2025Activity reportProject-Team​​​‌FACTAS

3.1 Elliptic PDEs‌​‌ and operators

Inverse problems​​ studied by Factas involve​​​‌ systems governed by an‌ equation of the form‌​‌ $ℒ\varphi =\mathrm{\Psi }\left(m\right)$,​​​‌ where $ℒ$ is an‌ elliptic partial differential operator‌​‌ and $\Psi \left(\mathrm{m}\right)$ a source term​​​‌ depending on some unknown‌ quantity $m$. The‌​‌ data consist of incomplete​​ measurements of the potential​​​‌ $\varphi$ or its gradient‌ (the field) in a‌​‌ portion of space, away​​ from the support of​​​‌ the source.

3.2 Inverse source problems​​​‌

Given an elliptic PDE​ of the form $\mathrm{ℒ‌}\varphi =\Psi (m)$ as in​​​‌ Section 3.1, the​ corresponding inverse source problem​‌ consists in recovering the​​ quantity $m$ from the​​​‌ data, that typically consist​ of measurements of the​‌ potential $\varphi$ or the​​ field away from the​​​‌ support of the source.​ Usually the support of​‌ $m$ is assumed to​​ be compact, and a​​​‌ super-set thereof is specified.​ This kind of issues​‌ may thus be seen​​ as parameterized inverse potential​​​‌ problems (parameterized by $\mathrm{m}$, that is). They​‌ arise naturally in non-destructive​​ testing, medical imaging, paleo-magnetism,​​​‌ gravimetry and geosciences, as​ well as in inverse​‌ scattering, see Section 4​​. The second and​​​‌ third application domains pertain​ to static electromagnetism, a​‌ framework in which source​​ terms typically occur in​​​‌ divergence form; that is,​ $\Psi (m)‌=\mathrm{div}m$ where​​ $m$ is a ${\mathrm{ℝ‌}}^3$-valued field or​ distribution. As a rule,​‌ the operator $ℒ$ is​​ then of the form​​​‌ $ℒ\varphi =\mathrm{div}\left(\Sigma \nabla \mathrm{\varphi ‌}\right)$, where $\mathrm{\Sigma }$ is valued in positive​​​‌ definite matrices satisfying fixed​ ellipticity bounds and relates​‌ to the electromagnetic characteristics​​ of the medium. In​​​‌ this case, the problem​ amounts to recover a​‌ vector field (namely: $\mathrm{m}/\det \left(\mathrm{\Sigma ‌}\right)$) knowing a​ super-set $S$ of its​‌ support and the gradient​​ summand of its Helmholtz​​ decomposition outside of $\mathrm{S‌}$, for the Riemannian‌ metric with tensor $(‌‌\det \Sigma ){\mathrm{\Sigma }}^{-1}$. The​​​‌ simplest case, of course,‌ occurs in the Euclidean‌​‌ setting, where $ℒ$ is​​ the ordinary Laplacian, which​​​‌ corresponds to homogeneous media.‌

3.3 Rational​​ approximation, behavior of poles​​​‌

Rational approximation to holomorphic‌ functions of one complex‌​‌ variable is a long​​ standing chapter of classical​​​‌ analysis, with notable applications‌ to number theory, spectral‌​‌ theory and numerical analysis.​​ Over the last decades,​​​‌ it has become a‌ cornerstone of modeling in‌​‌ Systems Engineering, and it​​ can also be construed​​​‌ as a technique to‌ regularize inverse source problems‌​‌ in the plane, where​​ the degree is the​​​‌ regularizing parameter. Indeed, by‌ partial fraction expansion, a‌​‌ rational function can be​​ viewed as the complex​​​‌ derivative of a discrete‌ logarithmic potential with as‌​‌ many masses as the​​ degree (assuming that the​​​‌ poles are simple); that‌ is, if $f$ is‌​‌ rational of degree $\mathrm{N}$, then $\stackrel{¯‌}{\partial }f={\sum }_{\mathrm{j‌}=1}^N{\mathrm{a‌‌}}_j{\delta }_{z_{\mathrm{j}}}$ where the $z_{\mathrm{j‌}}$ are its poles and‌ ${\delta }_{z_j}$ is‌​‌ a Dirac unit mass​​ at $z_j$.​​​‌ Moreover, a holomorphic function‌ is the complex derivative‌​‌ of the logarithmic potential​​​‌ of its own values​ on a curve encompassing​‌ the domain of analyticity​​ (this is the Cauchy​​​‌ formula); hence, rational approximation​ aims at representing as​‌ well as possible a​​ logarithmic potential by a​​​‌ discrete potential with prescribed​ number of masses, in​‌ the sense that their​​ derivatives should be close​​​‌ (a Sobolev-type approximation).

Predecessors​ of Factas (the Apics​‌ and Miaou project teams)​​ have designed a dedicated​​​‌ steepest-descent algorithm for quadratic​ approximation criteria whose convergence​‌ to a local minimum​​ is guaranteed. This gradient​​​‌ algorithm may either be​ initialized by a preliminary​‌ approximation method, or recursively​​ proceed with respect to​​​‌ the degree $N$ of​ the approximant, on a​‌ compactification of the parameter​​ space 29, as​​​‌ can be done with​ the RARL2 software (see​‌ Section 7.1.3). It​​ has proved to be​​​‌ effective in applications carried​ out by the team​‌ (see for instance 9​​ for the identification of​​​‌ micro-wave filters, and Sections​ 4.1 and 4.3).​‌

However, finding best rational​​ approximants of prescribed degree​​​‌ to a specific function,​ say in the uniform​‌ norm on a given​​ set, seems out of​​​‌ reach except in rare,​ particular cases. Instead, constructive​‌ rational approximation has focused​​ on estimating optimal convergence​​​‌ rates and deriving approximation​ schemes coming close to​‌ meet them, or studying​​ computationally appealing approximants like​​​‌ Padé interpolants and their​ variants. Two main issues​‌ are then the effective​​ computation of optimal or​​​‌ near optimal approximants of​ given degree, and the​‌ connection between the singularities​​ of the approximant (the​​​‌ poles) and those of​ the approximated function. Factas​‌ has contributed to both.​​

We studied in particular​​​‌ the behavior of best​ rational approximants of given​‌ degree, in the ${\mathrm{L}}^{\infty }$-sense on a​​​‌ compact subset of the​ domain of analyticity, to​‌ complex analytic functions $\mathrm{f}$ that can be continued​​​‌ analytically (possibly in a​ multi-valued manner) except perhaps​‌ over a set of​​ logarithmic capacity zero in​​​‌ the plane. When the​ continuation of $f$ has​‌ finitely many branches; that​​ is, if the Riemann​​​‌ surface to which this​ continuation extends analytically in​‌ a single-valued manner (except​​ perhaps on a polar​​​‌ subset thereof) is compact,​ then the behavior as​‌ $N$ goes large of​​ rational approximants of degree​​​‌ $N$ whose $N$-th​ root error is asymptotically​‌ smallest possible (in particular​​ the asymptotic behavior of​​​‌ best approximants) has recently​ been elucidated rather completely​‌ in this joint research​​ effort involving Factas.

As​​​‌ regards near-optimal approximants, their​ design requires a knowledge​‌ of optimal rates in​​ the situation at hand.​​​‌ In recent years, we​ were active determining lower​‌ bounds on that rate,​​ a piece of information​​​‌ which is crucial but​ difficult to obtain. Our​‌ methods are topological in​​ nature (Ljustenik-Schnirelman theory, genus​​​‌ of compact symmetric sets),​ like most techniques in​‌ the area, and in​​ collaboration with Tao Qian​​​‌ from the University of​ Macao, we devised algorithms​‌ to compute lower bounds​​ in best $L^{\mathrm{2‌}}$ approximation by stable rational​ function of given degree​‌ on the unit circle,​​ which is first of​​ this kind and sometimes​​​‌ quite precise, see 5‌. Research in this‌​‌ direction is still active,​​ in particular on best​​​‌ $L^2$ approximation of‌ functions of constant modulus,‌​‌ which is an old​​ issue in system theory​​​‌ (how to perform model‌ reduction of delay systems)‌​‌ that has received a​​ new lease of life​​​‌ from the heavy trend‌ of neural networks. Such‌​‌ functions cannot be approximated​​ in uniform norm, except​​​‌ when they are rational‌ of admissible degree, in‌​‌ which case they are,​​ of course, their own​​​‌ best approximant. Their ${\mathrm{L‌}}^2$ rational approximation is‌​‌ possible, though, but not​​ very efficient and the​​​‌ problem is to quantify‌ this fact by giving‌​‌ a lower bound on​​ the achievable approximation error​​​‌ by a rational function‌ of given degree.

Another‌​‌ classical technique to approximate​​ –more accurately: extrapolate– a​​​‌ function, given a set‌ of pointwise values, is‌​‌ to compute a rational​​ interpolant of minimal degree​​​‌ to match the values.‌ This method, known as‌​‌ Padé (or multi-point Padé)​​ approximation has been intensively​​​‌ studied for decades 27‌ but fails to produce‌​‌ pointwise convergence, even if​​ the data are analytic.​​​‌ The best it can‌ give in general, at‌​‌ least to functions whose​​ singular set has capacity​​​‌ zero, is convergence in‌ capacity which does not‌​‌ prevent poles of the​​ approximant from wandering about​​​‌ the domain of analyticity‌ of the approximated function,‌​‌ but does imply that​​ each pole of the​​​‌ approximated function attracts a‌ pole of the approximant‌​‌ 69. This phenomenon​​ is well-known in numerical​​​‌ analysis, and leads Physicists‌ and Engineers to distinguish‌​‌ between “mathematical” and “physical”​​ poles. A modification of​​​‌ the multi-point Padé technique,‌ in which the degree‌​‌ is kept much smaller​​ than the number of​​​‌ data and approximate interpolation‌ is performed in the‌​‌ least-square sense, has become​​ especially popular over the​​​‌ last decade under the‌ name vector fitting; it‌​‌ teams up with a​​ barycentric representation of rational​​​‌ functions satisfying prescribed interpolation‌ conditions, known as AAA‌​‌ (for Anderson-Antoulas Adaptive) scheme.​​ Though the behavior of​​​‌ this least square substitute‌ to Padé approximation, defined‌​‌ by Equation (1​​) in Section 4.4​​​‌, resembles the one‌ of multi-point Padé approximants‌​‌ from a numerical viewpoint,​​ there has been apparently​​​‌ no convergence result for‌ such approximate interpolants so‌​‌ far. Motivated by the​​ outcome of numerical schemes​​​‌ developed by our partners‌ to recover resonance frequencies‌​‌ of conductors under electromagnetic​​ inverse scattering, the PhD​​​‌ thesis 25 of Paul‌ Asensio started investigating the‌​‌ behavior of such least-square​​ rational approximants to functions​​​‌ with polar singular set,‌ and dwelling on this‌​‌ work, we were recently​​ able to show convergence​​​‌ in capacity thereof.

Regarding‌ complex rational approximation as‌​‌ a means to tackle​​ inverse source problems in​​​‌ the plane makes for‌ a unifying point of‌​‌ view on various deconvolution​​ techniques, from system identification​​​‌ and time series analysis‌ to frequency-wise inverse scattering‌​‌ and non-destructive testing. But​​ still more interestingly perhaps,​​​‌ it is suggestive of‌ similar approaches to problems‌​‌ in higher dimension, where​​​‌ holomorphic functions generalize to​ harmonic gradients and rational​‌ functions to finite linear​​ combinations of dipoles, see​​​‌ Section 3.1.2. This​ line of research is​‌ only starting, but seems​​ to offer new avenues​​​‌ in connection with applications.​

3.4 Asymptotic analysis

Asymptotic​‌ analysis deals with understanding​​ behavior or explicit construction​​​‌ of the solution when​ a parameter entering a​‌ problem is either small​​ or large. Factas has​​​‌ been involved in applications​ of asymptotic analysis in​‌ different contexts including both​​ formal constructions and their​​​‌ rigorous justifications.

One type​ of asymptotic analysis for​‌ dynamical problems is the​​ large-time behavior analysis. A​​​‌ rather classical issue here​ is that of limiting​‌ amplitude principle for wave​​ equation. This principle states​​​‌ that the solution of​ the time-dependent wave equation​‌ with a periodic-in-time monochromatic​​ source term $f{\mathrm{e‌}}^{i\omega t}$ necessarily​ stabilizes for large times​‌ $t$ to the solution​​ of the corresponding Helmholtz​​​‌ equation: $\mathrm{div}\left(\mathrm{\alpha }\nabla \varphi \right)+‌{\omega }^2\beta \mathrm{\varphi }=-f$ for​​​‌ suitable known functions $\mathrm{\alpha }$, $\beta$ and $\mathrm{f‌}$ depending on space variables.​​ Revisiting this topic is​​​‌ motivated by recent popularity​ of numerical time-domain approaches​‌ (see, e.g., 22​​, 23, 60​​​‌, 83) to​ elliptic PDE problems through​‌ efficient solution of auxiliary​​ time-dependent equations where, for​​​‌ example, computational effort needed​ to be concentrated only​‌ on wave front neighborhood​​ which is small for​​​‌ high-frequencies, a notoriously difficult​ regime for numerical solution​‌ of Helmholtz problems. With​​ this respect, not only​​​‌ the fact of the​ time convergence in the​‌ limiting amplitude principle is​​ important but also quantification​​​‌ of its rate. Dmitry​ Ponomarev was involved in​‌ the work 1 that​​ deals with the limiting​​​‌ amplitude principle for a​ non-homogeneous medium wave-equation in​‌ different dimensions (in the​​ dimension 1, a slight​​​‌ modification of the classical​ limiting amplitude principle was​‌ proposed). The analysis approach​​ relies on proving that​​​‌ the solution decays to​ zero for a source-free​‌ problem with localized initial​​ data and radiation boundary​​​‌ condition, an interesting problem​ of asymptotic analysis on​‌ its own, even in​​ dimension 1 24.​​​‌ In 78, convergence​ to the periodic motion​‌ was also shown in​​ a totally different context​​​‌ of one model of​ mechanical sliding contact problem​‌ with wear 77.​​

Previously, in a nonlinear​​​‌ context, rigorous asymptotic analysis​ 73 was instrumental to​‌ justify a parabolic model​​ of pulse propagation in​​​‌ photo-polymers by comparing solutions​ of that model with​‌ those of the original​​ Maxwell's system.

In the​​​‌ context of inverse problems,​ asymptotic analysis is useful​‌ when applied to the​​ magnitude of the regularization​​​‌ parameter. When the latter​ tends to its limiting​‌ value, a solution of​​ the regularized problem with​​​‌ ideal (noiseless) input data​ should tend to the​‌ exact solution. In presence​​ of noise, it is​​​‌ important to relate this​ convergence rate to the​‌ value of the problem's​​ constraint in the asymptotic​​​‌ regime of the regularization​ parameter.

The works 41​‌, 76 on convolution​​ integral equations on large​​ domains, mentioned in Section​​​‌ 3.1.3, are an‌ example of constructive asymptotic‌​‌ analysis. Here, one of​​ the difficulty comes from​​​‌ a singular perturbation. Indeed,‌ in the asymptotic limit‌​‌ of infinite size of​​ the region, the spectral​​​‌ problem solution cease to‌ exist since the integral‌​‌ operator loses the compactness​​ property.

In the context​​​‌ of net magnetization reconstruction‌ in the inverse magnetization‌​‌ problem (Sections 3.2 and​​ 4.2), situation when​​​‌ the measurement area size‌ is large leads to‌​‌ a different kind of​​ application of constructive asymptotic​​​‌ analysis 30, 38‌, 79. Here,‌​‌ explicit constructions of the​​ solution estimates are performed​​​‌ to different asymptotic orders‌ with respect to the‌​‌ measurement region size. The​​ higher-order estimates can give​​​‌ good accuracy already for‌ relatively small value of‌​‌ the measurement region but​​ are much more unstable​​​‌ with respect to the‌ perturbation of the measured‌​‌ data. This problem also​​ exhibits another interesting asymptotic​​​‌ phenomenon which is somewhat‌ similar to the “boundary‌​‌ layer” common for boundary-value​​ problem for differential equations​​​‌ with small or large‌ parameters. In particular, the‌​‌ solution (for tangential components​​ of net moment) is​​​‌ composed of a global‌ leading-order quantity (where formal‌​‌ passage to the asymptotic​​ limit can be performed)​​​‌ and a correction term‌ which is localized in‌​‌ a region that shrinks​​ in the asymptotic limit.​​​‌

4.1 Some‌​‌ inverse problems for cerebral​​ imaging

Solving over-determined Cauchy​​​‌ problems for the Laplace‌ equation on a spherical‌​‌ layer (in 3-D) in​​ order to extrapolate incomplete​​​‌ data (see Section 3.1.1‌) is an ingredient‌​‌ of the team's approach​​ to inverse source problems​​​‌ in electro-encephalography (EEG), see‌ 51. The model‌​‌ comes from Maxwell's equation​​ in the quasi-static regime,​​​‌ whence $\mathrm{div}\left(\mathrm{\sigma ‌}\nabla \varphi \right)=‌‌\mathrm{div}m$ in a​​ ball $\Omega$ which is​​​‌ the union of nested‌ layers ${\Omega }_i$ (brain,‌​‌ skull, scalp for $\mathrm{i}=0,\mathrm{1‌},2$), where‌ the singularities lie in‌​‌ ${\Omega }_0$ (i.e.​​, the current sources​​​‌ $m$ are supported in‌ $S\subset {\Omega }_{\mathrm{0‌‌}}$, with the notation​​ of Section 3),​​​‌ see Figure5 1‌. The inverse EEG‌​‌ source problem consists in​​ recovering $m$ from pointwise​​​‌ values of the electric‌ potential $\varphi$ measured on‌​‌ the scalp ${\Gamma }_{\mathrm{2}}$ (together with the assumption​​​‌ that the normal current‌ vanishes there). It first‌​‌ involves transmitting the data​​ from the scalp ${\mathrm{\Gamma ‌}}_2$ down to the‌ cortex ${\Gamma }_0$.‌​‌ Whenever the ${\Omega }_{\mathrm{i}}$ are of different constant​​​‌ conductivities, $\varphi$ satisfies Laplace‌ equation in the outermost‌​‌ shells ${\Omega }_2$ and​​ ${\Omega }_1$ and this​​​‌ “cortical mapping” step is‌ performed using integral representations‌​‌ of $\varphi$ on the​​ spheres ${\Gamma }_i$ (obtained​​​‌ through convolution by the‌ Poisson kernels of the‌​‌ balls and using Green​​ formula, a specific use​​​‌ of boundary element methods),‌ followed by expansions on‌​‌ spherical harmonic bases and​​​‌ Tikhonov regularization.

Assuming $\mathrm{m}$ to be a linear​‌ combination of Dirac masses​​ (dipolar sources), it turns​​​‌ out (by convolution with​ the fundamental solution of​‌ 3-D Laplace equation) that​​ traces of $\varphi$ on​​​‌ 2-D cross sections of​ ${\Gamma }_0$ coincide with​‌ functions with branched singularities​​ in the slicing plane​​​‌ 39, 45.​ These singularities are related​‌ to the actual location​​ of the sources. Hence​​​‌ we are back to​ the 2-D framework of​‌ Section 3.3, and​​ recovering these singularities can​​​‌ be performed via best​ rational approximation. The goal​‌ is to produce a​​ fast and sufficiently accurate​​​‌ initial guess on the​ number and location of​‌ the sources in order​​ to run heavier descent​​​‌ algorithms on the direct​ problem, which are more​‌ precise but computationally costly​​ and often fail to​​​‌ converge if not properly​ initialized. Such a localization​‌ process can add a​​ geometric, valuable piece of​​​‌ information to the standard​ temporal analysis of EEG​‌ signal records. It appears​​ that, in the rational​​​‌ approximation step, multiple poles​ possess a nice behavior​‌ with respect to branched​​ singularities. This is due​​​‌ to the very physical​ assumptions on the model​‌ from dipolar current sources,​​ for both EEG data​​​‌ and MEG (magneto-encephalography) data​ that correspond to measurements​‌ of the magnetic field,​​ as well as for​​​‌ (magnetic) field data produced​ by magnetic dipolar sources​‌ within rocks (see Section​​ 4.2). Though numerically​​​‌ observed in 51,​ there is no mathematical​‌ justification so far why​​ branched singularities generate such​​​‌ strong accumulation of the​ poles of the approximants.​‌ This intriguing property, however,​​ is definitely helping source​​​‌ recovery and will be​ the topic of further​‌ study. It is used​​ in order to automatically​​​‌ estimate the “most plausible”​ number of sources (numerically:​‌ up to 3, at​​ the moment). In this​​​‌ connection, a software FindSources3D​ (FS3D, see Section 7.1.4​‌) dedicated to pointwise​​ source estimation in EEG–MEG​​​‌ has been developed. We​ also studied the uniqueness​‌ of a critical point​​ of the quadratic criterion​​​‌ in the EEG source​ problem in ${\Omega }_{\mathrm{0‌}}$ for a single dipole​​ situation (see 25),​​​‌ an important issue for​ the use of descent​‌ algorithms.

Together with Marion​​ Darbas (LAGA, Université Sorbonne​​​‌ Paris Nord) and Pierre-Henri​ Tournier (JLL laboratory, Sorbonne​‌ Université), we recently considered​​ the EEG inverse problem​​​‌ with a variable conductivity​ in the intermediate skull​‌ layer ${\Omega }_1$,​​ in order to model​​​‌ hard / spongy bones,​ especially for neonates. The​‌ related transmission step is​​ then performed using a​​​‌ mixed variational regularization and​ finite elements on tetrahedral​‌ meshes, and the coupling​​ with FS3D for dipolar​​​‌ source estimation furnishes promising​ results 8.

Other​‌ approaches have been studied​​ for EEG, MEG and​​​‌ “Stereo” EEG (SEEG), where​ the potential is measured​‌ by deep electrodes and​​ sensors within the brain​​​‌ as in the scheme​ of Figure 1,​‌ and for more realistic​​ geometries of the head.​​​‌

Three nested ovoids illustrating​ the three layers of​‌ the head (denoted ${\mathrm{\Omega }}_i$ 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.

Assuming​​​‌ that the current source‌ term $m$ is a‌​‌ ${ℝ}^3$-valued vector​​ field (measure, or distribution)​​​‌ supported on a surface‌ $S\subset {\Omega }_{\mathrm{0‌‌}}$ (the gray / white​​ matters interface within the​​​‌ brain) and normally oriented‌ to $S$, they‌​‌ consist in regularizing the​​ inverse source problem by​​​‌ a total variation (TV)‌ constraint - to favor‌​‌ sparsity - on $\mathrm{m}$, added to the​​​‌ quadratic data approximation criterion‌ (see Section 3.2).‌​‌ This is similar to​​ the path that is​​​‌ taken for inverse magnetization‌ problems (see Section  4.2‌​‌). The approach follows​​ that of 7 and​​​‌ is implemented through algorithms‌ whose convergence properties were‌​‌ studied in 25,​​ 72. We are​​​‌ now able to handle‌ MEG, EEG, SEEG modalities,‌​‌ simultaneously or not. The​​ simultaneous handling of the​​​‌ different modalities is made‌ more straightforward by coupling‌​‌ the source localization problem​​ and the inverse transmission​​​‌ problem. Tests on synthetic‌ data provided good quality‌​‌ results, though they are​​ quite numerically costly to​​​‌ obtain. This opens up‌ the possibility to consider‌​‌ sources that may exhibit​​ properties usually associated with​​​‌ distributions rather than functions.‌

4.2 Inverse magnetization issues‌​‌ for planar and volumetric​​ samples

Among other things,​​​‌ geoscientists are interested in‌ understanding the magnetic characteristics‌​‌ of ancient rocks. Indeed,​​ ferro-magnetic particles in a​​​‌ rock carry a magnetization‌ that has been acquired‌​‌ when the rock was​​ hot, under the influence​​​‌ of the magnetic field‌ that was ambient at‌​‌ that time. For an​​ igneous rock for instance,​​​‌ and if no subsequent‌ event has heated it‌​‌ up, this corresponds to​​ the time when the​​​‌ rock was formed. If‌ the rock can be‌​‌ dated, recovering its magnetization​​ hence provides valuable information​​​‌ about the history of‌ the magnetic field. This‌​‌ gives elements for better​​ understanding to key questions​​​‌ such as: what was‌ the magnetic field of‌​‌ the sun when the​​ solar system was at​​​‌ the proto-planetary phase? when‌ did the magnetic dynamo‌​‌ of Mars stop? when​​ did the magnetic dynamo​​​‌ of Earth start?

The‌ magnetization of a rock‌​‌ is not directly measurable.​​ However, it produces a​​​‌ tiny magnetic field, which‌ can be measured if‌​‌ the sample is isolated​​​‌ from other sources of​ magnetic field. A category​‌ of instruments of particular​​ importance with that respect​​​‌ are the magnetic microscopes.​ They are used to​‌ measure the field produced​​ by a fairly small​​​‌ sample: either a simple​ grain, or a wider​‌ sample that has been​​ first prepared by gluing​​​‌ it on some support​ and polishing it until​‌ getting only a thin​​ slab. The microscope operates​​​‌ at some distance above​ the sample and measures​‌ the magnetic field. The​​ typical experimental set up​​​‌ is represented on Figure​ 2.

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 horizontal​ plane at height $\mathrm{z‌}$. Here, for instance,​​ it is a square,​​​‌ whose half-size is $\mathrm{R}$ and which is horizontally​‌ centered.

The magnetization $m$ is​ modeled as a ${\mathrm{ℝ‌}}^3$-valued vector field​​ defined on the rock​​​‌ sample which is assumed​ to be a subset​‌ of $S$. One​​ advantage of the magnetic​​​‌ microscopes is that they​ operate close to the​‌ sample, i.e., the​​ height $z$ of measurement​​​‌ is small compared to​ the horizontal characteristic lengths​‌ $s$ and $R$.​​ The thickness of the​​​‌ sample is also small​ compared to $s$.​‌ However, the ratio $\mathrm{r}/z$ is not​​​‌ necessarily negligible. In the​ cases when it is​‌ indeed negligible, the sample​​ can be supposed planar​​​‌ instead of being volumetric​ (i.e., $\mathrm{r‌}=0$) from​​ a practical point of​​​‌ view; in this case,​ $t$ becomes a planar​‌ variable $(t_{\mathrm{1}},t_2)‌\in S$ (a square)​ and $m$ is actually​‌ a magnetization density.

The​​ surface $Q_R$ of​​​‌ measurements is, most of​ the time, supposed to​‌ be a centered square​​ of half-size $R$,​​​‌ but in some situations​ it might be convenient​‌ to consider only the​​ data available on a​​​‌ centered disk $D_{\mathrm{R}}$ of radius $R$.​‌ The microscope provides a​​ map of the field​​​‌ on the whole surface:​ measurements are provided at​‌ many points of the​​ surface and, from a​​​‌ practical point of view,​ one may assume that​‌ the field is known​​ everywhere on $Q_{\mathrm{R‌}}$. In addition to​ the presence of noise​‌ in the measurements, an​​ important limitation is that,​​​‌ depending on the microscope​ technology, it is frequent​‌ that only one component​​ of the field be​​ measured.

The team has​​​‌ a long-standing collaboration with‌ the Earth and Planetary‌​‌ Sciences Laboratory at MIT.​​ They have a superconducting​​​‌ quantum interference device (SQUID).‌ The sensor is a‌​‌ tiny vertical coil maintained​​ at temperature close to​​​‌ 0 Kelvin, which provides‌ it with superconducting characteristics.‌​‌ In order to maintain​​ the sensor at very​​​‌ low temperature while the‌ microscope operates in a‌​‌ room at normal temperature,​​ the sensor is isolated​​​‌ behind a sapphire window.‌ Though thin, this window‌​‌ enforces a measurement height​​ $z$ such that $\mathrm{r‌}\ll z\ll \mathrm{s‌}$ and the sample can‌​‌ usually be supposed planar.​​ Also, because the coil​​​‌ only allows to measure‌ the field along its‌​‌ axis, the SQUID only​​ provides measurements of ${\mathrm{B‌}}_3$ and not the‌ whole field.

Another type‌​‌ of microscopes consists in​​ the quantum diamond microscopes​​​‌ (QDM). They use properties‌ of special diamonds which,‌​‌ when properly excited with​​ a laser and a​​​‌ microwave field, become luminescent‌ in the presence of‌​‌ a magnetic field. From​​ the difference of brightness​​​‌ of this luminescence under‌ slightly different frequencies of‌​‌ the microwave field, one​​ can recover the amplitude​​​‌ of a given component‌ of the magnetic field.‌​‌ This mechanism is already​​ in use to provide​​​‌ a magnetic microscope at‌ Harvard University (Massachusetts, USA).‌​‌ We are collaborating with​​ geoscientists of the Geophysics​​​‌ and Planetology Department of‌ Cerege (CNRS, Aix-en-Provence) and‌​‌ physicists from ENS Paris​​ Saclay to help them​​​‌ designing their own QDM.‌ The promises of the‌​‌ QDM are manifolds, see​​ also Section 8.2.​​​‌ First, they should allow‌ measurements of the field‌​‌ at a height $\mathrm{z}$ above the sample that​​​‌ is way smaller than‌ what is permitted with‌​‌ the SQUID. This improves​​ the spatial resolution and​​​‌ together with the conditioning‌ of the inverse magnetization‌​‌ problem. However, this also​​ imposes to model the​​​‌ sample as a 3-D‌ object, as its thickness‌​‌ $r$ becomes usually comparable​​ to $z$ in this​​​‌ case. Second, the technology‌ of the QDM could‌​‌ make it possible, in​​ principle, to measure the​​​‌ three components of the‌ field instead of only‌​‌ one, which opens the​​ way to new regularization​​​‌ techniques for the inverse‌ problems. However, the measurement‌​‌ of a field with​​ a QDM is more​​​‌ indirect than with a‌ SQUID and this raises‌​‌ specific issues: in particular,​​ it might be necessary​​​‌ to impose an external‌ field on the sample,‌​‌ called a bias field,​​ in order to resolve​​​‌ ambiguities when recovering the‌ field from the brightness‌​‌ of the luminescence, and​​ such a bias field​​​‌ can actually perturb the‌ magnetic properties of the‌​‌ rock sample under study.​​

The issues raised by​​​‌ the inverse magnetization problem‌ in the framework of‌​‌ magnetic microscopes such as​​ SQUIDs or QDMs are​​​‌ numerous and we got‌ several contributions on the‌​‌ subject over time. Particularly​​ important for the full​​​‌ recovery of the magnetization‌ are the silent sources,‌​‌ i.e., the magnetization​​ that belong to the​​​‌ kernel of the direct‌ operator, or in other‌​‌ terms, those magnetization that​​​‌ produce no field on​ the measurement area, see​‌ Section 3.2.2. We​​ fully characterized such magnetizations​​​‌ in the thin-plate hypothesis​ (i.e., when​‌ $r$ is assumed to​​ be 0), 36.​​​‌ Contrary to the full​ magnetization, the total net​‌ moment (i.e.,​​ the integral of the​​​‌ magnetization over the sample)​ is in principle a​‌ piece of information that​​ could be retrieved from​​​‌ the measurements, since silent​ sources have a null​‌ net moment. In order​​ to recover the total​​​‌ net moment, we proposed​ to use linear estimators​‌ and defined a bounded​​ extremal problem to find​​​‌ good such linear estimators​ 4. However, additional​‌ hypotheses must be added​​ in order to ensure​​​‌ the uniqueness of a​ solution for the full​‌ inversion problem. Such an​​ hypothesis is provided when​​​‌ the support of the​ magnetization inside the sample​‌ is supposed to be​​ sparse, e.g., composed​​​‌ of isolated points or​ 1-D curves. In this​‌ case, we proposed to​​ use total variation regularization​​​‌ to solve the inverse​ problem 7, 47​‌, see Section 3.2.3​​.

4.3 Inverse magnetization​​​‌ issues with the lunometer​

Measurements of the remanent​‌ magnetic field of the​​ Moon let geoscientists think​​​‌ that the Moon used​ to have a magnetic​‌ dynamo for some time,​​ but the exact process​​​‌ that triggered and fed​ this dynamo is not​‌ yet understood, much less​​ why it stopped. In​​​‌ particular, the Moon is​ too small to have​‌ a convecting dynamo like​​ the Earth has. In​​​‌ order to address this​ question, our geoscientists colleagues​‌ at Cerege decided to​​ systematically analyze the rock​​​‌ samples brought back from​ the Moon by Apollo​‌ missions.

The samples are​​ kept inside bags with​​​‌ a protective atmosphere, and​ geophysicists are not allowed​‌ to open the bags,​​ nor to take out​​​‌ the samples from NASA​ facilities. Moreover, the measurements​‌ must be performed with​​ a passive device in​​​‌ order to ensure that​ the samples would not​‌ be altered by the​​ measurements: in particular no​​​‌ cooling or heating is​ allowed, and neither is​‌ the use of anything​​ producing a magnetic field​​​‌ like, e.g., motors.​ Finally, since the measurements​‌ must be performed directly​​ at NASA, the instrument​​​‌ must be easy to​ take apart and to​‌ assemble once on site.​​ The overall time devoted​​​‌ to measuring all samples​ is limited and each​‌ sample must be analyzed​​ quickly (typically within a​​​‌ few minutes). For all​ these reasons, our colleagues​‌ from Cerege designed a​​ specific magnetometer called the​​​‌ “lunometer”: this device provides​ measurements of the components​‌ of the magnetic field​​ produced by the sample,​​​‌ at some discrete set​ of points located on​‌ disks belonging to three​​ cylinders (see Figure 3​​​‌). The goal was​ not to get a​‌ deep understanding of the​​ magnetic properties of the​​​‌ studied samples with such​ a rudimentary instrument but​‌ rather to help selecting​​ a few of them​​​‌ that seems really interesting​ to study in more​‌ details: this would be​​ used to file a​​ request to NASA to​​​‌ buy sub-samples of a‌ few grams on which‌​‌ more instructive (though possibly​​ destructive) experiments could be​​​‌ performed.

The three orthogonal‌ planes of the 3D‌​‌ canonical system of coordinates.​​ Perpendicular to each of​​​‌ the planes, there is‌ a cylinder and on‌​‌ each cylinder three sections​​ are highlighted as circles​​​‌ drawn in black, blue‌ and red. On each‌​‌ circle, regularly spaced dots​​ illustrate the points where​​​‌ measurements are performed.

The collaboration with Cerege​​​‌ on this topic started‌ in the framework of‌​‌ the MagLune ANR project​​ whose overall objective was​​​‌ to devise models to‌ explain how a dynamo‌​‌ phenomenon was possible on​​ the Moon. Our contribution​​​‌ is to design methods‌ to tell, from the‌​‌ measurements provided by the​​ Lunometer, whether the remanent​​​‌ magnetization of the sample‌ under study could be‌​‌ well modeled by a​​ single magnetic dipole, and​​​‌ if so, what would‌ be the position and‌​‌ magnetic moment of this​​ dipole. To this end,​​​‌ we use ideas similar‌ to those underlying the‌​‌ FindSources3D tool (see Sections​​ 3.3 and 7.1.4):​​​‌ we use rational approximation‌ techniques to recover the‌​‌ position of the dipole;​​ recovering the moment is​​​‌ then a rather simple‌ linear problem. The rational‌​‌ approximation solver gives, for​​ each circle of measurements,​​​‌ a partial information about‌ the position of the‌​‌ dipole. These partial informations​​ obtained on all nine​​​‌ circles must then be‌ combined in order to‌​‌ recover the exact position.​​ Theoretically speaking, the nine​​​‌ partial informations are redundant‌ and the position could‌​‌ be obtained by several​​ equivalent techniques. But in​​​‌ practice, due to the‌ fact that the field‌​‌ is not truly generated​​ by a single dipole,​​​‌ and also because of‌ noise in the measurements‌​‌ and numerical errors in​​ the rational approximation step,​​​‌ all methods do not‌ show the same reliability‌​‌ when combining the partial​​ results. We studied several​​​‌ approaches, testing them on‌ synthetic examples, with more‌​‌ or less noise, in​​ order to propose a​​​‌ good heuristic for the‌ reconstruction of the position‌​‌ 70.

4.4 Shape​​ identification of metallic objects​​​‌

We started an academic‌ collaboration with partners at‌​‌ LEAT (Laboratoire d'Électronique, Antennes,​​ Télécommunications, Université Côte d'Azur​​​‌ – CNRS) on the‌ topic of inverse scattering‌​‌ using frequency dependent measurements.​​ As opposed to classical​​​‌ electromagnetic imaging where several‌ spatially located sensors are‌​‌ used to identify the​​ shape of an object​​​‌ by means of scattering‌ data at a single‌​‌ frequency, a discrimination process​​ between different metallic objects​​​‌ is here being sought‌ for by means of‌​‌ a single, or a​​​‌ reduced number of sensors​ that operate on a​‌ whole frequency band. The​​ spatial multiplicity and complexity​​​‌ of antenna sensors is​ here traded against a​‌ simpler architecture performing a​​ frequency sweep.

The setting​​​‌ is shown on Figure​ 4. The total​‌ field $E_t$ is​​ the sum of the​​​‌ incident field $E_{\mathrm{i}n}$ (here a plane​‌ wave) and scattered field​​ $E_s$: at​​​‌ every point $X=(r,\mathrm{\theta ‌},\phi )$ in​​ space we have ${\mathrm{E‌}}_t=E_{\mathrm{i}n}+E_{\mathrm{s‌}}$. A harmonic time​​ dependency ($e^{\mathrm{i‌}\omega t})$,​ is supposed for the​‌ incident wave, so that​​ by linearity of Maxwell​​​‌ equations and after a​ transient state, the scattered​‌ field at the observation​​ point $X_o$ is​​​‌ related to the emitted​ planar wave field at​‌ the emission point ${\mathrm{X}}_e$ via the transfer​​​‌ function $H$: ${\mathrm{E}}_s\left(X_{\mathrm{o‌}}\right)=H(\omega ,X_{\mathrm{o‌}})E_{i\mathrm{n}}(X_e)‌$ (the dependency in ${\mathrm{X}}_e$ is omitted in​​​‌ $H$ since the emission​ point is fixed). Under​‌ regularity conditions on the​​ scatterer's boundary, the function​​​‌ $H$ can be shown​ to admit an analytic​‌ continuation into the complex​​ left half-plane for the​​​‌ $s$ variable, away from​ a discrete set (with​‌ a possible accumulation point​​ at infinity) where it​​​‌ admits poles. Thus, $\mathrm{H}$ is a meromorphic function​‌ in the variable $\mathrm{s}$. Its poles are​​​‌ called the resonating frequencies​ (resonances) of the scattering​‌ object. Recovering these resonating​​ frequencies from frequency scattering​​​‌ measurement, that is measurements​ of $H$ at particular​‌ ${\omega }_j^{\text{'}}\mathrm{s}$ (actually, $i{\omega }_{\mathrm{j‌}}^{\text{'}}s$) is​ the primary objective of​‌ this project.

A schematic​​ view of the problem:​​​‌ an antenna (horizontal and​ on the left) emits​‌ a planar wave (propagating​​ horizontally, towards the right​​​‌ and shown in red)​ with an emitted electric​‌ field $E$ perpendicular to​​ it (represented by a​​​‌ vertical arrow on the​ figure). On the right,​‌ a sphere of radius​​ $a$ reflects the wave.​​​‌ The reflected wave is​ radial to the sphere,​‌ directed towards its exterior,​​ and is shown in​​​‌ blue. A reception antenna​ points towards the center​‌ of the sphere, and​​ is located at a​​​‌ distance $r$ of the​ center of the sphere,​‌ while making an angle​​ $\theta$ with respect to​​​‌ the horizontal. It measures​ the reflected electric field,​‌ which is perpendicular to​​ it.

We started a​​​‌ study of the particular​ case when the scatterer​‌ is a spherical PEC​​ (Perfectly Electric Conductor). In​​​‌ this case, Maxwell equations​ can be solved by​‌ means of expansions in​​ series of vectorial spherical​​​‌ harmonics. We showed in​ particular that in this​‌ case $H$ admits a​​ simple structure involving a​​​‌ meromorphic function with poles​ at zeroes of the​‌ spherical Hankel functions and​​ their derivatives. Identification procedures,​​ surprisingly close to the​​​‌ ones we developed in‌ connection with amplifier stability‌​‌ analysis (Section 4.6),​​ were studied to gain​​​‌ information about the resonating‌ frequencies by means of‌​‌ a rational approximation of​​ this function 86.​​​‌

In order to perform‌ the rational approximation (see‌​‌ Section 3.3), the​​ behavior of $H$ outside​​​‌ the range of measured‌ frequencies, specifically at high‌​‌ frequencies, has been studied​​ for the particular case​​​‌ when the scatterer is‌ a spherical PEC (Perfectly‌​‌ Electric Conductor). In this​​ case, $H$ can be​​​‌ written as $H=‌H_O+{\mathrm{H‌‌}}_C$, where ${\mathrm{H}}_O$ and $H_{\mathrm{C‌}}$ are respectively the optic‌ and creeping wave parts.‌​‌ Their high-frequency behaviors are​​ given by series expansions​​​‌ whose coefficients are identified‌ when $X_o=‌‌X_e$. The​​ asymptotics of $H_{\mathrm{O‌}}$ is called the Luneberg-Kline‌ expansion; its first terms‌​‌ were analytically computed in​​ 25 (solving eikonal and​​​‌ transport equations).

Numerical simulations‌ showed that even though‌​‌ $H_C$ is negligible​​ with respect to ${\mathrm{H‌}}_O$ at high frequencies,‌ it needs to be‌​‌ taken into account around​​ the band of measured​​​‌ frequencies for the rational‌ approximation. Furthermore, the physical‌​‌ interpretation of these two​​ terms leads to consider​​​‌ that $H_C$ should‌ carry more information about‌​‌ the scatterer and we​​ want to investigate the​​​‌ conjecture that the poles‌ of $H$ are those‌​‌ of $H_C$ hence​​ that $H_O$ is​​​‌ analytic.

The rational approximation‌ of the transfer function‌​‌ $H$ is performed with​​ a least-squares substitute to​​​‌ multi-point Padé approximation: the‌ approximant $R_{k,‌‌n}[H]=p_{k,‌n}/q_{\mathrm{k‌},n}$ is given‌​‌ by:

where $n$, $\mathrm{k}$ and $N$ are integers​​​‌ such that $k+‌n+1\le ‌‌N$ and ${\left({\mathrm{z}}_j\right)}_{j=‌1...‌N}$ is a collection‌​‌ of points at which​​ $H$ is analytic. Here,​​​‌ ${𝒫}_n$ is the‌ set of polynomials of‌​‌ degree less than $\mathrm{n}$ and ${𝒫}_k^{\mathrm{1‌}}$ is the set of‌ monic polynomials of degree‌​‌ $k$. An analog​​ of the Nuttal-Pommerenke theorem​​​‌ for a least square‌ version of classical Padé‌​‌ approximants was obtained in​​ 25, where the​​​‌ values $(p-‌Hq)(‌‌z_j)$ in​​ Equation (1)​​​‌ get replaced by the‌ $j$-th coefficients of‌​‌ the Taylor expansion of​​ $(p-\mathrm{H‌}q)$ at a‌ given point in the‌​‌ domain of analyticity of​​ $H$. It says​​​‌ that if $H$ is‌ holomorphic and single-valued on‌​‌ $ℂ$, except perhaps​​ on a polar set,​​​‌ then $p/\mathrm{q‌}$ converges to $H$ in‌​‌ capacity as $k$,​​​‌ $n$ go to infinity​ as well as $\mathrm{N‌}$, in such a​​ way that $n/‌k$ remains bounded and​ $N\le C(‌k+n)$ for some constant $\mathrm{C‌}>0$; convergence​ in capacity means that​‌ for each $\epsilon >0$, the capacity​​​‌ of the set where​ the pointwise error is​‌ bigger than $\epsilon$ goes​​ to zero.

Dwelling on​​​‌ this work, we recently​ showed under similar conditions​‌ on $k$, $\mathrm{n}$, $N$, and​​​‌ provided that the interpolation​ points $z_i$ remain​‌ in a bounded set,​​ that convergence in capacity​​​‌ still holds for the​ solutions to Equation (​‌1); this is​​ a least square analog​​​‌ of Wallin's theorem in​ multi-point Padé approximation.

This​‌ result is interesting as​​ it entails that poles​​​‌ of $H$ can indeed​ be retrieved as limit​‌ points of certain poles​​ of $p_{k,‌n}/q_{\mathrm{k},n}$, while​‌ explaining the chaotic behavior​​ of other, so-called spurious​​​‌ poles that wander about​ the domain of analyticity.​‌ We plan to investigate​​ other PEC scatterers.

4.5​​​‌ Inverse problems in orthopedic​ surgery

Apart from more​‌ classical medical imaging domains,​​ inverse problems find a​​​‌ rather surprising application in​ the field of orthopedics,​‌ see 58, 68​​, 71 and Section​​​‌ 9.2.

We are​ concerned, in particular, with​‌ a hip prosthetic surgery​​ when an insertion of​​​‌ an acetabular cup (AC)​ implant into a bone​‌ by press-fit is performed​​ with the use of​​​‌ an instrumented hammer. Such​ a hammer is equipped​‌ with a sensor capable​​ to measure impact momentum​​​‌ (force) and hence yield​ important information about the​‌ bone quality and the​​ stability of an AC​​​‌ implant. These are, indeed,​ crucial pieces of information​‌ to have in real-time​​ during a surgery. On​​​‌ the one hand, if​ the achieved bone-implant contact​‌ area is not sufficiently​​ large, osseointegration may fail​​​‌ eventually leading to an​ aseptic loosening of the​‌ implant and a necessity​​ of a revision surgery.​​​‌ On the other hand,​ if the insertion of​‌ the implant is too​​ deep, the generated stresses​​​‌ may induce fractures or​ bone tissue necrosis.

The​‌ mathematical side of the​​ process is far from​​​‌ trivial. First of all,​ contact mechanics is a​‌ highly nonlinear problem due​​ to geometrical constraint on​​​‌ the solution. Already a​ basic problem of an​‌ elastic body on a​​ rigid foundation is a​​​‌ free-boundary problem with an​ unknown effective contact surface​‌ which is characterized by​​ the so-called Signorini conditions,​​​‌ nonlinear constraints of Karush-Kuhn-Tucker​ type involving stress and​‌ displacements. In the present​​ case, several coupled problems​​​‌ have to be solved​ since regions corresponding to​‌ the bone, the implant​​ and the hammer all​​​‌ possess different material properties.​ Moreover, the deformations cannot​‌ be considered small, consequently,​​ a hypo-elastic description is​​​‌ more appropriate than that​ of linear elasticity. In​‌ such a formulation, a​​ rate relation between Cauchy​​​‌ stress and strain tensors​ replaces a linear stress-strain​‌ constitutive law, hence its​​ integration induces additional non-linearity.​​ Finally, the bone is​​​‌ a porous multi-scale medium,‌ and appropriate homogenization model‌​‌ should be deduced, with​​ adequate parameter fitting.

For​​​‌ tackling inverse problems (‌e.g., that of‌​‌ determining material parameters), the​​ direct formulation has to​​​‌ be solved in such‌ an effective way that‌​‌ iterative approaches are not​​ prohibitively expensive. This motivates​​​‌ exploration of model-order reduction‌ strategies that would, in‌​‌ particular, allow efficient integration​​ of the system in​​​‌ time.

4.6 Stability and‌ design of active devices‌​‌

Through contacts with CNES​​ (Toulouse) and UPV (Bilbao),​​​‌ the team got involved‌ in the design of‌​‌ amplifiers which, unlike filters,​​ are active devices. A​​​‌ prominent issue here is‌ stability. Twenty years ago,‌​‌ it was not possible​​ to simulate unstable responses,​​​‌ and only after building‌ a device could one‌​‌ detect instability. The advent​​ of so-called harmonic balance​​​‌ techniques, which compute steady‌ state responses of linear‌​‌ elements in the frequency​​ domain and look for​​​‌ a periodic state in‌ the time domain of‌​‌ a network connecting these​​ linear elements via static​​​‌ non-linearities made it possible‌ to compute the harmonic‌​‌ response of a (possibly​​ nonlinear and unstable) device​​​‌ 84. This has‌ had tremendous impact on‌​‌ design, and there is​​ a growing demand for​​​‌ software analyzers. In this‌ connection, there are two‌​‌ types of stability involved.​​ The first is stability​​​‌ of a fixed point‌ around which the linearized‌​‌ transfer function accounts for​​ small signal amplification. The​​​‌ second is stability of‌ a limit cycle which‌​‌ is reached when the​​ input signal is no​​​‌ longer small and truly‌ nonlinear amplification is attained‌​‌ (e.g., because​​ of saturation).

Initial applications​​​‌ by the team have‌ been concerned with the‌​‌ first type of stability,​​ and emphasis was put​​​‌ on defining and extracting‌ the “unstable part” of‌​‌ the response. We showed​​ that under realistic dissipativity​​​‌ assumptions at high frequency‌ for the building blocks‌​‌ of the circuit, the​​ linearized transfer functions are​​​‌ meromorphic in the complex‌ frequency variable $s$,‌​‌ with at most finitely​​ many unstable poles in​​​‌ the right half-plane 3‌. Dwelling on the‌​‌ unstable/stable decomposition in Hardy​​ Spaces, we developed a​​​‌ procedure to assess the‌ stability or instability of‌​‌ the transfer functions at​​ hand, from their evaluation​​​‌ on a finite frequency‌ grid 54, that‌​‌ was further improved in​​ 53 to address the​​​‌ design of oscillators. This‌ has resulted in the‌​‌ development of a software​​ library called Pisa6​​​‌, aiming at making‌ these techniques available to‌​‌ practitioners. Extending this methodology​​ to the strong signal​​​‌ case, where linearization is‌ considered around a periodic‌​‌ trajectory, is considerably more​​ difficult and has received​​​‌ much attention by the‌ team in recent years.‌​‌ The exponential stability of​​ the high frequency limit​​​‌ of a circuit was‌ established in 6,‌​‌ implying that there are​​ at most finitely many​​​‌ unstable poles and no‌ other unstable singularity for‌​‌ the monodromy operator around​​ the cycle. Furthermore, the​​​‌ links between the monodromy‌ operator and the (operator-valued)‌​‌ “harmonic transfer function” (HTF)​​​‌ of the linearized system​ along the trajectory were​‌ brought to light in​​ 10: the system​​​‌ is exponentially stable if​ and only if its​‌ HTF is bounded an​​ analytic in a right​​​‌ half-plane of $ℂ$ of​ the form $\{\mathrm{\Re ‌}z>-\mathrm{\epsilon }\}$ for some $\mathrm{\epsilon ‌}>0$. We​ deal here with input-output​‌ system of the form:​​

where ${\tau }_1<‌\cdots <{\tau }_{\mathrm{N}}$ are positive delays and​‌ $D_1\left(\mathrm{t}\right),...,‌D_N\left(\mathrm{t}\right)$ real $d\times ‌d$ matrices depending periodically​​ on time $t$,​​​‌ while $u$ is the​ ${ℝ}^d$-valued input​‌ and $y\left(\mathrm{t}\right)$ the ${ℝ}^{\mathrm{d‌}}$-valued output (complex coefficients​ can of course be​‌ handled in the same​​ way). The HTF, that​​​‌ generalizes the usual transfer​ function of linear constant​‌ systems, is an analytic​​ function of the Laplace-Fourier​​​‌ variable which is valued​ in the space of​‌ operators on $L^{\mathrm{2}}([0,‌T),{\mathrm{ℂ}}^d)$ ($\mathrm{T‌}$ is the period of​​ the system). It can​​​‌ be defined as follows:​ if a periodic linear​‌ control system at rest​​ is fed from initial​​​‌ instant $t_i=-\infty$ with an​‌ input signal $v(t)e^{\mathrm{x‌}t}{e}^{i\mathrm{\nu }t}$ where $v$ is​‌ $T$-periodic and $\mathrm{x}>0$ is large​​​‌ enough, then the output​ is of the form​‌ $𝐇(x+i\nu )(‌v)e^{\mathrm{x}t}{e}^{i\mathrm{\nu ‌}t}$, where $\mathrm{𝐇}(x+\mathrm{i‌}\nu )$ is the​ harmonic transfer evaluated at​‌ $x+i\mathrm{\nu }$ (an operator) and $\mathrm{𝐇‌}(x+\mathrm{i}\nu )\left(\mathrm{v‌}\right)$ is the $\mathrm{T}$-periodic function which is​​​‌ the image of the​ $T$-periodic function $\mathrm{v‌}$ under $𝐇(\mathrm{x}+i\nu )‌$. That is to​ say, an exponentially modulated​‌ input wave of frequency​​ $\nu$ carried by a​​​‌ $T$-periodic signal is​ mapped to an output​‌ of the same form,​​ and the HTF maps​​​‌ the input carrier to​ the output carrier. For​‌ stable systems, this is​​ asymptotically true if the​​​‌ initial time is a​ finite instant; otherwise, the​‌ unstable transient will prevent​​ this mathematical solution from​​​‌ being physical. Other definitions​ are given in 10​‌.

We were able​​ to construct a simple​​​‌ nonlinear circuit whose linearization​ around a periodic trajectory​‌ has a spectrum containing​​ a whole circle; we​​​‌ currently investigate whether the​ singularities of the Fourier​‌ coefficients of the HTF​​ (that are themselves analytic​​​‌ functions) also contain that​ circle. Indeed, just like​‌ a series of functions​​ may diverge even though​​​‌ the summands are smooth,​ it is a priori​‌ possible that the HTF​​ has a singularity at​​ a point whereas its​​​‌ Fourier coefficients do not.‌ Note that these coefficients‌​‌ are all one can​​ estimate by harmonic balance​​​‌ techniques, and therefore the‌ above question is of‌​‌ great practical relevance. We​​ also investigate the relation​​​‌ between our stability criterion‌ (that the HTF should‌​‌ be bounded and analytic​​ on a “vertical” half-plane​​​‌ containing the origin), and‌ the weaker requirement that‌​‌ the HTF exists pointwise​​ in a half-plane. Connections​​​‌ with the representation of‌ Volterra equations with jumps‌​‌ in the kernel are​​ also a motivation for​​​‌ such a study, see‌ 33.

4.7 Tools‌​‌ for numerically guaranteed computations​​

The overall and long-term​​​‌ goal is to enhance‌ the quality of numerical‌​‌ computations. This includes developing​​ algorithms whose convergence is​​​‌ proved not only when‌ assuming that the numerical‌​‌ computations are performed in​​ exact real or complex​​​‌ arithmetic, but rather when‌ really accounting for the‌​‌ fact that the computations​​ are performed with an​​​‌ inexact arithmetic (usually floating-point‌ arithmetic). A numerical result‌​‌ alone is of little​​ interest if no rigorous​​​‌ bound is provided together‌ with it, in order‌​‌ to ensure that the​​ real theoretical result is​​​‌ proved to be not‌ to far from the‌​‌ computed result.

A specific​​ way of contributing to​​​‌ this objective is to‌ develop efficient numerical implementations‌​‌ of mathematical functions with​​ rigorous bounds. We do​​​‌ sometimes provide such implementations.‌ The software tool Sollya‌​‌ (see Section 7.1.1),​​ developed together with Christoph​​​‌ Lauter (University of Texas‌ at El Paso, UTEP)‌​‌ is also an achievement​​ of the team in​​​‌ that respect. This tool‌ intends to provide an‌​‌ interactive environment for performing​​ numerically rigorous computations. Sollya​​​‌ comes as a standalone‌ tool and also as‌​‌ a C library that​​ allows one to benefit​​​‌ from all the features‌ of the tool in‌​‌ C programs.

6.1 Awards​‌

Mubasharah Khalid Omer was​​ awarded a poster prize​​​‌ at the 5th edition​ of Complex Days,​‌ Nice (February), see Section​​ 10.1.4 and 21.​​​‌

6.2 Working conditions

One​ of our PhD students​‌ got serious health problems​​ in 2024, that led​​​‌ to a temporary interruption​ of his progress. As​‌ he was being back​​ to work beginning of​​​‌ 2025, after recovering thanks​ to medical care, he​‌ has been subject to​​ a decision7 from​​​‌ the prefecture (Alpes-Maritimes) to​ leave the French territory​‌ (“obligation de quitter le​​ territoire français, OQTF”), hence​​​‌ forced back to Morocco​ (his birth country). Granted​‌ such poor conditions, it​​ turned out that the​​​‌ completion of his PhD​ was impossible, despite the​‌ fact that we all​​ had hoped for a​​​‌ more favorable end of​ his stay.

7.1 Latest software​​​‌ developments

8.1 Field extrapolation for​​​‌ inverse magnetization problem and​ beyond

Participants: Axel Knecht​‌, Juliette Leblond,​​ Mubasharah Khalid Omer,​​ Dmitry Ponomarev.

Motivation​​​‌ of the field extrapolation‌ problem has been described‌​‌ in the last year's​​ activity report of the​​​‌ team.

One of the‌ approaches, based on regularized‌​‌ deconvolution (see Section 4.2​​), is the topic​​​‌ of the PhD thesis‌ work of Mubasharah Khalid‌​‌ Omer . Over the​​ last year not only​​​‌ different basis functions were‌ explored, but also discretization,‌​‌ various ways of choosing​​ the regularization parameter and​​​‌ alternative projection and iterative‌ formulations for the solution‌​‌ of the normal equation.​​ Further, numerical tests validated​​​‌ the applicability of this‌ approach for volumetric magnetizations.‌​‌

A second method, the​​ so-called double-spectral vector approach​​​‌ 75, which preserves‌ the magnetization localization, has‌​‌ been revised, with its​​ altered version explored (when​​​‌ the role of the‌ kernel functions entering it‌​‌ was swapped). Moreover, alternative​​ scalar spectral approaches have​​​‌ been proposed based on‌ a similar idea of‌​‌ individual continuation of eigenfunctions​​ of appropriate integral operators.​​​‌ Comparison has been made‌ and the paper outlining‌​‌ extrapolation methods based on​​ such spectral methods is​​​‌ currently in preparation.

Furthermore,‌ we investigated a possibility‌​‌ of taking advantage of​​ the auxiliary quantities that​​​‌ were used in each‌ case to generate the‌​‌ extrapolated field. Namely, it​​ became clear that, in​​​‌ the case of planar‌ source support (or very‌​‌ thin sample), both the​​ divergence of the tangential​​​‌ magnetization and the vertical‌ component of magnetization can‌​‌ indeed be reconstructed, either​​ directly (as for the​​​‌ double-spectral approach) or by‌ means of solving an‌​‌ additional ill-posed problem (as​​ with the deconvolution approach).​​​‌ The latter indicates that‌ the extrapolation approach can‌​‌ indeed be considered as​​ a flexible first step​​​‌ of the full inverse‌ magnetization problem. The aforementioned‌​‌ auxiliary quantities can also​​ be used to directly​​​‌ calculate the magnetization net‌ moment components, which provides‌​‌ an alternative bypassing the​​ use of the asymptotic​​​‌ formulas with the extrapolated‌ field.

We note that‌​‌ the knowledge of the​​ tangential (two-dimensional) divergence of​​​‌ the planar magnetization allows‌ the reconstruction of the‌​‌ magnetization itself modulo silent​​ sources whose structure is​​​‌ well-understood (see Sections 3.2.2‌ and 4.2 and 31‌​‌). Two constructive strategies​​ for recovering the unique​​​‌ magnetization of minimal quadratic‌ norm equivalent to the‌​‌ given one (i.e.​​ with the same tangential​​​‌ divergence and the normal‌ component) have been investigated‌​‌ in the internship work​​ of Axel Knecht .​​​‌ In particular, one of‌ these strategies was shown‌​‌ to be especially computationally​​ efficient due to the​​​‌ local formulation of the‌ problem (avoiding repetitive evaluations‌​‌ of auxiliary integral operators​​ over the entire plane).​​​‌

8.2 Net moment estimation‌

Participants: Sylvain Chevillard,‌​‌ Juliette Leblond, Jean-Paul​​ Marmorat, Dmitry Ponomarev​​​‌, Fatima Swaydan,‌ Anass Yousfi.

We‌​‌ continued the work started​​ in the past years​​​‌ to establish formulas for‌ the integrals of the‌​‌ form

where $P$ is a​​ polynomial, $j\in \{‌1,2,‌3\}$, and‌​‌ the domain of integration​​​‌ is either the square​ $Q_R$ (and more​‌ generally, any rectangle) or​​ the disk $D_{\mathrm{R‌}}$ of radius $R$ (the​ notations are those of​‌ Section 4.2, see​​ especially Figure 2).​​​‌ While we were historically​ focusing on the case​‌ $j=3$ because​​ it corresponds to the​​​‌ measurements obtained with the​ SQUID, we now try​‌ to get as much​​ results as possible to​​​‌ be valid for any​ component $j\in \{‌1,2,3\}$, in​​​‌ view of the measurements​ that could be soon​‌ obtained with the QDM.​​

In the case of​​​‌ the disk, Equation (​2) does not​‌ admit an exact explicit​​ expression. However Dmitry Ponomarev​​​‌ described in 80,​ a few years ago,​‌ asymptotic expansions of moderate​​ order with respect to​​​‌ variable $R$ as it​ grows large in the​‌ case when $j=3$ and $P$ has​​​‌ degree less than 2.​ On the one hand,​‌ these results have been​​ extended to higher orders​​​‌ and higher degrees and​ published this year in​‌ 13. On the​​ other hand, Anass Yousfi​​​‌ contributed a considerably simpler​ proof of these formulas​‌ during his PhD, while​​ extending the results to​​​‌ any component $B_{\mathrm{j}}$. They are available​‌ in the research report​​ 18.

Even though​​​‌ a technique for obtaining​ estimates of arbitrary high​‌ order was devised, it​​ was also observed in​​​‌ 13 that such high-order​ estimates are more sensitive​‌ to measurement noise and​​ thus might be of​​​‌ a little value in​ practice. Mid-order estimates were​‌ numerically shown to be​​ the best in a​​​‌ situation mimicking the realistic​ set-up where the data​‌ are corrupted by noise​​ and are not available​​​‌ over a large region.​ Remarkable features of the​‌ derived asymptotic estimates are​​ that they are applicable​​​‌ for fairly general magnetization​ distributions (say, a combination​‌ of dipoles) and that​​ their application does not​​​‌ require the knowledge of​ the distance from the​‌ sample to the measurement​​ plane.

The case when​​​‌ the domain is a​ rectangle allows for more​‌ detailed results than just​​ asymptotic estimates. More specifically,​​​‌ denoting by abuse of​ notations $B_j$ the​‌ (forward) operator from ${\mathrm{L}}^2(S)‌$ to $L^2(Q_R)$ that​‌ maps a magnetization to​​ the corresponding component of​​​‌ the field ($\mathrm{j}\in \{1,‌2,3\}$), its adjoint operator​​​‌ $B_j^{☆}$ plays​ an important role for​‌ the estimation of the​​ net moment from measured​​​‌ data since the quality​ of a linear estimator​‌ is good when its​​ image by the adjoint​​​‌ operator is close to​ be a characteristic function.​‌ This year, we completely​​ wrote down computations that​​​‌ were only sketched in​ a conference paper in​‌ 2024: we showed that​​ the adjoint operator admits​​​‌ nice integral representations and,​ pushing forward similar computations​‌ done by Jung for​​ small degrees 63,​​​‌ we derived exact and​ explicit expressions for the​‌ image of any polynomial​​ by the adjoint operator​​ 17. These expressions​​​‌ can be used either‌ to straightforwardly get asymptotic‌​‌ formulas of the same​​ kind as those described​​​‌ above in the disk‌ geometry, or more generally,‌​‌ can be handy in​​ any context when the​​​‌ evaluation of the adjoint‌ operator is important, such‌​‌ as, e.g., to​​ numerically determine the solution​​​‌ of the extremal problem‌ presented in 4.‌​‌

We also started to​​ take advantage of the​​​‌ 3-component field measurements provided‌ by QDM for the‌​‌ reconstruction of net moments​​ through the solution of​​​‌ an auxiliary bounded extremal‌ problem, see 4.‌​‌ This is the topic​​ of the PhD thesis​​​‌ of Fatima Swaydan .‌ Some preliminary results are‌​‌ reported in 20,​​ where a linear combination​​​‌ of 3 components of‌ the magnetic field is‌​‌ considered as data, and​​ a numerical study of​​​‌ the spectral properties of‌ the corresponding operator was‌​‌ performed. The follow-up considerations​​ will concern further exploration​​​‌ of such linear combinations‌ for net moment estimation,‌​‌ depending also on the​​ considered constraints or regularization​​​‌ term and parameters, in‌ the underlying bounded extremal‌​‌ problem.

8.3 Inverse conductivity​​ estimation problems in cerebral​​​‌ imaging

Participants: Juliette Leblond‌, Jean-Paul Marmorat,‌​‌ Rui Martins, Dmitry​​ Ponomarev.

For the​​​‌ spherical and piecewise constant‌ conductivity head model (see‌​‌ Section 4.1), we​​ now consider together with​​​‌ Marion Darbas (LAGA, Université‌ Sorbonne Paris Nord) the‌​‌ issue of simultaneous recovery​​ of dipolar source terms​​​‌ and of the intermediate‌ (skull) conductivity value from‌​‌ partial Dirichlet-Neumann data on​​ the outermost boundary (EEG​​​‌ setup), following 51,‌ 52. We were‌​‌ able to establish a​​ uniqueness result and an​​​‌ iterative alternate estimation procedure.‌ Preliminary numerical computations from‌​‌ synthetic data using the​​ software FS3D (see Section​​​‌ 7.1.4) are encouraging.‌ Stability, robustness, and convergence‌​‌ properties, remain under study.​​

A quite new imaging​​​‌ modality is MDEIT (Magnetic‌ Detection Electrical Impedance Tomography),‌​‌ where upon injection of​​ an electrical current pattern​​​‌ through surface electrodes, the‌ response of the media‌​‌ is probed through a​​ measurement of the produced​​​‌ magnetic field (instead of‌ measuring the electrical potential).‌​‌ Preliminary simulations predict better​​ results than the conventional​​​‌ electrical impedance tomography (EIT)‌ approach. Moreover, it was‌​‌ already shown that measuring​​ all three components of​​​‌ the magnetic field leads‌ to much better conditioning‌​‌ and thus much faster​​ convergence of iterative schemes​​​‌ for the conductivity reconstruction.‌ This was the topic‌​‌ of the internship of​​ Rui Martins and remains​​​‌ an on-going work with‌ him and colleagues at‌​‌ University College London (UCL).​​

8.4 General Inverse Source​​​‌ Problems in Divergence Form‌

Participants: Laurent Baratchart.‌​‌

Inverse source problems with​​ source term in divergence​​​‌ form consist, roughly speaking,‌ in finding a vector‌​‌ field with prescribed support​​ whose divergence is the​​​‌ Laplacian of a potential,‌ given some measurements of‌​‌ that potential (see Section​​ 3.2). Here the​​​‌ Laplacian could be either‌ the usual Euclidean one‌​‌ or a more general,​​ elliptic operator in divergence​​​‌ form arising as the‌ Laplacian of some Riemannian‌​‌ metric. From the point​​​‌ of view of geometric​ analysis, such questions amount​‌ to retrieving the divergence​​ free term in the​​​‌ Helmholtz decomposition of a​ vector field, knowing the​‌ gradient term on some​​ observation set $Q$ and​​​‌ a superset $S$ of​ the support of the​‌ unknown vector field. Such​​ problems appear in various​​​‌ contexts including Geomagnetism, Paleomagnetism,​ as well as Medical​‌ Imaging from EEG and​​ Electro-Cardiography (ECG) and are​​​‌ severely ill-posed with non-unique​ solutions, making regularization techniques​‌ an essential aspect of​​ every approach.

If the​​​‌ unknown vector field is​ denoted by $m$,​‌ the potential $\varphi =\varphi (m)‌$ satisfies $\mathrm{div}\left(\mathrm{\Sigma }\nabla \varphi \right)=‌\mathrm{div}m$ on ${\mathrm{ℝ}}^3$ with $\Sigma$ an​​​‌ elliptic symmetric matrix-valued function,​ and it vanishes at​‌ infinity. When $m$ is​​ assumed to be a​​​‌ ${ℝ}^3$-valued compactly​ supported measure (a standard​‌ model in electro/magnetostatics as​​ it includes dipoles), this​​​‌ equation has a wild​ right hand side and​‌ is not standard. We​​ showed this year that​​​‌ it has a unique​ very weak solution vanishing​‌ at infinity, provided $\mathrm{\Sigma }$ is uniformly elliptic and​​​‌ Dini-continuous in a neighborhood​ of the support of​‌ $m$; more generally,​​ we proved such well-posedness​​​‌ when $\Sigma$ is merely​ uniformly elliptic, in some​‌ even weaker sense where​​ test functions depend on​​​‌ $m$. Under slightly​ stronger assumptions, namely when​‌ $\Sigma$ is Dini-continuous in​​ a neighborhood of $\mathrm{S‌}$ and $Q$, then​ the regularized minimization problem​‌

has a unique solution​‌ for fixed $\lambda >0$, with $\mathrm{f‌}$ arbitrary data in ${\mathrm{L}}^2(Q)‌$. This result supersedes​​ a uniqueness result dealing​​​‌ with the Euclidean Laplacian​ and planar samples obtained​‌ in 46.

8.5​​ Rational and meromorphic approximation​​​‌

Participants: Laurent Baratchart,​ Sylvain Chevillard.

Rational​‌ approximation on the circle​​ or the line to​​​‌ functions of constant modulus​ is an old issue;​‌ in the language of​​ system identification and control,​​​‌ it is akin to​ ask how well stationary​‌ delay systems can be​​ represented by finite-dimensional ones,​​​‌ and it has long​ been surmised that such​‌ a representation should be​​ inefficient for all reasonable​​​‌ input-output criteria. In recent​ years, approximation in ${\mathrm{L‌}}^2$-norm, which can​​ be interpreted as minimum​​​‌ variance approximation in a​ probabilistic context, received renewed​‌ interest due to the​​ advent of (linear) neural​​​‌ nets. Still, it seems​ that no quantitative lower​‌ bound for the approximation​​ error is known in​​​‌ terms of the respective​ degrees of the approximated​‌ function and its approximant​​ (see Section 3.3).​​​‌ Through a collaborative research​ effort with Alexander Borichev,​‌ Claire Coiffard and Rachid​​ Zarouf (Aix-Marseille Université), triggered​​​‌ by a question of​ Alexandre François (Sierra team,​‌ centre Inria de Paris),​​ we established such a​​​‌ bound by combining a​ formula expressing the degree​‌ in terms of the​​ $W^{1/\mathrm{2},2}$-norm due​​​‌ to Brézis and new‌ estimates of the Fourier‌​‌ coefficients of Blaschke products.​​

Let us also mention​​​‌ that an article is‌ still under writing on‌​‌ convergence in capacity of​​ least square substitutes to​​​‌ multi-point Padé approximants (‌1) to functions‌​‌ with polar singular set​​ on $ℂ$, a​​​‌ topic that underwent advances‌ last year after groundbreaking‌​‌ work in 25.​​

9.1 International research visitors‌

9.2​​ National initiatives

10.1​‌ Promoting scientific activities

10.2​​​‌ Teaching - Supervision -​ Juries - Educational and​‌ pedagogical outreach

10.3 Popularization‌

• Juliette Leblond and Martine‌​‌ Olivi are members of​​ Terra Numerica. In​​​‌ this capacity, they design‌ workshops (geosciences, fractals, exponential‌​‌ growth, and eco-responsible digital​​ technologies), and participate in​​​‌ several outreach events (science‌ festivals, MathsC2+ training).

10.4 Community​​ services

• Sylvain Chevillard is​​​‌ an elected member of​ the local board of​‌ the works council (AGOS)​​ of Inria, with local​​​‌ treasurer duty. He benefits​ from an officially reduced​‌ working load of 30​​ hours per month for​​​‌ this purpose.
• Juliette Leblond​ is an elected member​‌ of the “Commission Administrative​​ Paritaire (CAP)” and an​​​‌ associated member of the​ “Comité Égalité et Parité​‌ des chances” of Inria.​​

11.1​​​‌ Major publications

• 1 article​A.Anton Arnold,​‌ S.Sjoerd Geevers,​​ I.Ilaria Perugia and​​​‌ D.Dmitry Ponomarev.​ On the limiting amplitude​‌ principle for the wave​​ equation with variable coefficients​​​‌.Communications in Partial​ Differential Equations494​‌April 2024, 333-380​​HALDOIback to​​​‌ text
• 2 articleL.​Laurent Baratchart, A.​‌Alexander Borichev and S.​​Slah Chaabi. Pseudo-holomorphic​​​‌ functions at the critical​ exponent.Journal of​‌ the European Mathematical Society​​1892016,​​​‌ 1919 - 1960HAL​DOIback to text​‌back to text
• 3​​ articleL.Laurent Baratchart​​​‌, S.Sylvain Chevillard​, A.Adam Cooman​‌, M.Martine Olivi​​ and F.Fabien Seyfert​​​‌. Linearized Active Circuits:​ Transfer Functions and Stability​‌.Mathematics in Engineering​​45October 2021​​​‌, 1-18HALDOI​back to text
• 4​‌ articleL.Laurent Baratchart​​, S.Sylvain Chevillard​​​‌, D. P.Douglas​ P. Hardin, J.​‌Juliette Leblond, E.​​ A.Eduardo Andrade Lima​​​‌ and J.-P.Jean-Paul Marmorat​. Magnetic moment estimation​‌ and bounded extremal problems​​.Inverse Problems and​​​‌ Imaging 131February​ 2019, 29HAL​‌DOIback to text​​back to textback​​​‌ to text
• 5 article​L.Laurent Baratchart,​‌ S.Sylvain Chevillard and​​ T.Tao Qian.​​​‌ Minimax principle and lower​ bounds in $H^{\mathrm{2‌}}$-rational approximation.Journal​​ of Approximation Theory206​​​‌2015, 17--47back​ to text
• 6 article​‌L.Laurent Baratchart,​​ S.Sébastien Fueyo,​​​‌ G.Gilles Lebeau and​ J.-B.Jean-Baptiste Pomet.​‌ Sufficient Stability Conditions for​​ Time-varying Networks of Telegrapher's​​​‌ Equations or Difference Delay​ Equations.SIAM Journal​‌ on Mathematical Analysis53​​2021, 1831–1856HAL​​​‌DOIback to text​
• 7 articleL.Laurent​‌ Baratchart, C.Cristobal​​ Villalobos Guillén, D.​​​‌Doug Hardin, M.​Michael Northington and E.​‌Edward Saff. Inverse​​ Potential Problems for Divergence​​ of Measures with Total​​​‌ Variation Regularization.Foundations‌ of Computational Mathematics20‌​‌2020, 1273-1307HAL​​DOIback to text​​​‌back to text
• 8‌ 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​‌, 115003HALDOI​​back to text
• 9​​​‌ articleM.Martine Olivi‌, F.Fabien Seyfert‌​‌ and J.-P.Jean-Paul Marmorat​​. Identification of microwave​​​‌ filters by analytic and‌ rational H2 approximation.‌​‌Automatica492January​​ 2013, 317-325HAL​​​‌DOIback to text‌

11.2 Publications of the‌​‌ year

11.3​​ Cited publications

• 22 article​​​‌D.Daniel Appelo,​ F.Fortino Garcia and​‌ O.Olof Runborg.​​ WaveHoltz: Iterative solution of​​​‌ the Helmholtz equation via​ the wave equation.​‌SIAM Journal on Scientific​​ Computing4242020​​​‌, A1950--A1983back to​ text
• 23 articleA.​‌Anton Arnold, S.​​Sjoerd Geevers, I.​​​‌Ilaria Perugia and D.​Dmitry Ponomarev. An​‌ adaptive finite element method​​ for high-frequency scattering problems​​​‌ with smoothly varying coefficients​.Computers & Mathematics​‌ with Applications1092022​​, 1--14back to​​​‌ text
• 24 articleA.​Anton Arnold, S.​‌Sjoerd Geevers, I.​​Ilaria Perugia and D.​​​‌Dmitry Ponomarev. On​ the exponential time-decay for​‌ the one-dimensional wave equation​​ with variable coefficients.​​​‌Communications on Pure and​ Applied Analysis2110​‌2022, 3389back​​ to text
• 25 thesis​​​‌P.Paul Asensio.​ Inverse problems of source​‌ localization with applications to​​ EEG and MEG.​​​‌Université Côte d'AzurSeptember​ 2023HALback to​‌ textback to text​​back to textback​​​‌ to textback to​ textback to text​‌
• 26 articleB.Bilal​​ Atfeh, L.Laurent​​​‌ Baratchart, J.Juliette​ Leblond and J. R.​‌Jonathan R. Partington.​​ Bounded extremal and Cauchy-Laplace​​​‌ problems on the sphere​ and shell.J.​‌ Fourier Anal. Appl.16​​2Published online Nov.​​​‌ 20092010, 177--203​URL: http://dx.doi.org/10.1007/s00041-009-9110-0back to​‌ text
• 27 bookG.​​ A.George A. Baker​​​‌ and P.Peter Graves-Morris​. Padé approximants.​‌Cambridge University Press2010​​back to text
• 28​​​‌ articleL.Laurent Baratchart​, L.Laurent Bourgeois​‌ and J.Juliette Leblond​​. Uniqueness results for​​​‌ inverse Robin problems with​ bounded coefficient.Journal​‌ of Functional Analysis2016​​HALDOIback to​​​‌ textback to text​back to text
• 29​‌ articleL.Laurent Baratchart​​, M.M. Cardelli​​​‌ and M.Martine Olivi​. Identification and rational​‌ $L^2$ approximation: a​​ gradient algorithm.Automatica​​​‌271991, 413--418​back to text
• 30​‌ articleL.Laurent Baratchart​​, S.Sylvain Chevillard​​​‌, J.Juliette Leblond​, E. A.Eduardo​‌ Andrade Lima and D.​​Dmitry Ponomarev. Asymptotic​​​‌ method for estimating magnetic​ moments from field measurements​‌ on a planar grid​​.HAL preprint: hal-01421157​​​‌2018back to text​back to text
• 31​‌ articleL.Laurent Baratchart​​, S.Sylvain Chevillard​​​‌ and J.Juliette Leblond​. Silent and equivalent​‌ magnetic distributions on thin​​ plates.Theta Series​​ in Advanced Mathematics, The​​​‌ Theta Foundationhttp://hal.archives-ouvertes.fr/hal-012861172017‌, 11-27back to‌​‌ text
• 32 articleL.​​Laurent Baratchart, Y.​​​‌Yannick Fischer and J.‌Juliette Leblond. Dirichlet/Neumann‌​‌ problems and Hardy classes​​ for the planar conductivity​​​‌ equation.Complex Variables‌ and Elliptic Equations2014‌​‌, 41HALDOI​​back to textback​​​‌ to text
• 33 article‌L.Laurent Baratchart,‌​‌ S.Sébastien Fueyo and​​ J.-B.Jean-Baptiste Pomet.​​​‌ Integral representation formula for‌ linear non-autonomous difference-delay equations‌​‌.Journal of Integral​​ Equations and Applications36​​​‌42024HALDOI‌back to text
• 34‌​‌ articleL.Laurent Baratchart​​, C.Christian Gerhards​​​‌ and A.Alexander Kegeles‌. Decomposition of L2-vector‌​‌ fields on Lipschitz surfaces:​​ characterization via null-spaces of​​​‌ the scalar potential.‌SIAM Journal on Mathematical‌​‌ Analysis5342021​​, 4096 - 4117​​​‌HALDOIback to‌ textback to text‌​‌
• 35 articleL.Laurent​​ Baratchart, C.Christian​​​‌ Gerhards, A.Alexander‌ Kegeles and P.Peter‌​‌ Menzel. Unique reconstruction​​ of simple magnetizations from​​​‌ their magnetic potential.‌Inverse Problems3710‌​‌9 2021, 105006​​HALDOIback to​​​‌ text
• 36 articleL.‌Laurent Baratchart, D.‌​‌ P.Douglas P. Hardin​​, E. A.Eduardo​​​‌ Andrade Lima, E.‌ d.Edwar dB. Saff‌​‌ and B.Benjamin Weiss​​. Characterizing kernels of​​​‌ operators related to thin-plate‌ magnetizations via generalizations of‌​‌ Hodge decompositions.Inverse​​ Problems2912013​​​‌, URL: https://inria.hal.science/hal-00919261DOI‌back to textback‌​‌ to textback to​​ textback to text​​​‌
• 37 articleL.Laurent‌ Baratchart and J.Juliette‌​‌ Leblond. Hardy approximation​​ to $L^p$ functions​​​‌ on subsets of the‌ circle with $1\mathrm{p‌‌}<$.Constructive Approximation​​141998, 41--56​​​‌back to textback‌ to text
• 38 inproceedings‌​‌L.Laurent Baratchart,​​ J.Juliette Leblond,​​​‌ E. A.Eduardo Andrade‌ Lima and D.Dmitry‌​‌ Ponomarev. Magnetization moment​​ recovery using Kelvin transformation​​​‌ and Fourier analysis.‌Journal of Physics: Conference‌​‌ Series9041IOP​​ Publishing2017, 012011​​​‌back to textback‌ to text
• 39 article‌​‌L.Laurent Baratchart,​​ J.Juliette Leblond and​​​‌ J.-P.Jean-Paul Marmorat.‌ Sources identification in 3D‌​‌ balls using meromorphic approximation​​ in 2D disks.​​​‌Electronic Transactions on Numerical‌ Analysis (ETNA)252006‌​‌, 41--53back to​​ text
• 40 articleL.​​​‌Laurent Baratchart, J.‌Juliette Leblond and J.‌​‌ n.Jonatha nR. Partington​​. Hardy approximation to​​​‌ $L^{}$ functions on subsets‌ of the circle.‌​‌Constructive Approximation121996​​, 423--435back to​​​‌ text
• 41 inproceedingsL.‌Laurent Baratchart, J.‌​‌Juliette Leblond and D.​​Dmitry Ponomarev. Solution​​​‌ of a homogeneous version‌ of Love type integral‌​‌ equation in different asymptotic​​ regimes.International Conference​​​‌ on Integral Methods in‌ Science and EngineeringSpringer‌​‌2019, 67--79back​​ to textback to​​​‌ text
• 42 articleL.‌Laurent Baratchart, J.‌​‌Juliette Leblond, S.​​Stéphane Rigat and E.​​​‌Emmanuel Russ. Hardy‌ spaces of the conjugate‌​‌ Beltrami equation.Journal​​​‌ of Functional Analysis259​22010, 384-427​‌URL: http://dx.doi.org/10.1016/j.jfa.2010.04.004back to​​ textback to text​​​‌
• 43 unpublishedL.Laurent​ Baratchart, D.Dang​‌ Pei and T.Tao​​ Qian. Hardy-Hodge decomposition​​​‌ of vector fields on​ compact Lipschitz hypersurfaces.,​‌ In preparationback to​​ text
• 44 articleL.​​​‌Laurent Baratchart and F.​Fabien Seyfert. An​‌ $L^p$ analog to​​ AAK theory for $\mathrm{p‌}2$.Journal of​ Functional Analysis1911​‌2002, 52--122back​​ to text
• 45 article​​​‌L.Laurent Baratchart,​ H.Herbert Stahl and​‌ M.Maxim Yattselev.​​ Weighted Extremal Domains and​​​‌ Best Rational Approximation.​Advances in Mathematics229​‌2012, 357-407URL:​​ http://hal.inria.fr/hal-00665834back to text​​​‌
• 46 articleL.Laurent​ Baratchart, C.Cristobal​‌ Villalobos Guillén and D.​​Doug Hardin. Inverse​​​‌ potential problems in divergence​ form for measures in​‌ the plane.ESAIM:​​ Control, Optimisation and Calculus​​​‌ of Variations2787​August 2021HALDOI​‌back to textback​​ to text
• 47 article​​​‌L.Laurent Baratchart,​ C.Cristobal Villalobos-Guillén,​‌ D.Douglas Hardin,​​ M.Michael Northington and​​​‌ E.Edward Saff.​ Inverse Potential Problems for​‌ Divergence of Measures with​​ Total Variation Regularization.​​​‌FoCM2020back to​ textback to text​‌
• 48 bookA.Albrecht​​ Böttcher and S. M.​​​‌Sergei M Grudsky.​ Toeplitz matrices, asymptotic linear​‌ algebra and functional analysis​​.67Springer2000​​​‌back to text
• 49​ articleS.Slim Chaabane​‌, I.I. Fellah​​, M.Mohamed Jaoua​​​‌ and J.Juliette Leblond​. Logarithmic stability estimates​‌ for a Robin coefficient​​ in 2D Laplace inverse​​​‌ problems.Inverse Problems​2012004,​‌ 49--57URL: https://dx.doi.org/10.1088/0266-5611/20/1/003back​​ to text
• 50 phdthesis​​​‌S.Slah Chaabi.​ Analyse complexe et problèmes​‌ de Dirichlet dans le​​ plan : équation de​​​‌ Weinstein et autres conductivités​ non bornées.Mathématiques​‌ et Informatique de Marseille​​2013back to text​​​‌
• 51 articleM.Maureen​ Clerc, J.Juliette​‌ Leblond, J.-P.Jean-Paul​​ Marmorat and T.Théodore​​​‌ Papadopoulo. Source localization​ using rational approximation on​‌ plane sections.Inverse​​ Problems285May​​​‌ 2012, Article number​ 055018 - 24 pages​‌HALDOIback to​​ textback to text​​​‌back to text
• 52​ articleM.Maureen Clerc​‌, J.Juliette Leblond​​, J.-P.Jean-Paul Marmorat​​​‌ and C.Christos Papageorgakis​. Uniqueness result for​‌ an inverse conductivity recovery​​ problem with application to​​​‌ EEG.Rendiconti dell'Istituto​ di Matematica dell'Università di​‌ Trieste48http://hal.archives-ouvertes.fr/hal-013036402016​​back to text
• 53​​​‌ inproceedingsA.Adam Cooman​, F.Fabien Seyfert​‌ and S.Smain Amari​​. Estimating unstable poles​​​‌ in simulations of microwave​ circuits.IMS 2018​‌Philadelphia, United States6​​ 2018HALback to​​​‌ text
• 54 articleA.​Adam Cooman, F.​‌Fabien Seyfert, M.​​Martine Olivi, S.​​​‌Sylvain Chevillard and L.​Laurent Baratchart. Model-Free​‌ Closed-Loop Stability Analysis: A​​ Linear Functional Approach.​​​‌IEEE Transactions on Microwave​ Theory and Techniqueshttps://arxiv.org/abs/1610.03235​‌2017HALback to​​ text
• 55 articleV.​​V. Duval and G.​​​‌G. Peyré. Exact‌ support recovery for sparse‌​‌ spikes deconvolution.FoCM​​2015back to text​​​‌
• 56 phdthesisY.Yannick‌ Fischer. Approximation des‌​‌ des classes de fonctions​​ analytiques généralisées et résolution​​​‌ de problèmes inverses pour‌ les tokamaks.Université‌​‌ Nice Sophia Antipolis2011​​, URL: https://tel.archives-ouvertes.fr/tel-00643239/back​​​‌ to text
• 57 article‌A.A. Friedman and‌​‌ M.M. Vogelius.​​ Determining cracks by boundary​​​‌ measurements.Indiana Univ.‌ Math. J.383‌​‌1989, 527--556back​​ to text
• 58 article​​​‌X.Xing Gao,‌ M.Manon Fraulob and‌​‌ G.Guillaume Haïat.​​ Biomechanical behaviours of the​​​‌ bone--implant interface: a review‌.Journal of The‌​‌ Royal Society Interface16​​1562019, 20190259​​​‌back to text
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• 75 inproceedingsD.Dmitry​​ Ponomarev. A Method​​​‌ to Extrapolate the Data​ for the Inverse Magnetisation​‌ Problem with a Planar​​ Sample.International conference​​​‌ Inverse problems: modelling and​ simulationIPMS 2024 -​‌ Inverse Problems: Modeling and​​ SimulationSpringerCirkewwa, Malta​​​‌May 2024, 293--299​HALback to text​‌back to text
• 76​​ articleD.Dmitry Ponomarev​​​‌. Asymptotic solution to​ convolution integral equations on​‌ large and small intervals​​.Proceedings of the​​​‌ Royal Society A477​22482021, 20210025​‌back to textback​​ to text
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• 86 inproceedingsY.​​Yasmina Zaky, N.​​​‌Nicolas Fortino, J.-Y.‌Jean-Yves Dauvignac, F.‌​‌Fabien Seyfert, M.​​Martine Olivi and L.​​​‌Laurent Baratchart. Comparison‌ of SEM Methods for‌​‌ Poles Estimation from Scattered​​ Field by Canonical Objects​​​‌.Renaissance meets advancing‌ technologyFlorence, Italy9‌​‌ 2020, 6URL:​​ https://hal.archives-ouvertes.fr/hal-02941637DOIback to​​​‌ text