Publications

2024

• Deterministic optimal control on Riemannian manifolds under probability knowledge of the initial condition
  
  • Jean Frédéric
  • Jerhaoui Othmane
  • Zidani Hasnaa
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2024, 56 (3), pp.3326-3356. In this article, we study an optimal control problem on a compact Riemannian manifold M with imperfect information on the initial state of the system. The lack of information is modelled by a Borel probability measure along which the initial state is distributed. The state space of this problem is the space of Borel probability measures over M. We define a notion of viscosity in this space by taking as test functions a subset of the set of functions that can be written as a difference of two semi-convex functions. With this choice of test functions, we extend the notion of viscosity solution to Hamilton-Jacobi-Bellman equations in Wasserstein space, we also establish that the value function of the control problem with imperfect information is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the space of Borel probability measures. (10.1137/23M1575251)
  DOI : 10.1137/23M1575251
• Combined field-only boundary integral equations for PEC electromagnetic scattering problem in spherical geometries
  
  • Faria Luiz
  • Pérez-Arancibia Carlos
  • Turc Catalin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (1). We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIE. In this paper we use the Combined Field methodology of Burton and Miller within the field-only BIE approach and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are non zero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one -- a property similar to second kind integral equations. (10.1137/23M1561865)
  DOI : 10.1137/23M1561865
• Optimization of a domestic microgrid equipped with solar panel and battery: Model Predictive Control and Stochastic Dual Dynamic Programming approaches
  
  • Pacaud François
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  Energy Systems, Springer, 2024, 15 (1), pp.115-139. In this study, a microgrid with storage (battery, hot water tank) and solar panel is considered. We benchmark two algorithms, MPC and SDDP, that yield online policies to manage the microgrid, and compare them with a rule based policy. Model Predictive Control (MPC) is a well-known algorithm which models the future uncertainties with a deterministic forecast. By contrast, Stochastic Dual Dynamic Programming (SDDP) models the future uncertainties as stagewise independent random variables with known probability distributions. We present a scheme, based on out-of-sample validation, to fairly compare the two online policies yielded by MPC and SDDP. Our numerical studies put to light that MPC and SDDP achieve significant gains compared to the rule based policy, and that SDDP overperforms MPC not only on average but on most of the out-of-sample assessment scenarios. (10.1007/s12667-022-00522-7)
  DOI : 10.1007/s12667-022-00522-7
• A complex-scaled boundary integral equation for time-harmonic water waves
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Faria Luiz
  • Pérez‐Arancibia Carlos
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (4), pp.1532-1556. This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent. (10.1137/23M1607866)
  DOI : 10.1137/23M1607866
• 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
• Asymptotic Expansion of Transmission Eigenvalues for Anisotropic Thin Layers
  
  • Boujlida Hanen
  • Haddar Houssem
  • Khenissi Moez
  Applicable Analysis, Taylor & Francis, 2024, 103. We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [19]. We also provide formal derivation in more general cases validating the obtained theoretical result. (10.1080/00036811.2023.2187788)
  DOI : 10.1080/00036811.2023.2187788
• Modified error-in-constitutive-relation (MECR) framework for the characterization of linear viscoelastic solids
  
  • Bonnet Marc
  • Salasiya Prasanna
  • Guzina Bojan B.
  Journal of the Mechanics and Physics of Solids, Elsevier, 2024, 190, pp.105746. We develop an error-in-constitutive-relation (ECR) approach toward the full-field characterization of linear viscoelastic solids described within the framework of standard generalized materials. To this end, we formulate the viscoelastic behavior in terms of the (Helmholtz) free energy potential and a dissipation potential. Assuming the availability of full-field interior kinematic data, the constitutive mismatch between the kinematic quantities (strains and internal thermodynamic variables) and their ``stress'' counterparts (Cauchy stress tensor and that of thermodynamic tensions), commonly referred to as the ECR functional, is established with the aid of Legendre-Fenchel gap functionals linking the thermodynamic potentials to their energetic conjugates. We then proceed by introducing the modified ECR (MECR) functional as a linear combination between its ECR parent and the kinematic data misfit, computed for a trial set of constitutive parameters. The affiliated stationarity conditions then yield two coupled evolution problems, namely (i) the forward evolution problem for the (trial) displacement field driven by the constitutive mismatch, and (ii) the backward evolution problem for the adjoint field driven by the data mismatch. This allows us to establish compact expressions for the MECR functional and its gradient with respect to the viscoelastic constitutive parameters. For generality, the formulation is established assuming both time-domain (i.e. transient) and frequency-domain data. We illustrate the developments in a two-dimensional setting by pursuing the multi-frequency MECR reconstruction of (i) piecewise-homogeneous standard linear solid, and (b) smoothly-varying Jeffreys viscoelastic material. (10.1016/j.jmps.2024.105746)
  DOI : 10.1016/j.jmps.2024.105746
• Computing singular and near-singular integrals over curved boundary elements: The strongly singular case
  
  • Montanelli Hadrien
  • Collino Francis
  • Haddar Houssem
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2024, 46 (6), pp.A3756-A3778. (10.1137/23M1605594)
  DOI : 10.1137/23M1605594
• 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
• Fair Energy Allocation for Collective Self-Consumption
  
  • Jorquera-Bravo Natalia
  • Elloumi Sourour
  • Kedad-Sidhoum Safia
  • Plateau Agnès
  , 2024, 14594. This study explores a collective self-consumption community with several houses, a shared distributed energy resource (DER), and a common energy storage system, as a battery. Each house has an energy demand over a discrete planning horizon, met by using the DER, the battery, or purchasing electricity from the main power grid. Excess energy can be stored in the battery or sold back to the main grid. The objective is to determine a supply plan ensuring a fair allocation of renewable energy while minimizing the overall microgrid cost. We investigate and discuss the formulation of these optimization problems using mixed integer linear programming. We show some dominance properties that allow to reformulate the model into a linear program. We study some fairness metrics like the proportional allocation rule and max-min fairness. Finally, we illustrate our proposal in a real case study in France with up to seven houses and a one-day time horizon with 15minute intervals. (10.1007/978-3-031-60924-4_29)
  DOI : 10.1007/978-3-031-60924-4_29
• 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
• 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
• Construction of transparent conditions for electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2024. We are interested in the numerical resolution of diffraction problems in closed electromagnetic waveguides by means of finite elements methods. To proceed, we need to truncate the domain and design adapted transparent conditions on the artificial boundary to avoid spurious reflections. When the guide is homogeneous in the transverse section, this can be done by writing an Electric-to-Magnetic condition based on a modal decomposition of the field. The latter takes a rather simple form thanks to the orthogonality of transverse modes. For guides that are heterogeneous in the transverse section, the transverse modes are no longer orthogonal but satisfy bi-orthogonality relations linked to the Poynting energy flux. Modal decompositions are more delicate to derive and it may happen that certain modes have phase and group velocities of different sign, which prevents the use of Perfectly Matched Layers. Adapting techniques already developed in elasticity, we derive a new transparent condition based on a Poynting-to-Magnetic operator with overlap. To illustrate the method, we present numerical results obtained with Nédélec finite elements using the XLiFE++ library.
• 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
• 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
• Stochastic incremental mirror descent algorithms with Nesterov smoothing
  
  • Bitterlich Sandy
  • Grad Sorin-Mihai
  Numerical Algorithms, Springer Verlag, 2024, 95, pp.351–382. For minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in an Euclidean space we propose a stochastic incremental mirror descent algorithm constructed by means of the Nesterov smoothing. Further we modify the algorithm in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Next a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing is proposed in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions and a prox-friendly proper, convex and lower semicontinuous function. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements. (10.1007/s11075-023-01574-1)
  DOI : 10.1007/s11075-023-01574-1
• 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/die037-0910-585)
  DOI : 10.57262/die037-0910-585
• 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