial \Omega }$ vanishes, which is but $(x_0,p_0)$ (the absolute minimum). For the spherical case where $\Omega =𝔹$ is the ball of center 0 and radius 1, we only obtained partial results so far, and managed to prove the uniqueness of the critical point in the specific case where $x_0=0$ is located at the center of $\Omega$. Observe that the problem is more difficult because no translation invariance holds anymore. We also studied the uniqueness of the critical point of the quadratic criterion in the charge localization problem for a single charge situation. This leads to the study of the following criterion:

where $x$ and $x_0$ (actual charge position) are in a smooth open set $\Omega$ included in ${ℝ}^3$ and $q$, $q_0$ (actual charge quantity) belong to $ℝ$. In both cases (the half-space case and the spherical one), we have proven that there is a unique critical point.

We considered a different class of source models, not necessarily dipolar, and related estimation algorithms. Such models may be supported on the surface of the cortex or in the volume of the encephalon. We represent sources by vector-valued measures, and in order to favor sparsity in this infinite-dimensional setting we use a TV (i.e., total variation) regularization term as in Section 6.1.1. The approach follows that of 8 and is implemented through two different algorithms, whose convergence properties were studied in the PhD thesis of M. Nemaire 20. Tests on synthetic data provided good quality results, though they are quite numerically costly to obtain.

Show image description

Three nested ovoids illustrating the three layers of the head (denoted ${\Omega }_i$ with $i=0,1,2$, 0 being the index of the inner layer). The conductivity of the $i$-th layer is denoted with ${\sigma }_i$ and its boundary is denoted with ${\Gamma }_i$. Two red straight lines illustrate how deep electrodes take measurements in the inner layer. Little segments perpendicular to each electrode illustrate the different sensors that lie along the electrode. A blue closed curve, denoted $S$, in the inner layer illustrates the surface where the sources lie, and little arrows show that their orientation is normal to the curve.

Progresses were made on the inverse problem of “Stereo” EEG (SEEG), where the potential is measured by deep electrodes and sensors within the brain as in the scheme of Figure 3. Assuming that the current source term $\mu$ is a ${ℝ}^3$-valued vector field (or a measure, or a dipolar distribution) supported on a surface $S$ and normally oriented to $S$, makes it possible to describe the potential $\varphi \left(\mu \right)$ generated by $\mu$ as a double layer potential:

The associated forward and inverse problems were solved for both an infinite medium conductor and a more realistic single model of the brain ${\Omega }_0$ (as first step towards nested domains for the head). Concerning the latter, it is preliminary assumed that $S\subset \partial {\Omega }_0$, whence $\mu$ consists in patches on the cortex surface.

The numerical implementation was done by approximating the density $\mu$ of the double layer potential with linear shape functions on triangulations of the involved surface $S$. For the discretized version, $\mu$ is then approximated on every triangle by a piecewise linear function, and the associated moment is taken to be located at the center of the triangle, perpendicular to the triangle. Note that explicit expressions of the double layer potential are also available at points located on $S$ (that supports $\mu$).

The inverse problem for SEEG is ill-posed and a Tychonov regularization is used in order to solve the problem: find a distribution $\mu$ defined on the prescribed surface $S$ such that $\parallel \varphi \left(\mu \right)-{\varphi }^d{\parallel }_{L^2\left({\Omega }_0\right)}+\lambda R\left(\mu \right)$ is minimized, where ${\varphi }^d$ is the given data in ${\Omega }_0$, $\varphi \left(\mu \right)$ is the “forward model” (measurements of the double layer potential), $R\left(\mu \right)$ is an appropriate regularization term that results in $\mu$ having some desired properties (being bounded in $L^2$ norm or $TV$ or $L^1$ on $S$), and $\lambda$ a (Lagrange) parameter. Because ${\varphi }^d$ is only furnished at sensor positions, the $L^2$ term in the above criterion becomes an $l^2$ term.

We now consider $S\subset {\Omega }_0$ within the brain, at the gray / white matters interface and we handle outer layers ${\Omega }_1$ and ${\Omega }_2$ for the skull and the scalp, as illustrated in Figure 3. Such configurations involve a data transmission / “cortical mapping” step relating the potential and normal current values at the boundaries ${\Gamma }_i$, that the present double layer formulation allows us to solve with the source localization issue. It only requires to solve the transmission problem for the surface electrical potentials hence this reduces the complexity of the problem. Numerically, this is a feasible implementation of the problem because the expressions for double layer potentials do not suffer from numerical singularities, hence this suffices to give reasonable solutions.

We are now able to handle MEG, EEG, SEEG modalities, simultaneously or not. The simultaneous handling of the different modalities is made the more straightforward by coupling the source localization problem and the inverse transmission problem.

The data transmission needs to be considered in the source localization problem so that the problem remains faithful to the data (electric or magnetic) when realistic head geometries are used in the computations. This also allows one to exploit properties of the problem that bring more regularity to the solutions. These considerations warrant that the transmission problem be solved simultaneously with the source localization problem, but in this connection the vector spaces used to model the sources and electrical properties need to be handled with care. The mathematical formulation allows for the source to be a distribution whereas the electric potential is always a function. This leads to the consideration that the optimal methods to recover the source and electric potential may not be the same. We proposed an alternating minimization procedure to solve the problem. We managed to show that this procedure converges to the desired solution and converges to the Tychonov functional linearly.

For more general source terms (vector valued measures, $L^p$ vector-fields), an alternating minimization procedure between refining the cortical mapping and the source localization is being studied and implemented. It takes advantage of the fact that both the source and the surface potentials are assumed to be elements of Banach spaces that have certain smoothness properties which enable the convergence to a solution as soon as the problem is convex. This alleviates problems that may be encountered when the regularizing term is non-smooth. This opens up the possibility to consider sources that may exhibit properties usually associated with distributions rather than functions 13.

7.1.1 Visits of international scientists

7.2.1 Other european programs/initiatives

Participants: Paul Asensio, Laurent Baratchart, Masimba Nemaire, Martine Olivi, Cristóbal Villalobos Guillén.

Factas is part of the European Research Network on System Identification (ERNSI) since 1992. System identification deals with the derivation, estimation and validation of mathematical models of dynamical phenomena from experimental data.

ANR REPKA

ANR-18-CE40-0035, “REProducing Kernels in Analysis and beyond” (2019–2023).

Led by Aix-Marseille Univ. (IMM), involving Factas team, together with Bordeaux (IMB), Paris-Est, Toulouse Universities.

The project consists of several interrelated tasks dealing with topical problems in modern complex analysis, operator theory and their important applications to other fields of mathematics including approximation theory, probability, and control theory. The project is centered around the notion of the so-called reproducing kernel of a Hilbert space of holomorphic functions. Reproducing kernels are very powerful objects playing an important role in numerous domains such as determinantal point processes, signal theory, Sturm-Liouville and Schrödinger equations.

This project supported the PhD of M. Nemaire within Factas, co-advised by IMB partners.

GDR AFHP

GDR “Analyse Fonctionnelle, Harmonique et Probabilités”.

Led by Gustave Eiffel Univ. (LAMA), involving Factas team, together with several universities.

The GDR is concerned with five main axes: linear dynamics, Banach spaces and their operators, holomorphic dynamics, harmonic analysis, analysis and probability, and with the interactions between them.

8.1.1 Scientific events: organisation

8.1.2 Journal

8.1.3 Invited talks

• L. Baratchart was an invited speaker at the workshop “Orthogonal polynomials and special functions” of the Foundations of Computational Mathematics (FoCM) conference, June 19-21, 2023, Paris.
• L. Baratchart and D. Ponomarev were invited speakers at the GDR AFHP days, October 24-27, 2023, Porquerolles.
• D. Ponomarev was an invited speaker at Journée Analyse Harmonique et EDP, 27 March 2023, Bordeaux, and at the minisymposium “Inverse Problems in Imaging and Tomographic Issues, Theory and Practice” of SIAM GS23 conference, June 19-22, Bergen, Norway.

8.1.4 Contributed talks

• S. Chevillard gave a presentation at the “Rencontres nationales du groupe de travail Arith du GDR IM” (RAIM 2023), November 2023, Nancy.
• M. Nemaire gave a communication at the workshop Mathematics of electrical imaging: modeling, theory and implementation, June 2023, Toulouse.
• D. Ponomarev gave presentations (poster and oral, respectively) at the conference IEEE AIM 2023 Advances in Magnetics (January 15-18, 2023, Moena, Italy) and at the workshop Complex Analysis: Techniques, Applications and Computations - Perspectives 2023 (July 24-28, 2023, Cambridge, UK).
• C. Villalobos Guillén gave presentations at the Journées du GDR AFHP, October 2023, Porquerolles, and at the Winter-School Sound and Fury of Modeling, November 2023, Arpino, Italy. She also presented a poster at the Conference: 30 ans de mathématiques pour l'imagerie optique, September 2023, Marseille.

8.1.5 Research administration

• J. Leblond is a member of the “Conseil Scientifique” and of the “Commission Administrative Paritaire” of Inria.
• M. Olivi is a member of the CLDD (Commission Locale de Développement Durable) and in charge of coordination.
• During the period March-November 2023, the team members were actively involved in an experimental project led by the Direction of the Research Center for the design of new Inria office spaces. This included participation in a series of workshops “Ateliers de co-conception de l'Espace UX” (S. Chevillard, D. Ponomarev), complemented by discussions within the “Comité de pilotage” (D. Ponomarev) and general meetings with the other concerned teams.

8.2.1 Teaching

• S. Chevillard gives “Colles” (oral examination preparing undergraduate students for the competitive examination to enter French Engineering Schools) at Centre International de Valbonne (CIV) (2 hours per week). In 2023, he also gave a course on Carbon Footprints at Polytech Sophia Antipolis, as a participant of the course Environmental Issues addressed to all students of Polytech, whatever their pathway (level L3; the course on Carbon Footprints was 2 hours long and represented a fifth of the whole course; it was repeated 4 times, due to the year group being divided into four sub-groups).
• S. Chevillard and M. Olivi gave a one-day Doctoral Workshop on Science, Environment and Society (in collaboration with G. Urvoy-Keller, S. Icart and L. Deneire, I3S). The first half-day was inspired by the workshop SEnS, created by S. Quinton (Inria, SPADES) and E. Tannier (Inria, BEAGLE). The second half-day was more specifically dedicated to the environmental crisis and the study of prospective scenarios.

8.2.2 Supervision

• PhD: P. Asensio, Inverse source estimation problems in EEG and MEG, defended September 2023, advisers: L. Baratchart, J. Leblond.
• PhD: M. Nemaire, Inverse potential problems with application to quasi-static electromagnetics, defended March 2023, advisers: L. Baratchart, J. Leblond, S. Kupin (IMB, Univ. Bordeaux).
• PhD in progress: A. Yousfi, Methods to estimate the net magnetic moment of rocks, since October 2022, advisers: S. Chevillard, J. Leblond.
• PhD in progress: M. Khalid Omer, Field preprocessing and treatment of complex samples in the paleomagnetic context, since October 2023, advisers: J. Leblond, D. Ponomarev.
• Post-doc.: C. Villalobos Guillén, Inverse source problems for magnetization, primary current and inverse scattering, adviser: L. Baratchart.
• Internship: M. Khalid Omer, Field preprocessing in inverse magnetisation problems, April-August 2023, advisers: J. Leblond, D. Ponomarev.

8.2.3 Juries

• J. Leblond was a member of the examining committees for the defense of the PhD theses of B. Laville (Univ. Côte d'Azur, ED STIC, September) and of N. Nasr (Univ. Bordeaux, ED MI, December).
• M. Olivi was a member of the “Comité de sélection MCF61” of Centrale Lyon EEA/Ampère.

8.3.1 Internal or external Inria responsibilities

J. Leblond and M. Olivi are members of Terra Numerica.

8.3.2 Interventions

• M. Olivi participated in a meeting with scholars at Collège la Chênaie, in the context of “la cordée de la réussite” : Un parcours scientifique sans préjugés : brisons le plafond de verre !. She gave four conferences “CHICHE” at Lycée Simone Veil in Valbonne. She gave a the c@fé-in of the research center, Martine à la COP : au delà des slogans, quelles solutions ?, and a general audience conference at Terra Numerica on the same subject.
• D. Ponomarev gave a presentation at C@fé ADSTIC, 30 November 2023, Sophia Antipolis.

International journals

• 13 articleP.Paul Asensio, J.-M.Jean-Michel Badier, J.Juliette Leblond, J.-P.Jean-Paul Marmorat and M.Masimba Nemaire. A layer potential approach to inverse problems in brain imaging.Journal of Inverse and Ill-posed ProblemsFebruary 2023HALDOIback to text
• 14 articleM.Marion Darbas, J.Juliette Leblond, J.-P.Jean-Paul Marmorat and P.-H.Pierre-Henri Tournier. Numerical resolution of the inverse source problem for EEG using the quasi-reversibility method.Inverse Problems39112023, 115003HALDOIback to text
• 15 articleE.Eduardo Lima, B.Benjamin Weiss, C.Caue Borlina, L.Laurent Baratchart and D.Douglas Hardin. Estimating the Net Magnetic Moment of Geological Samples From Planar Field Maps Using Multipoles.Geochemistry, Geophysics, Geosystems247July 2023, 38HALDOIback to text

Scientific book chapters

• 16 inbookL.Leonid Golinskii, S.Stanislas Kupin, M.Masimba Nemaire and J.Juliette Leblond. On discrete spectra of Bergman-Toeplitz operators with harmonic symbols.291From Complex Analysis to Operator Theory: A PanoramaIn Memory of Sergey NabokoOperator Theory: Advances and ApplicationsSpringer International Publishing2023HALDOIback to text
• 17 inbookJ.Juliette Leblond and E.Elodie Pozzi. An operator theoretical approach of some inverse problems.Function Spaces, Theory and ApplicationsFields Institute Communications ; vol 87Springer2023HALDOIback to text
• 18 inbookD.Dmitry Ponomarev. Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load.Integral Methods in Science and Engineering: Analytic and Computational ProceduresBirkhäuserNovember 2023HALDOIback to text

Doctoral dissertations and habilitation theses

• 19 thesisP.Paul Asensio. Inverse problems of source localization with applications to EEG and MEG.Université cote d'azurSeptember 2023HALback to textback to text
• 20 thesisM.Masimba Nemaire. Inverse potential problems, with applications to quasi-static electromagnetics.Université de BordeauxMarch 2023HALback to textback to text

Reports & preprints

• 21 miscA.Anton Arnold, S.Sjoerd Geevers, I.Ilaria Perugia and D.Dmitry Ponomarev. On the limiting amplitude principle for the wave equation with variable coefficients.December 2023HALback to text
• 22 miscP.Paul Asensio and J.Juliette Leblond. Critical points for least-squares estimation of dipolar sources in inverse problems for Poisson equation.January 2023HALback to text
• 23 miscL.Laurent Baratchart, S.Sébastien Fueyo and J.-B.Jean-Baptiste Pomet. Exponential stability of linear periodic difference-delay equations.2023HALback to textback to text
• 24 miscL.Laurent Baratchart, S.Sébastien Fueyo and J.-B.Jean-Baptiste Pomet. Integral representation formula for linear non-autonomous difference-delay equations.2023HALback to text
• 25 miscL.Laurent Baratchart, H.Houssem Haddar and C.Cristóbal Villalobos Guillén. Silent sources on a surface for the Helmholtz equation and decomposition of L² vector fields.December 2023HALback to text
• 26 miscL.Laurent Baratchart, J.Juliette Leblond and M.Masimba Nemaire. Silent sources in $L^p$ and Helmholtz-type decompositions.2023HALback to textback to text
• 27 miscD.Dmitry Ponomarev. Magnetisation Moment of a Bounded 3D Sample: Asymptotic Recovery from Planar Measurements on a Large Disk Using Fourier Analysis.September 2023HALback to text