Publications

2015

• Solutions of the time-harmonic wave equation in periodic waveguides : asymptotic behaviour and radiation condition
  
  • Fliss Sonia
  • Joly Patrick
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 219 (1), pp.10.1007/s00205-015-0897-3. In this paper, we give the expression and the asymptotic behaviour of the physical solution of a time harmonic wave equation set in a periodic waveguide. This enables us to define a radiation condition and show well-posedness of the Helmholtz equation set in a periodic waveguide. (10.1007/s00205-015-0897-3)
  DOI : 10.1007/s00205-015-0897-3
• Asymptotic analysis for the multiscale modeling of defects in mechanical structures
  
  • Marenić Eduard
  • Brancherie Delphine
  • Bonnet Marc
  , 2015. This research is a first step towards designing a numerical strategy capable of assessing the nocivity of a small defect in terms of its size and position in the structure with low computational cost, using only a mesh of the defect-free reference structure. The proposed strategy aims at taking into account the modification induced by the presence of a small defect through displacement field correction using an asymptotic analysis. Such an approach would allow to assess the criticality of defects by introducing trial micro-defects with varying positions, sizes and mechanical properties.
• A Wideband Fast Multipole Method for the Helmholtz Kernel: Theoretical Developments
  
  • Chaillat Stéphanie
  • Collino Francis
  Computers & Mathematics with Applications, Elsevier, 2015, pp.to appear. This work presents a new Fast Multipole Method (FMM) based on plane wave expansions (PWFMM), combining the advantages of the low and high frequency formulations. We revisit the method of Greengard et al. [1] devoted to the low frequency regime and based on the splitting of the Green's function into a propagative and an evanescent part. More precisely, we give an explicit formula of the filtered translation function for the propagative part, we derive a new formula for the evanescent part and we provide a new interpolation algorithm. At all steps, we check the accuracy of the method by providing error estimates. These theoretical developments are used to propose a wideband FMM based entirely on plane wave expansions. The numerical efficiency and accuracy of this broadband PWFMM are illustrated with a numerical example. (10.1016/j.camwa.2015.05.019)
  DOI : 10.1016/j.camwa.2015.05.019
• Local controllability of the two-link magneto-elastic swimmer
  
  • Giraldi Laetitia
  • Pomet Jean-Baptiste
  , 2015. A recent promising technique for moving a robotic micro-swimmers is to apply an external magnetic field. This paper focuses on a simple micro-swimmer model with two magnetized segments connected by an elastic joint, which is able to move in a plane by using a magnetic field. By considering the latter as control functions, we prove that the swimmer is locally controllable around the straight position.
• Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  , 2015. We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.
• On some stochastic control problems with state constraints
  
  • Picarelli Athena
  , 2015. This thesis deals with Hamilton-Jacobi-Bellman (HJB) approach for some stochastic control problems in presence of state-constraints. This class of problems arises in many challenging applications, and a wide literature has already analysed such problems under some strong compatibility conditions. The main features of the present thesis is to provide new ways to face the presence of constraints without assuming any controllability condition. The first contribution of the thesis in this direction is obtained by exploiting the existing link between backward reachability and optimal control problems. It is shown that by considering a suitable auxiliary unconstrained optimal control problem, the level set approach can be extended to characterize the backward reachable sets under state-constrained. On the other hand the value function associated with a general state constrained stochastic optimal control problem is characterized by means of a state constrained backward reachable set, enabling the application of the level set method for handling the presence of the state constraints. This link between optimal control problems and reachability sets led to the theoretical and numerical analysis of HJB equations with oblique derivative boundary conditions and problems with unbounded controls. Error estimates for Markov-chain approximation represent another contribution of this manuscript. Furthermore, the properties of asymptotic controllability of a stochastic system have also been studied. A generalization of the Zubov method to state constrained stochastic systems is presented. In the last part of the thesis an ergodic optimal control problems in presence of state-constraints are considered.
• Complexity of control-affine motion planning
  
  • Jean Frédéric
  • Prandi Dario
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2015, 53 (2), pp.816-844. In this paper we study the complexity of the motion planning problem for control- affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time- rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities. As a byproduct, we are able to treat the long time local controllability problem, giving quanti- tative estimates on the cost of stabilizing the system near a non-equilibrium point of the drift. (10.1137/130950793)
  DOI : 10.1137/130950793
• Perfectly matched layers in negative index metamaterials and plasmas
  
  • Bécache Eliane
  • Joly Patrick
  • Kachanovska Maryna
  • Vinoles Valentin
  ESAIM: Proceedings, EDP Sciences, 2015, pp.Vol. 50, p. 113-132. This work deals with the stability of Perfectly Matched Layers (PMLs). The first part is a survey of previous results about the classical PMLs in non-dispersive media (construction and necessary condition of stability). The second part concerns some extensions of these results. We give a new necessary criterion of stability valid for a large class of dispersive models and for more general PMLs than the classical ones. This criterion is applied to two dispersive models: negative index metamaterials and uniaxial anisotropic plasmas. In both cases, classical PMLs are unstable but the criterion allows us to design new stable PMLs. Numerical simulations illustrate our purpose. (10.1051/proc/201550006)
  DOI : 10.1051/proc/201550006
• Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Nazarov Sergei
  Inverse Problems, IOP Publishing, 2015. We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce inci-dent plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coeffi-cients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain D, we present a construc-tive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which is described at the end of the paper and which allows to provide examples of (approximated) invisible inclusions.
• A Wideband Fast Multipole Method for the Helmholtz kernel: Theoretical developments
  
  • Chaillat Stéphanie
  • Collino Francis
  , 2015, pp.28. This work presents a new Fast Multipole Method (FMM) based on plane wave expansions, combining the advantages of the low and high frequency formulations. We revisit the method of Greengard et al. devoted to the low frequency regime and based on the splitting of the Green's function into a propagative and an evanescent part. More precisely, we give an explicit formula of the filtered translation function for the propagative part, we derive a new formula for the evanescent part and we provide a new interpolation algorithm. At all steps, we check the accuracy of the method by providing error estimates. These theoretical developments are used to propose a wideband FMM based entirely on plane wave expansions. The numerical efficiency and accuracy of this broadband are illustrated with a numerical example.
• Pattern selection in a biomechanical model for the growth of walled cells
  
  • Calvez Vincent
  • Giraldi Laetitia
  , 2015. In this paper, we analyse a model for the growth of three-dimensional walled cells. In this model the biomechanical expansion of the cell is coupled with the geometry of its wall. We consider that the density of building material depends on the curvature of the cell wall, thus yield-ing possible anisotropic growth. The dynamics of the axisymmetric cell wall is described by a system of nonlinear PDE including a nonlin-ear convection-diffusion equation coupled with a Poisson equation. We develop the linear stability analysis of the spherical symmetric config-uration in expansion. We identify three critical parameters that play a role in the possible instability of the radially symmetric shape, namely the degree of nonlinearity of the coupling, the effective diffusion of the building material, and the Poisson's ratio of the cell wall. We also investigate numerically pattern selection in the nonlinear regime. All the results are also obtained for a simpler, but similar, two-dimensional model.
• Infinite horizon problems on stratifiable state-constraints sets
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2015, 258 (4), pp.1430–1460. This paper deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the value function has not enough regularity, or can fail to be the unique constrained viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. Here, we consider the case of a set of constraints having a stratified structure. Under this circumstance, the interior of this set may be empty or disconnected, and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints. In such situations, the classical pointing qualification hypothesis are not relevant. The discontinuous value function is then characterized by means of a system of HJB equations on each stratum that composes the state constraints. This result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur. (10.1016/j.jde.2014.11.001)
  DOI : 10.1016/j.jde.2014.11.001
• Comparison of mean and osculating stability in the vicinity of the (2:1) tesseral resonant surface
  
  • Daquin Jérôme
  • Deleflie Florent
  • Perez Jérôme
  Acta Astronautica, Elsevier, 2015, 111, pp.170-177. We confront stability results over long time scales, considering alternately the averaged and the non-averaged theory to propagate the equations of motion of a celestial body orbiting the vicinity of the (2:1) tesseral resonant surface. This confrontation is performed using Fast Lyapunov Indicator stability maps. The benefit of such maps is threefold: (i) to reveal the whole phase space architecture and the consequences of the resonance overlap when several combinations of tesseral resonant parameters are accounted for, (ii) to perform a stability analysis on a whole phase space region, and (iii) to have a clear view of the possible impacts of the short-periodic effects removed during the averaging procedure. Our detailed numerical investigations conclude that the tesseral chaos is robust to the averaging procedure and the numerical methods used to propagate the equations of motion over such long time scales. (10.1016/j.actaastro.2015.02.014)
  DOI : 10.1016/j.actaastro.2015.02.014
• Optimal control problems on well-structured domains and stratified feedback controls
  
  • Hermosilla Cristopher
  , 2015. The aim of this dissertation is to study some issues in Control Theory of ordinary differential equations. Optimal control problems with tame state-constraints and feedback controls with stratified discontinuities are of special interest. The techniques employed along the manuscript have been chiefly taken from control theory, nonsmooth analysis, variational analysis, tame geometry, convex analysis and differential inclusions theory. The first part of the thesis is devoted to provide general results and definitions required for a good understanding of the entire manuscript. In particular, a strong invariance criterion adapted to manifolds is presented. Moreover, a short insight into manifolds and stratifications is done. The notions of relatively wedged sets is introduced and in addition, some of its properties are stated. The second part is concerned with the characterization of the Value Function of an optimal control problem with state-constraints. Three cases have been taken into account. The first one treats stratifiable state-constraints, that is, sets that can be decomposed into manifolds of different dimensions. The second case is focused on linear systems with convex state-constraints, and the last one considers convex state-constraints as well, but from a penalization point of view. In the latter situation, the dynamics are nonlinear and verify an absorbing property at the boundary. The third part is about discontinuous feedbacks laws whose singularities form a stratified set on the state-space. This type of controls yields to consider stratified discontinuous ordinary differential equations, which motivates an analysis of existence of solutions and robustness with respect to external perturbation for these equations. The construction of a suboptimal continuous feedback from an optimal one is also addressed in this part. The fourth part is dedicated to investigate optimal control problems on networks. The main feature of this contribution is that no controllability assumption around the junctions is imposed. The results can also be extended to generalized notions of networks, where the junction is not a single point but a manifold.
• On the Convergence of Decomposition Methods for Multistage Stochastic Convex Programs
  
  • Girardeau Pierre
  • Leclere Vincent
  • Philpott A. B.
  Mathematics of Operations Research, INFORMS, 2015, 40 (1). We prove the almost-sure convergence of a class of sampling-based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions , and uncertainty is modelled by a scenario tree. As special cases, our results imply the almost-sure convergence of SDDP, CUPPS and DOASA when applied to problems with general convex cost functions. (10.1287/moor.2014.0664)
  DOI : 10.1287/moor.2014.0664
• Formulations par équations intégrales de surface pour la simulation numérique du contrôle non destructif par courants de Foucault
  
  • Vigneron Audrey
  , 2015. Cette thèse s'inscrit dans le contexte de la simulation numérique pour le contrôle non destructif (CND) par courants de Foucault et concerne le calcul des champs électromagnétiques induits par un capteur émetteur dans une pièce saine. Ce calcul constitue la première étape de la modélisation complète d'un procédé de contrôle dans la plateforme logicielle CIVA développée au CEA LIST. Aujourd'hui les modèles intégrés dans CIVA sont restreints à des pièces de géométrie canonique (calcul modal) ou axisymétriques. La demande de configurations plus diverses et complexes nécessite l'introduction de nouveaux outils numériques de modélisation. En pratique les capteurs peuvent être constitués d'éléments aux propriétés physiques et aux formes variées. Quant aux pièces à contrôler, elles sont conductrices et peuvent contenir des éléments diélectriques ou magnétiques. Du fait des différents matériaux présents dans une même configuration, différents régimes de modélisation (statique, quasi-statique, voire dynamique) peuvent cohabiter. Sous l'hypothèse de travail de milieux à propriétés linéaires, isotropes et homogènes par morceaux, l'approche par équations intégrales de surface (SIE) permet de ramener le problème volumique à un problème surfacique équivalent. Cependant les formulations SIE usuelles pour le problème de Maxwell souffrent en général d'un problème de robustesse numérique pour certains cas asymptotiques, en particulier à basse fréquence. L'objectif de cette étude est de déterminer une version stable pour une gamme de paramètres physique typique du CND. C'est dans ce cadre qu’un schéma itératif par blocs basé sur une décomposition liée à la physique du problème est proposé. Ce schéma est précis et bien conditionné pour le calcul des champs primaires. Une étude asymptotique du problème intégral de Maxwell est de plus effectuée. Celle-ci permet de formuler le problème intégral de l'approximation courants de Foucault comme une forme asymptotique de celui de Maxwell.
• A regularization approach to functional Itô calculus and strong-viscosity solutions to path-dependent PDEs
  
  • Cosso Andrea
  • Russo Francesco
  , 2015. First, we revisit functional Itô/path-dependent calculus started by B. Dupire, R. Cont and D.-A. Fournié, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus introduced by C. Di Girolami and the second named author are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of classical solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness. The notion of strong-viscosity solution motivates the last part of the paper which is devoted to explore this new concept of solution for general semilinear PDEs in the finite dimensional case. We prove an equivalence result between the classical viscosity solution and the new one. The definition of strong-viscosity solution for semilinear PDEs is inspired by the notion of "good" solution, and it is based again on an approximating procedure.
• Éléments de physique statistique - 2e édition
  
  • Perez Jérôme
  • Chardin Gabriel
  • Debu Pascal
  , 2015, pp.268 pages. Cet ouvrage aborde le thème classique de la physique statistique par la méthode pédagogique du dénombrement des états d’énergie microscopiques. Après avoir passé en revue les divers résultats de la théorie des systèmes sans interactions, divers cas plus généraux sont abordés comme la transition gaz-liquide, le ferromagnétisme ou la théorie du proche équilibre. Ce cours s’insère parfaitement dans la suite logique de l’enseignement de la physique de premier cycle et met en œuvre les résultats essentiels de la théorie quantique. Il permet d’appréhender l’origine microscopique d’un grand nombre de propriétés macroscopiques essentielles d’un système qui caractérisent son état d’équilibre (température, énergie, pression, etc.). La prise en compte des interactions à l’échelle microscopique et l’étude du proche équilibre viennent compléter ce panorama pour préparer des cours plus avancés comme l’étude physique des solides ou celle des plasmas. Cet ouvrage est le fruit d’un cours donné par les auteurs à L’École Nationale Supérieure de Techniques Avancées (ENSTA ParisTech). Il contient de nombreux exercices et une synthèse des points essentiels en fin de chaque chapitre.
• Solving mutizone and muticrack elastostatic problems: a Fast multipole symmetric Galerkin Boundary element method approach
  
  • Trinh Quoc Tuan
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Engineering Analysis with Boundary Elements, Elsevier, 2015, 50, pp.486-495. Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully populated (albeit symmetric) matrix equations, entailing high solution times for large models. This paper is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdomain numbering. This formulation (i) automatically enforces the perfectbonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by assembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (Pham et al., Eng Anal Bound Elem 2012;36:1838–47 [36]). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N 106. (10.1016/j.enganabound.2014.10.004)
  DOI : 10.1016/j.enganabound.2014.10.004
• Qualitative modeling of the dynamics of detonations with losses
  
  • Faria Luiz
  • Kasimov Aslan
  Proceedings of the Combustion Institute, Elsevier, 2015, 35 (2), pp.2015-2023. (10.1016/j.proci.2014.07.006)
  DOI : 10.1016/j.proci.2014.07.006
• Solving multizone and multicrack elastostatic problems: a fast multipole symmetric Galerkin boundary element method approach
  
  • Trinh Quoc Tuan
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Engineering Analysis with Boundary Elements, Elsevier, 2015, 50, pp.486-495. Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully-populated (albeit symmetric) matrix equations, entailing high solution times for large models. This article is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdo-main numbering. This formulation (i) automatically enforces the perfect-bonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by as-sembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (EABE, 36:1838-1847, 2012). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N ≈ 10 6 . (10.1016/j.enganabound.2014.10.004)
  DOI : 10.1016/j.enganabound.2014.10.004
• Gaussian and non-Gaussian processes of zero power variation
  
  • Russo Francesco
  • Viens Frederi
  ESAIM: Probability and Statistics, EDP Sciences, 2015, 19 (9), pp.414-439. This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô's formula is proved to hold for all functions of class $C^{6}$. (10.1051/ps/2014031)
  DOI : 10.1051/ps/2014031
• Stochastic Multi-Stage Optimization
  
  • Carpentier Pierre
  • Cohen Guy
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2015, 75. (10.1007/978-3-319-18138-7)
  DOI : 10.1007/978-3-319-18138-7
• The topological derivative of stress-based cost functionals in anisotropic elasticity
  
  • Delgado Gabriel
  • Bonnet Marc
  Computers & Mathematics with Applications, Elsevier, 2015, 69, pp.1144-1166. The topological derivative of cost functionals J that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative dJ(z) of J quantifies the asymptotic behavior of J under the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius a at a specified location z. The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small a makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals. The asymptotic perturbation of J is shown to be of order O(a^3) for a wide class of stress-based cost functionals having smooth densities. The topological derivative of J, i.e. the coefficient of the O(a^3) perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of J is mathematically justified (i.e. its remainder is proved to be of order o(a^3)). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of \dJ on cases involving anisotropic elasticity and non-quadratic cost functionals. (10.1016/j.camwa.2015.03.010)
  DOI : 10.1016/j.camwa.2015.03.010
• Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1d case
  
  • Bécache Eliane
  • Bourgeois Laurent
  • Franceschini Lucas
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2015. In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical La-grange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations. 1. Introduction. The method of quasi-reversibility has now a quite long history since the pioneering book of Latt es and Lions in 1967 [1]. The original idea of these authors was, starting from an ill-posed problem which satisfies the uniqueness property, to introduce a perturbation of such problem involving a small positive parameter ε. This perturbation has essentially two effects. Firstly the perturbation transforms the initial ill-posed problem into a well-posed one for any ε, secondly the solution to such problem converges to the solution (if it exists) to the initial ill-posed problem when ε tends to 0. Generally, the ill-posedness in the initial problem is due to unsuitable boundary conditions. As typical examples of linear ill-posed problems one may think of the backward heat equation, that is the initial condition is replaced by a final condition, or the heat or wave equations with lateral Cauchy data, that is the usual Dirichlet or Neumann boundary condition on the boundary of the domain is replaced by a pair of Dirichlet and Neumann boundary conditions on the same subpart of the boundary, no data being prescribed on the complementary part of the boundary. (10.3934/ipi.2015.9.971)
  DOI : 10.3934/ipi.2015.9.971