Publications

2024

• Towards high-performance linear potential flow BEM solver with low-rank compressions
  
  • Ancellin Matthieu
  • Marchand Pierre
  • Dias Frédéric
  Energies, MDPI, 2024, 17 (2), pp.372. The interaction of water waves with floating bodies can be modelled with linear potential flow theory, numerically solved with the Boundary Element Method (BEM). This method requires the construction of dense matrices and the resolution of the corresponding linear systems. The cost in time and memory of the method grows at least quadratically with the size of the mesh and the resolution of large problems (such as large farms of wave energy converters) can thus be very costly. Approximating some blocks of the matrix by data-sparse matrices can limit this cost. While matrix compression with low-rank blocks has become a standard tool in the larger BEM community, the present paper provides its first application (to our knowledge) to linear potential flows. In this paper, we assess that low-rank blocks can efficiently approximate interaction matrices between distant meshes when using the Green function of linear potential flow. Due to the complexity of this Green function, a theoretical study is difficult and numerical experiments are used to test the approximation method. Typical results on large arrays of floating bodies show that 99% of the accuracy can be reached with 10% of the coefficients of the matrix. (10.3390/en17020372)
  DOI : 10.3390/en17020372
• Bi-objective finite horizon optimal control problems with Bolza and maximum running cost
  
  • Chorobura Ana Paula
  • Zidani Hasnaa
  , 2024. In this paper, we investigate optimal control problems with two objective functions of different nature that need to be minimized simultaneously. One objective is in the classical Bolza form and the other one is defined as a maximum running cost. Our approach is based on the Hamilton-Jacobi-Bellman framework. In the problem considered here the existence of Pareto solutions is not guaranteed. So first, we consider the bi-objective problem to be minimized over the convexified dynamical system. We show that if a vector is (weak) Pareto optimal solution for the convexified problem, then there exists an (weak) ε-Pareto optimal solution of the original problem that is in the neighborhood of this vector. After we define an auxiliary optimal control problem and show that the weak Pareto front of the convexified problem is a subset of the zero level set of the corresponding value function. Moreover, with a geometrical approach we establish a characterization of the Pareto front. It is also proved that the (weak) ε-Pareto front is contained in the negative level set of the auxiliary optimal control problem that is less or equal ε. Some numerical examples are considered to show the relevance of our approach.
• A new class of uniformly stable time-domain Foldy-Lax models for scattering by small particles. Acoustic sound-soft scattering by circles. Extended version
  
  • Kachanovska Maryna
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2024. In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem valid when the size of the particles tends to zero. We present a systematic approach to constructing such models, based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. For the case of circular obstacles, we prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments. (10.1137/22M149551)
  DOI : 10.1137/22M149551
• 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
• 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
• 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
• 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. 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.
• 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
• 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 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
• 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
• 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
• 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
• Construction of polynomial particular solutions of linear constant-coefficient partial differential equations
  
  • Anderson Thomas G.
  • Bonnet Marc
  • Faria Luiz
  • Pérez-Arancibia Carlos
  Computers & Mathematics with Applications, Elsevier, 2024, 162C, pp.94-103. This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell’s equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, ElementaryPDESolutions.jl, which implements the proposed methodology in an efficient and user-friendly format. (10.1016/j.camwa.2024.02.045)
  DOI : 10.1016/j.camwa.2024.02.045
• Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation
  
  • Anderson Thomas G.
  • Bonnet Marc
  • Faria Luiz
  • Pérez‐Arancibia Carlos
  Journal of Computational Physics, Elsevier, 2024, 511, pp.113091. This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented. (10.1016/j.jcp.2024.113091)
  DOI : 10.1016/j.jcp.2024.113091