Publications

2019

• Analysis of topological derivative as a tool for qualitative identification
  
  • Bonnet Marc
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2019, 35 (104007). The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al. (Inverse Problems 29:075012, 2013) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (J. Integral Equ. Appl. 29:271-295, 2017). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region. (10.1088/1361-6420/ab0b67)
  DOI : 10.1088/1361-6420/ab0b67
• On the well-posedness of a class of McKean Feynman-Kac equations
  
  • Lieber Jonas
  • Oudjane Nadia
  • Russo Francesco
  Markov Processes And Related Fields, Polymat Publishing Company, 2019, 25 (5), pp.821-862. We analyze the well-posedness of a so called McKean Feynman-Kac Equation (MFKE), which is a McKean type equation with a Feynman-Kac perturbation. We provide in particular weak and strong existence conditions as well as pathwise uniqueness conditions without strong regularity assumptions on the coefficients. One major tool to establish this result is a representation theorem relating the solutions of MFKE to the solutions of a nonconservative semilinear parabolic Partial Differential Equation (PDE).
• Trapped modes in thin and infinite ladder like domains. Part 2 : asymptotic analysis and numerical application
  
  • Delourme Bérangère
  • Fliss Sonia
  • Joly Patrick
  • Vasilevskaya Elizaveta
  Asymptotic Analysis, IOS Press, 2019. We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter ε > 0) whose limit (when ε tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter ε) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.
• Pareto Front Characterization for Multiobjective Optimal Control Problems Using Hamilton--Jacobi Approach
  
  • Desilles Anna
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (6), pp.3884-3910. (10.1137/18M1176993)
  DOI : 10.1137/18M1176993
• On the convergence in $H^1$-norm for the fractional Laplacian
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2019, 57, pp.1723-1743. We consider the numerical solution of the fractional Laplacian of index $s \in (1/2, 1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space $\widetilde{H}^s(\Omega)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(\Omega)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1(\Omega)$. A natural question is then whether one can obtain error estimates in $H^1(\Omega)$-norm, in addition to the classical ones that can be derived in the $\widetilde{H}^s(\Omega)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes. (10.1137/18M1221436)
  DOI : 10.1137/18M1221436
• On projective and affine equivalence of sub-Riemannian metrics
  
  • Jean Frédéric
  • Maslovskaya Sofya
  • Zelenko I.
  Geometriae Dedicata, Springer Verlag, 2019, 203 (1), pp.279-319. Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and affinely equivalent metrics is classical (Levi-Civita, Eisenhart). In particular, a Riemannian metric which is not rigid satisfies the following two special properties: its geodesic flow possesses nontrivial integrals and the metric induces certain canonical product structure on the ambient manifold. These classification results were extended to contact and quasi-contact distributions by Zelenko. Our general goal is to extend these results to arbitrary sub-Riemannian manifolds, and we establish two types of results toward this goal: if a sub-Riemannian metric is not projectively conformally rigid, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results: first, we show that a generic sub-Riemannian metric on a fixed pair $(M,D)$ is projectively conformally rigid. Second, we prove that, except for special pairs $(m,n)$, every sub-Riemannian metric on a rank $m$ generic distribution in an $n$-dimensional manifold is projectively conformally rigid. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity. (10.1007/s10711-019-00437-1)
  DOI : 10.1007/s10711-019-00437-1
• Optimality and modularity in human movement: from optimal control to muscle synergies
  
  • Berret B.
  • Delis Ioannis
  • Gaveau Jeremie
  • Jean Frédéric
  , 2019, 124. In this chapter, we review recent work related to the optimal and modular control hypotheses for human movement. Optimal control theory is often thought to imply that the brain continuously computes global optima for each motor task it faces. Modular control theory typically assumes that the brain explicitly stores genuine synergies in specific neural circuits whose combined recruitment yields task-effective motor inputs to muscles. Put this way, these two influential motor control theories are pushed to extreme positions. A more nuanced view, framed within Marr’s tri-level taxonomy of a computational theory of movement neuroscience, is discussed here. We argue that optimal control is best viewed as helping to understand “why” certain movements are preferred over others but does not say much about how the brain would practically trigger optimal strategies. We also argue that dimensionality reduction found in muscle activities may be a by-product of optimality and cannot be attributed to neurally hardwired synergies stricto sensu, in particular when the synergies are extracted from simple factorization algorithms applied to electromyographic data; their putative nature is indeed strongly dictated by the methodology itself. Hence, more modeling work is required to critically test the modularity hypothesis and assess its potential neural origins. We propose that an adequate mathematical formulation of hierarchical motor control could help to bridge the gap between optimality and modularity, thereby accounting for the most appealing aspects of the human motor controller that robotic controllers would like to mimic: rapidity, efficiency, and robustness. (10.1007/978-3-319-93870-7_6)
  DOI : 10.1007/978-3-319-93870-7_6
• The eddy current model as a low-frequency, high-conductivity asymptotic form of the Maxwell transmission problem
  
  • Bonnet Marc
  • Demaldent Edouard
  Computers & Mathematics with Applications, Elsevier, 2019, 77 (8), pp.2145-2161. We study the relationship between the Maxwell and eddy current (EC) models for three-dimensional configurations involving bounded regions with high conductivity $\sigma$ in air and with sources placed remotely from the conducting objects, which typically occur in the numerical simulation of eddy current nondestructive testing (ECT) experiments. The underlying Maxwell transmission problem is formulated using boundary integral formulations of PMCHWT type. In this context, we derive and rigorously justify an asymptotic expansion of the Maxwell integral problem with respect to the non-dimensional parameter $\gamma:=\sqrt{\omega\varepsilon_{0}/\sigma}$. The EC integral problem is shown to constitute the limiting form of the Maxwell integral problem as $\gamma\to0$, i.e. as its low-frequency and high-conductivity limit. Estimates in $\gamma$ are obtained for the solution remainders (in terms of the surface currents, which are the primary unknowns of the PMCHWT problem, and the electromagnetic fields) and the impedance variation measured at the extremities of the excitating coil. In particular, the leading and remainder orders in $\gamma$ of the surface currents are found to depend on the current component (electric or magnetic, charge-free or not). These theoretical results are demonstrated on three-dimensional illustrative numerical examples, where the mathematically established estimates in $\gamma$ are reproduced by the numerical results. (10.1016/j.camwa.2018.12.006)
  DOI : 10.1016/j.camwa.2018.12.006
• Global solution of non-convex quadratically constrained quadratic programs
  
  • Elloumi Sourour
  • Lambert Amélie
  Optimization Methods and Software, Taylor & Francis, 2019, 34 (1), pp.98-114. The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QC-QPs with the following restriction : all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P *). This equivalent problem (P *) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y = xx T , where x are the initial variables of problem (P). We then propose to solve (P *) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO. (10.1080/10556788.2017.1350675)
  DOI : 10.1080/10556788.2017.1350675
• Some loci of rational cubic fourfolds
  
  • Bolognesi Michele
  • Russo Francesco
  • Staglianò Giovanni
  Mathematische Annalen, Springer Verlag, 2019, 373, pp.165-190. In this paper we investigate the geography of some codimension two loci inside the moduli space of smooth cubic hypersurfaces in $\mathbb{P}^5$ and the rationality of their elements. In particular, we study the loci of cubics which contain a plane and another surface, whose ideal is generated by quadratic equations. This is the case of cubic and quartic scrolls and Veronese surfaces - and some of their degenerations. Using the fact that all degenerations of quartic scrolls in $\mathbb{P}^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we also show that every cubic hypersurface belonging to the divisor $\mathcal{C}_{14}$ in the moduli space is rational. (10.1007/s00208-018-1707-7)
  DOI : 10.1007/s00208-018-1707-7
• McKean Feynman-Kac probabilistic representations of non-linear partial differential equations
  
  • Izydorczyk Lucas
  • Oudjane Nadia
  • Russo Francesco
  , 2019. This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations. (10.1007/978-3-030-87432-2)
  DOI : 10.1007/978-3-030-87432-2
• STRONG-VISCOSITY SOLUTIONS: SEMILINEAR PARABOLIC PDEs AND PATH-DEPENDENT PDEs
  
  • Cosso Andrea
  • Russo Francesco
  Osaka Journal of Mathematics, Osaka University, 2019, 56 (2), pp.323-373. The aim of the present work is the introduction of a viscosity type solution, called strong-viscosity solution to distinguish it from the classical one, with the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.
• Analysis of the error in constitutive equation approach for time-harmonic elasticity imaging
  
  • Aquino Wilkins
  • Bonnet Marc
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2019, 79, pp.822-849. We consider the identification of heterogeneous linear elastic moduli in the context of time-harmonic elastodynamics. This inverse problem is formulated as the minimization of the modified error in constitutive equation (MECE), an energy-based cost functional defined as an weighted additive combination $\mathcal{E}+\kappa\mathcal{D}$ of the error in constitutive equation (ECE) $\mathcal{E}$, expressed using an energy seminorm, and a quadratic error term $\mathcal{D}$ incorporating the kinematical measurements. MECE-based identification are known from existing computational evidence to enjoy attractive properties such as improved convexity, robustness to resonant frequencies, and tolerance to incompletely specified boundary conditions (BCs). The main goal of this work is to develop theoretical foundations, in a continuous setting, allowing to explain and justify some of the aforementioned beneficial properties, in particular addressing the general case where BCs may be underspecified. A specific feature of MECE formulations is that forward and adjoint solutions are governed by a fully coupled system, whose mathematical properties play a fundamental role in the qualitative and computational aspects of MECE minimization. We prove that this system has a unique and stable solution at any frequency, provided data is abundant enough (in a sense made precise therein) to at least compensate for any missing information on BCs. As a result, our formulation leads in such situations to a well-defined solution even though the relevant forward problem is not \emph{a priori} clearly defined. This result has practical implications such as applicability of MECE to partial interior data (with important practical applications including ultrasound elastography), convergence of finite element discretizations and differentiability of the reduced MECE functional. In addition, we establish that usual least squares and pure ECE formulations are limiting cases of MECE formulations for small and large values of $\kappa$, respectively. For the latter case, which corresponds to exact enforcement of kinematic data, we furthermore show that the reduced MECE Hessian is asymptotically positive for any parameter perturbation supported on the measurement region, thereby corroborating existing computational evidence on convexity improvement brought by MECE functionals. Finally, numerical studies that support and illustrate our theoretical findings, including a parameter reconstruction example using interior data, are presented. (10.1137/18M1231237)
  DOI : 10.1137/18M1231237
• Path dependent equations driven by Hölder processes
  
  • Andretto Castrequini Rafael
  • Russo Francesco
  Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 37 (3), pp.480-498. This paper investigates existence results for path-dependent differential equations driven by a Hölder function where the integrals are understood in the Young sense. The two main results are proved via an application of Schauder theorem and the vector field is allowed to be unbounded. The Hölder function is typically the trajectory of a stochastic process. (10.1080/07362994.2019.1585263)
  DOI : 10.1080/07362994.2019.1585263
• A Graph-based Heuristic for Variable Selection in Mixed Integer Linear Programming
  
  • Ethève Marc
  • Alès Zacharie
  • Bissuel Côme
  • Juan Olivier
  • Kedad-Sidhoum Safia
  , 2019.