Publications

2024

• Characteristics and Itô's formula for weak Dirichlet processes: an equivalence result
  
  • Bandini Elena
  • Russo Francesco
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2024, 97 (8), pp.992-1015. The main objective consists in generalizing a well-known Itô formula of J. Jacod and A. Shiryaev: given a càdlàg process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C, ν) and a Itô formula type expansion of F (S), where F is a bounded function of class C2. This result connects weak solutions of path-dependent SDEs and related martingale problems. We extend this to the case when S is a weak Dirichlet process. A second aspect of the paper consists in discussing some untreated features of stochastic calculus for finite quadratic variation processes. (10.1080/17442508.2024.2397984)
  DOI : 10.1080/17442508.2024.2397984
• SDEs WITH SINGULAR COEFFICIENTS: THE MARTINGALE PROBLEM VIEW AND THE STOCHASTIC DYNAMICS VIEW
  
  • Issoglio Elena
  • Russo Francesco
  Journal of Theoretical Probability, Springer, 2024. We consider SDEs with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, like continuity with respect to the drift and the link with the Fokker-Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem. (10.1007/s10959-024-01325-5)
  DOI : 10.1007/s10959-024-01325-5
• Solvability results for the transient acoustic scattering by an elastic obstacle
  
  • Bonnet Marc
  • Chaillat Stéphanie
  • Nassor Alice
  Journal of Mathematical Analysis and Applications, Elsevier, 2024, 536 (128198). The well-posedness of the linear evolution problem governing the transient scattering of acoustic waves by an elastic obstacle is investigated. After using linear superposition in the acoustic domain, the analysis focuses on an equivalent causal transmission problem. The proposed analysis provides existence and uniqueness results, as well as continuous data-to-solution maps. Solvability results are established for three cases, which differ by the assumed regularity in space on the transmission data on the acoustic-elastic interface Γ. The first two results consider data with "standard" H −1/2 (Γ) and improved H 1/2 (Γ) regularity in space, respectively, and are established using the Hille-Yosida theorem and energy identities. The third result assumes data with L 2 (Γ) regularity in space and follows by Sobolev interpolation. Obtaining the latter result was motivated by the key role it plays (in a separate study) in the justification of an iterative numerical solution method based on domain decomposition. A numerical example is presented to emphasize the latter point. (10.1016/j.jmaa.2024.128198)
  DOI : 10.1016/j.jmaa.2024.128198
• ROUGH PATHS AND SYMMETRIC-STRATONOVICH INTEGRALS DRIVEN BY SINGULAR COVARIANCE GAUSSIAN PROCESSES
  
  • Ohashi Alberto
  • Russo Francesco
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2024, 30 (2), pp.1197-1230. We examine the relation between a stochastic version of the rough path integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough path and symmetric-Stratonovich integrals driven by a class of Gaussian processes. As a by-product, we show that solutions of multi-dimensional rough differential equations driven by a large class of Gaussian rough paths they are actually solutions to Stratonovich stochastic differential equations. We obtain almost sure convergence rates of the first-order Stratonovich scheme to rough paths integrals in the sense of Gubinelli. In case the time-increment of the Malliavin derivative of the integrands is regular enough, the rates are essentially sharp. The framework applies to a large class of Gaussian processes whose the second-order derivative of the covariance function is a sigma-finite non-positive measure on ${\mathbb R}^2$ + off diagonal. (10.3150/23-BEJ1629.short)
  DOI : 10.3150/23-BEJ1629.short
• The T-coercivity approach for mixed problems
  
  • Barré Mathieu
  • Ciarlet Patrick
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2024, 362, pp.1051-1088. Classically, the well-posedness of variational formulations of mixed linear problems is achieved through the inf-sup condition on the constraint. In this note, we propose an alternative framework to study such problems by using the T-coercivity approach to derive a global inf-sup condition. Generally speaking, this is a constructive approach that, in addition, drives the design of suitable approximations. As a matter of fact, the derivation of the uniform discrete inf-sup condition for the approximate problems follows easily from the study of the original problem. To support our view, we solve a series of classical mixed problems with the T-coercivity approach. Among others, the celebrated Fortin Lemma appears naturally in the numerical analysis of the approximate problems. (10.5802/crmath.590)
  DOI : 10.5802/crmath.590
• Study of a degenerate non-elliptic equation to model plasma heating
  
  • Ciarlet Patrick
  • Kachanovska Maryna
  • Peillon Etienne
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2024. In this manuscript, we study solutions to resonant Maxwell's equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω 2 u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [13] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series a numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.
• On the Accessibility and Controllability of Statistical Linearization for Stochastic Control: Algebraic Rank Conditions and their Genericity
  
  • Bonalli Riccardo
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  Mathematical Control and Related Fields, AIMS, 2024, 14 (2). Statistical linearization has recently seen a particular surge of interest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control complex stochastic systems, accessibility and controllability properties of statistical linearization, which are key to make the robust control problem well-posed, have not been investigated yet. In this paper, we bridge this gap by providing sufficient conditions for the accessibility and controllability of statistical linearization. Specifically, we establish simple sufficient algebraic conditions for the accessibility and controllability of statistical linearization, which involve the rank of the Lie algebra generated by the drift only. In addition, we show these latter algebraic conditions are essentially sharp, by means of a counterexample, and that they are generic with respect to the drift and the initial condition. (10.3934/mcrf.2023020)
  DOI : 10.3934/mcrf.2023020
• On the convergence analysis of one-shot inversion methods
  
  • Bonazzoli Marcella
  • Haddar Houssem
  • Vu Tuan Anh
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (6), pp.2440-2475. When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same time on the inverse problem unknown and on the forward and adjoint problem solutions yields the concept of one-shot inversion methods. We are especially interested in the case where the inner iterations for the direct and adjoint problems are incomplete, that is, stopped before achieving a high accuracy on their solutions. Here, we focus on general linear inverse problems and generic fixed-point iterations for the associated forward problem. We analyze variants of the so-called multi-step one-shot methods, in particular semi-implicit schemes with a regularization parameter. We establish sufficient conditions on the descent step for convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical gradient descent, where the forward and adjoint problems are solved exactly by a direct solver instead. We observe that very few inner iterations are enough to guarantee good convergence of the inversion algorithm, even in the presence of noisy data. (10.1137/23M1585866)
  DOI : 10.1137/23M1585866
• Coupling of discontinuous Galerkin and pseudo-spectral methods for time-dependent acoustic problems
  
  • Meyer Rose-Cloé
  • Bériot Hadrien
  • Gabard Gwenael
  • Modave Axel
  Journal of Theoretical and Computational Acoustics, World Scientific, 2024, 32 (4), pp.2450017. Many realistic problems in computational acoustics involve complex geometries and sound propagation over large domains, which requires accurate and efficient numerical schemes. It is difficult to meet these requirements with a single numerical method. Pseudo-spectral (PS) methods are very efficient, but are limited to rectangular shaped domains. In contrast, the nodal discontinuous Galerkin (DG) method can be easily applied to complex geometries, but can become expensive for large problems. In this paper, we study a coupling strategy between the PS and DG methods to efficiently solve time-domain acoustic wave problems. The idea is to combine the strengths of these two methods: the PS method is used on the part of the domain without geometric constraints, while the DG method is used around the PS region to accurately represent the geometry. This combination allows for the rapid and accurate simulations of large-scale acoustic problems with complex geometries, but the coupling and the parameter selection require great care. The coupling is achieved by introducing an overlap between the PS and DG regions. The solutions are interpolated on the overlaps, which allows the use of unstructured finite element meshes. A standard explicit Runge-Kutta time-stepping scheme is used with the DG scheme, while implicit schemes can be used with the PS scheme due to the peculiar structure of this scheme. We present one-and two-dimensional results to validate the coupling technique. To guide future implementations of this method, we extensively study the influence of different numerical parameters on the accuracy of the schemes and the coupling strategy. (10.1142/S2591728524500178)
  DOI : 10.1142/S2591728524500178
• Adaptive solution of the domain decomposition+ $L^2$ -jumps method applied to the neutron diffusion equation on structured meshes
  
  • Gervais Mario
  • Madiot François
  • Do Minh-Hieu
  • Ciarlet Patrick
  EPJ Web of Conferences, EDP Sciences, 2024, 302, pp.02011. At the core scale, neutron deterministic calculations are usually based on the neutron diffusion equation. Classically, this equation can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. The goal is to extend the Adaptive Mesh Refinement (AMR) strategy previously proposed in [1] to the Domain Decomposition+ $L^2$ jumps which allows non conformity at the interface between subdomains. We are able to refine each subdomain independently, which eventually leads to a more optimal refinement. We numerically investigate the improvements made to the AMR strategy. (10.1051/epjconf/202430202011)
  DOI : 10.1051/epjconf/202430202011
• A PDE WITH DRIFT OF NEGATIVE BESOV INDEX AND LINEAR GROWTH SOLUTIONS
  
  • Issoglio Elena
  • Russo Francesco
  Differential and integral equations, Khayyam Publishing, 2024, 37 (9-10), pp.585-622. This paper investigates a class of PDEs with coefficients in negative Besov spaces and whose solutions have linear growth. We show existence and uniqueness of mild and weak solutions, which are equivalent in this setting, and several continuity results. To this aim, we introduce ad-hoc Besov-Hölder type spaces that allow for linear growth, and investigate the action of the heat semigroup on them. We conclude the paper by introducing a special subclass of these spaces which has the useful property to be separable. (10.57262/die037-09-10-585)
  DOI : 10.57262/die037-09-10-585
• Active Design of Diffuse Acoustic Fields in Enclosures
  
  • Aquino Wilkins
  • Rouse Jerry
  • Bonnet Marc
  Journal of the Acoustical Society of America, Acoustical Society of America, 2024, 155, pp.1297-1307. This paper presents a numerical framework for designing diffuse fields in rooms of any shape and size, driven at arbitrary frequencies. That is, we aim at overcoming the Schroeder frequency limit for generating diffuse fields in an enclosed space. We formulate the problem as a Tikhonov regularized inverse problem and propose a lowrank approximation of the spatial correlation that results in significant computational gains. Our approximation is applicable to arbitrary sets of target points and allows us to produce an optimal design at a computational cost that grows only linearly with the (potentially large) number of target points. We demonstrate the feasibility of our approach through numerical examples where we approximate diffuse fields at frequencies well below the Schroeder limit. (10.1121/10.0024770)
  DOI : 10.1121/10.0024770
• The linear sampling method for data generated by small random scatterers
  
  • Garnier Josselin
  • Haddar Houssem
  • Montanelli Hadrien
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2024, 17 (4), pp.2142-2173. (10.1137/24M1650417)
  DOI : 10.1137/24M1650417