Publications

2025

• Solving numerically the two-dimensional time harmonic Maxwell problem with sign-changing coefficients
  
  • Chaaban Farah
  • Ciarlet Patrick
  • Rihani Mahran
  , 2025. We are investigating the numerical solution to the 2D time-harmonic Maxwell equations in the presence of a classical medium and a metamaterial, that is with sign-changing coefficients. As soon as the problem has a unique solution, we are able to build a converging numerical approximation based on the finite element method, for which there is no constraint on the meshes related to the sign-changing behavior. Numerical examples illustrate the theory.
• Alpha Mesh Swc: automatic and robust surface mesh generation from the skeleton description of brain cells
  
  • Mcsweeney-Davis Alex
  • Fang Chengran
  • Caruyer Emmanuel
  • Kerbrat Anne
  • Li Jing-Rebecca
  Briefings in Bioinformatics, Oxford University Press (OUP), 2025, 26 (3). In recent years, there has been a significant increase in publicly available skeleton descriptions of real brain cells from laboratories all over the world. In theory, this should make it possible to perform large scale realistic simulations on brain cells. However, currently there is still a gap between the skeleton descriptions and high quality simulation-ready surface and volume meshes of brain cells. We propose and implement a tool called {\it Alpha\_Mesh\_Swc} to generate automatically and efficiently triangular surface meshes that are optimized for finite elements simulations. We use an Alpha Wrapping method with an offset parameter on component surface meshes to efficiently generate a global watertight mesh. Then mesh simplification and re-meshing are used to produce an optimal surface mesh. Our methodology limits the number of surface triangles while preserving geometrical accuracy, permits cutting and gluing of cell components, is robust to imperfect skeleton descriptions, and allows mixed cell descriptions (surface meshes combined with skeletons). We compared the robustness, performance and accuracy of {\it Alpha\_Mesh\_Swc} against existing tools and found significant improvement in terms of mesh accuracy. We show, on average, we can generate fully automatically a brain cell (neurons or glia) surface mesh in a couple of minutes on a laptop computer resulting in a simplified surface mesh with only around 10k nodes. The resulting meshes were used to perform diffusion MRI simulations in neurons and microglia. The code and a number of sample brain cell surface meshes have been made publicly available. (10.1093/bib/bbaf258)
  DOI : 10.1093/bib/bbaf258
• A Two-Timescale Decision-Hazard-Decision Formulation for Storage Usage Values Calculation
  
  • Martinez Parra Camila
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Janin Jean-Marc
  • Ruiz Manuel
  , 2024. The penetration of renewable energies requires additional storages to deal with intermittency. Accordingly, there is growing interest in evaluating the opportunity cost (usage value) associated with stored energy in large storages, a cost obtained by solving a multistage stochastic optimization problem. Today, to compute usage values under uncertainties, an adequacy resource problem is solved using stochastic dynamic programming assuming a hazard-decision information structure. This modelling assumes complete knowledge of the coming week uncertainties, which is not adapted to the system operation as the intermittency occurs at smaller timescale. We equip the twotimescale problem with a new information structure considering planning and recourse decisions: decision-hazard-decision. This structure is used to decompose the multistage decision-making process into a nonanticipative planning step in which the on/off decisions for the thermal units are made, and a recourse step in which the power modulation decisions are made once the uncertainties have been disclosed. In a numerical case, we illustrate how usage values are sensitive as how the disclosure of information is modelled. (10.48550/arXiv.2408.17113)
  DOI : 10.48550/arXiv.2408.17113
• Differentiable Optimisation: Theory and Algorithms -- Part II: Algorithms
  
  • Simonetto Andrea
  , 2025. This course follows naturally OPT201, which covers the theory part of continuous optimisation. OPT201 focuses on optimality conditions, convexity, and duality. In OPT202, we will look at how to use these notions to build algorithms that solve the problems. In particular, the aim of the course is to be able to answer the questions, 1. Given an optimisation problem, which algorithm do I use to solve it? 2. Which properties and theoretical guarantees does the algorithm that I have chosen have? 3. Conversely, if I want to use a certain algorithm, which characteristics does the optimisation problem need to have? In order to answer to these three questions, we will need to build a theory of algorithms, and ultimately understand what we really mean by solving an optimisation problem.
• Energy stable and linearly well-balanced numerical schemes for the non-linear Shallow Water equations with Coriolis force
  
  • Audusse Emmanuel
  • Dubos Virgile
  • Gaveau Noémie
  • Penel Yohan
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2025, 47 (01), pp.A1-A23. We analyse a class of energy-stable and linearly well-balanced numerical schemes dedicated to the nonlinear Shallow Water equations with Coriolis force. The proposed algorithms rely on colocated finite-difference approx- imations formulated on cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semi-discrete energy estimates. Among the proposed algorithms a colocated finite-volume scheme is described. Numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes. (10.1137/22M1515707)
  DOI : 10.1137/22M1515707
• The non-intrusive reduced basis two-grid method applied to sensitivity analysis
  
  • Grosjean Elise
  • Simeon Bernd
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2025, 59 (1), pp.101-135. This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly ans thus Reduced Basis Methods (RBMs) may be a viable option. We propose new NIRB two-grid algorithms for both the direct and adjoint state methods in the context of parabolic equations. The NIRB two-grid method uses the HF code solely as a “black-box”, requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage employs coarser grids and thus, is significantly less expensive than a fine HF evaluation. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in L∞(0, T ; H10(Ω)), and then establish that these rates are recovered by the NIRB two-grid approximation. These results are supported by numerical simulations. We then propose a new procedure that further reduces the computational costs of the online step while only computing a coarse solution of the state equations. On the adjoint state method, we propose a new algorithm that reduces both the state and adjoint solutions. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system. (10.1051/m2an/2024044)
  DOI : 10.1051/m2an/2024044
• Contributions on complexity bounds for Deterministic Partially Observed Markov Decision Process
  
  • Vessaire Cyrille
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Rodríguez-Martínez Alejandro
  Annals of Operations Research, Springer Verlag, 2025, 344 (1), pp.345-382. Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of optimization problems. The Mdp model consists of sets of states, actions, time steps, rewards, and probability transitions. When in a given state and at a given time, the decision maker's action generates a reward and determines the state at the next time step according to the probability transition function. However, Mdps assume that the decision maker knows the state of the controlled dynamical system. Hence, when one needs to optimize controlled dynamical systems under partial observation, one often turns toward the formalism of Partially Observed Markov Decision Processes (Pomdp). Pomdps are often untractable in the general case as Dynamic Programming suffers from the curse of dimensionality. Instead of focusing on the general Pomdps, we present a subclass where transitions and observations mappings are deterministic: Deterministic Partially Observed Markov Decision Processes (Det-Pomdp). That subclass of problems has been studied by (Littman, 1996) and (Bonet, 2009). It was first considered as a limit case of Pomdps by Littman, mainly used to illustrate the complexity of Pomdps when considering as few sources of uncertainties as possible. In this paper, we improve on Littman's complexity bounds. We then introduce and study an even simpler class: Separated Det-Pomdps and give some new complexity bounds for this class. This new class of problems uses a property of the dynamics and observation to push back the curse of dimensionality. (10.1007/s10479-024-06282-0)
  DOI : 10.1007/s10479-024-06282-0
• A posteriori error estimates for the DD+$L^2$ jumps method on the Neutron Diffusion equations
  
  • Ciarlet Patrick
  • Do Minh-Hieu
  • Gervais Mario
  • Madiot François
  Computers & Mathematics with Applications, Elsevier, 2025, 195, pp.349-365. We analyse a posteriori error estimates for the discretization of the neutron diffusion equations with a Domain Decomposition Method, the so-called DD+$L^2$ jumps method. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. Classically, one introduces a Lagrange multiplier to account for the jumps on the interface. This Lagrange multiplier is used for the reconstruction of the physical variables. Remarkably, no reconstruction of the Lagrange multiplier is needed to achieve the optimal a posteriori estimates. (10.1016/j.camwa.2025.07.026)
  DOI : 10.1016/j.camwa.2025.07.026
• 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, 2025, 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
• Machine Learning for Scientific Computing and Numerical Analysis
  
  • Montanelli Hadrien
  , 2025. These notes are for third-year students at École Polytechnique (Palaiseau, France), who have completed two years of classes préparatoires, making the material equivalent to a graduate-level course in the UK or US. The development of a Scientific Machine Learning (SciML) course for third-year students bridges the gap between scientific computing and machine learning. SciML, an emerging field, enables efficient methods for solving complex problems in science and engineering, like partial differential equations (PDEs). This course offers a new, valuable specialization, helping students stand out in the evolving job market, where demand for ML experts with specialized skills is growing.
• Adaptive mesh refinement on Cartesian meshes applied to the mixed finite element discretization of the multigroup neutron diffusion equations
  
  • Ciarlet Patrick
  • Do Minh-Hieu
  • Madiot François
  , 2025, 60. The multigroup neutron diffusion equations are often used to model the neutron density at the nuclear reactor core scale. Classically, these equations can be recast in a mixed variational form.This chapter presents an adaptive mesh refinement approach based on a posteriori estimators. We focus on refinement strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications.
• Averaged Steklov Eigenvalues, Inside Outside Duality and Application to Inverse Scattering
  
  • Audibert Lorenzo
  • Haddar Houssem
  • Pourre Fabien
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2025. <div><p>We introduce a new family of artificial backgrounds corresponding to averaged impedance boundary conditions formulated in an abstract framework. These backgrounds are used to define a finite number of averaged Steklov eigenvalues, which are associated with inverse scattering problems from inhomogeneous media. We prove that these special eigenvalues can be determined from full-aperture, fixed-frequency far-fields using the inside-outside duality method. We then show and numerically demonstrate how this method can be used to reconstruct averaged values of the refractive index.</p></div>
• Temporal and Spatial Decomposition for Prospective Studies in Energy Systems under Uncertainty
  
  • Martinez Parra Camila
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Janin Jean-Marc
  , 2025. The increasing penetration of renewable energy requires greater use of storage resources to manage system intermittency. As a result, there is growing interest in evaluating the opportunity cost of stored energy, or usage values, which can be derived by solving a multistage stochastic optimization problem. Stochasticity arises from net demand (the aggregation of demand and non-dispatchable generation), the availability of dispatchable generation, and inflows when the storage facilities considered are hydroelectric dams. We aim to compute these usage values for each market zone of the interconnected European electricity system, in the context of prospective studies currently conducted by RTE, the French TSO. The energy system is mathematically modelled as a directed graph, where nodes represent market zones and arcs represent interconnection links. In large energy systems, spatial complexity (thirty nodes in the system, each with at most one aggregated storage unit) compounds temporal complexity (a one-year horizon modelled with two timescales: weekly subproblems with hourly time steps). This work addresses three main sources of complexity: temporal, spatial, and stochastic. We tackle the multinode multistage stochastic optimisation problem by incorporating a spatio-temporal decomposition scheme. To efficiently compute usage values, we apply Dual Approximate Dynamic Programming (DADP), which enables tractable decomposition across both time and space. This approach yields nodal usage values that depend solely on the local state of each node, independently of the others. We conduct numerical studies on a realistic system composed of thirty nodes (modelling part of Europe) and show that DADP obtains competitive results when comparing with traditional methods like Stochastic Dual Dynamic Programming (SDDP).
• Explicit T-coercivity for the Stokes problem: a coercive finite element discretization
  
  • Ciarlet Patrick
  • Jamelot Erell
  Computers & Mathematics with Applications, Elsevier, 2025, 188, pp.137-159. Using the T -coercivity theory as advocated in Chesnel-Ciarlet [Numer. Math., 2013], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small (10.1016/j.camwa.2025.03.028)
  DOI : 10.1016/j.camwa.2025.03.028
• Continuous-Time Nonlinear Optimal Control Problem Under Signal Temporal Logic Constraints
  
  • Lai En
  • Bonalli Riccardo
  • Girard Antoine
  • Jean Frédéric
  , 2025. This work introduces a novel method for solving optimal control problems under Signal Temporal Logic (STL) constraints, by implementing STL constraints into the dynamics. Our approach reformulates the original problem as a classical continuous-time optimal control problem. Specifically, we extend the original dynamics by introducing auxiliary variables that encode STL satisfaction through their evolution and boundary conditions. Numerical simulations are realized to demonstrate the feasibility of our method, highlighting its potential for practical applications.
• Shape optimization of slip-driven axisymmetric microswimmers
  
  • Liu Ruowen
  • Zhu Hai
  • Guo Hanliang
  • Bonnet Marc
  • Veerapaneni Shravan
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2025, 47 (2), pp.A1065-A1090. In this work, we develop a computational framework that aims at simultaneously optimizing the shape and the slip velocity of an axisymmetric microswimmer suspended in a viscous fluid. We consider shapes of a given reduced volume that maximize the swimming efficiency, i.e., the (size-independent) ratio of the power loss arising from towing the rigid body of the same shape and size at the same translation velocity to the actual power loss incurred by swimming via the slip velocity. The optimal slip and efficiency (with shape fixed) are here given in terms of two Stokes flow solutions, and we then establish shape sensitivity formulas of adjoint-solution that provide objective function derivatives with respect to any set of shape parameters on the sole basis of the above two flow solutions. Our computational treatment relies on a fast and accurate boundary integral solver for solving all Stokes flow problems. We validate our analytic shape derivative formulas via comparisons against finite-difference gradient evaluations, and present several shape optimization examples. (10.1137/24M1659649)
  DOI : 10.1137/24M1659649
• On the breathing of spectral bands in periodic quantum waveguides with inflating resonators
  
  • Chesnel Lucas
  • Nazarov Sergei A
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2025, 59 (4). We are interested in the lower part of the spectrum of the Dirichlet Laplacian A^ε in a thin waveguide Π^ε obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry Ω by a small factor ε &gt; 0. The Floquet-Bloch theory ensures that the spectrum of A^ε has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to +∞ as O(ε^{-2}) when ε → 0^+. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in Ω that we denote by X_†. Generically, i.e. for most Ω, there holds X_† = {0} and the lower part of the spectrum of A^ε is very sparse, made of bands of length at most O(ε) as ε → 0^+. For certain Ω however, we have dim X_† = 1 and then there are bands of length O(1) which allow for wave propagation in Π^ε. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing Ω around a particular Ω_⋆ where dim X_† = 1. We show a breathing phenomenon for the spectrum of A^ε : when inflating Ω around Ω_⋆ , the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold π^2 /ε^2 , stops breathing and becomes extremely short as Ω continues to inflate.
• Maxwell's equations with hypersingularities at a negative index material conical tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2025, 7 (1), pp.127–169. We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity ε and the permeability µ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in ε and in µ, the corresponding scalar operators are not of Fredholm type in the usual H^1 spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical L^2 framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the T-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.
• Fading regularization method for an inverse boundary value problem associated with the biharmonic equation
  
  • Boukraa Mohamed Aziz
  • Caillé Laëtitia
  • Delvare Franck
  Journal of Computational and Applied Mathematics, Elsevier, 2025, 457, pp.116285. In this paper, we propose a numerical algorithm that combines the fading regularization method with the method of fundamental solutions (MFS) to solve a Cauchy problem associated with the biharmonic equation. We introduce a new stopping criterion for the iterative process and compare its performance with previous criteria. Numerical simulations using MFS validate the accuracy of this stopping criterion for both compatible and noisy data and demonstrate the convergence, stability, and efficiency of the proposed algorithm, as well as its ability to deblur noisy data. (10.1016/j.cam.2024.116285)
  DOI : 10.1016/j.cam.2024.116285
• SpinDoctor-IVIM: A virtual imaging framework for intravoxel incoherent motion MRI
  
  • Lashgari Mojtaba
  • Yang Zheyi
  • Bernabeu Miguel O
  • Li Jing-Rebecca
  • Frangi Alejandro F
  Medical Image Analysis, Elsevier, 2025, 99, pp.103369. <div><p>Intravoxel incoherent motion (IVIM) imaging is increasingly recognised as an important tool in clinical MRI, where tissue perfusion and diffusion information can aid disease diagnosis, monitoring of patient recovery, and treatment outcome assessment. Currently, the discovery of biomarkers based on IVIM imaging, similar to other medical imaging modalities, is dependent on long preclinical and clinical validation pathways to link observable markers derived from images with the underlying pathophysiological mechanisms. To speed up this process, virtual IVIM imaging is proposed. This approach provides an efficient virtual imaging tool to design, evaluate, and optimise novel approaches for IVIM imaging. In this work, virtual IVIM imaging is developed through a new finite element solver, SpinDoctor-IVIM, which extends SpinDoctor, a diffusion MRI simulation toolbox. SpinDoctor-IVIM simulates IVIM imaging signals by solving the generalised Bloch-Torrey partial differential equation. The input velocity to SpinDoctor-IVIM is computed using HemeLB, an established Lattice Boltzmann blood flow simulator. Contrary to previous approaches, SpinDoctor-IVIM accounts for volumetric microvasculature during blood flow simulations, incorporates diffusion phenomena in the intravascular space, and accounts for the permeability between the intravascular and extravascular spaces. The above-mentioned features of the proposed framework are illustrated with simulations on a realistic microvasculature model.</p></div> (10.1016/j.media.2024.103369)
  DOI : 10.1016/j.media.2024.103369
• Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (II) A modal approach
  
  • Cassier Maxence
  • Joly Patrick
  • Martínez Luis Alejandro Rosas
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2025, 201. This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in [Cassier, Joly, Rosas Martínez, Z. Angew. Math. Phys. 74 (2023), 115], we develop a method based on the spectral analysis of the underlying non-self-adjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation. (10.48550/arXiv.2312.12231)
  DOI : 10.48550/arXiv.2312.12231
• Scattering of transient waves by an interface with time-modulated jump conditions
  
  • Michaël Darche
  • Assier Raphaël
  • Guenneau S
  • Lombard Bruno
  • Touboul Marie
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2025, 335, pp.923-951. Time modulation of the physical parameters offers interesting new possibilities for wave control. Examples include amplification of waves, harmonic generation and non-reciprocity, without resorting to non-linear mechanisms. Most of the recent studies focus on the time-modulation of the bulk physical properties. However, as the temporal modulation of these properties is difficult to achieve experimentally, we will concentrate here on the special case of an interface with time-varying jump conditions, which is simpler to implement. This work is focused on wave propagation in a one-dimensional medium containing one modulated interface. Properties of the scattered waves are investigated theoretically: energy balance, generation of harmonics, impedance matching and non-reciprocity. A fourth-order numerical method is also developed to simulate transient scattering. Numerical experiments are conducted to validate the numerical scheme and to illustrate the theoretical findings.
• Withdrawal of: Solution of the Ovals problem
  
  • Chitour Yacine
  • Denzler Jochen
  • Jean Frédéric
  • Trélat Emmanuel
  , 2025. In the previous version of the preprint, we made a mistake in our proposed solution to the Ovals problem (formulated in [3, 24]). The erroneous claim is that the operator A_T, used in the proof of Lemma 2.4.1, is selfadjoint. But this fact is wrong, as kindly pointed out to us by Matthias Baur, Rupert L. Frank, Larry Read and Timo Weidl, whom we warmly thank. At the moment, unfortunately, this mistake seems fatal to us, so the Ovals conjecture remains open. Nevertheless, since our work contains arguments that may be useful to address the conjecture, we let it available as a preprint, with a warning on Section 2.4.
• Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging
  
  • Garnier Josselin
  • Giovangigli Laure
  • Goepfert Quentin
  • Millien Pierre
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025 (1930-8337). <div><p>In this paper, we present a mathematical model and analysis for a new experimental method [Bureau and al., arXiv:2409.13901, 2024] for effective sound velocity estimation in medical ultrasound imaging. We perform a detailed analysis of the point spread function of a medical ultrasound imaging system when there is a mismatch between the effective sound speed in the medium and the one used in the backpropagation imaging functional. Based on this analysis, an estimator for the speed of sound error is introduced. Using recent results on stochastic homogenization of the Helmholtz equation, we provide a representation formula for the field scattered by a random multi-scale medium (whose acoustic behavior is similar to a biological tissue) in the time-harmonic regime. We then prove that statistical moments of the imaging function can be accessed from data collected with only one realization of the medium. We show that it is possible to locally extract the point spread function from an image constituted only of speckle and build an estimator for the effective sound velocity in the micro-structured medium. Some numerical illustrations are presented at the end of the paper.</p></div> (10.3934/ipi.2026001)
  DOI : 10.3934/ipi.2026001
• ABOUT SEMILINEAR LOW DIMENSION BESSEL PDEs
  
  • Ohashi Alberto
  • Russo Francesco
  • Teixeira Alan
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2025. We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator $L^\delta$ with low dimension $0 &lt; \delta &lt; 1$. $L^\delta$ is a local operator, whose drift is the derivative of $x \mapsto \log (\vert x\vert)$: in particular it is a Schwartz distribution, which is not the derivative of a continuous function. The solutions are intended in a duality ("weak") sense with respect to state space $L^2(\R_+, d\mu),$ $\mu$ being an invariant measure for the Bessel semigroup. (10.1007/s40072-025-00386-9)
  DOI : 10.1007/s40072-025-00386-9