Publications

2023

• Scattering in a partially open waveguide: the inverse problem
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2023, 17 (2), pp.463-469. In this paper we consider an inverse scattering problem which consists in retrieving obstacles in a partially embedded waveguide in the acoustic case, the measurements being located on the accessible part of the structure. Such accessible part can be considered as a closed waveguide (with a finite cross section), while the embedded part can be considered as an open waveguide (with an infinite cross section). We propose an approximate model of the open waveguide by using Perfectly Matched Layers in order to simplify the resolution of the inverse problem, which is based on a modal formulation of the Linear Sampling Method. Some numerical results show the efficiency of our approach. This paper can be viewed as a continuation of the article [11], which was focused on the forward problem. (10.3934/ipi.2022052)
  DOI : 10.3934/ipi.2022052
• Robust capacitated Steiner trees and networks with uniform demands
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Poirion Pierre‐louis
  • Ridremont Thomas
  Networks, Wiley, 2023, 82 (1), pp.3-31. We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in the case of terminals with uniform demands. Formally, we are given a graph, capacity, and cost functions on the edges, a root, a subset of vertices called terminals, and a bound k on the number of possible edge failures. We first study the problem where k=1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Secondly, we consider the problem of computing a minimum-cost survivable network, that is, a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a bilevel formulation (with an attacker and a defender), a cutset-based formulation and a flow-based one. We compare the formulations from a theoretical point of view, and we propose algorithms to solve them and compare their efficiency in practice. (10.1002/net.22143)
  DOI : 10.1002/net.22143
• Ultrasonic imaging in highly heterogeneous backgrounds
  
  • Pourahmadian Fatemeh
  • Haddar Houssem
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2023, 479 (2271). This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time t◦ contains (i) a multiply connected set of viscoelastic, anisotropic, and piece-wise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties, and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step t◦, t1, . . ., multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scattering equation. The incident fields affiliated with distinct pairs of obtained wavefronts are analyzed over the stationary and evolving scatterers for a suit of geometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basis for theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features. (10.1098/rspa.2022.0721)
  DOI : 10.1098/rspa.2022.0721
• Asymptotic models of the diffusion MRI signal accounting for geometrical deformations
  
  • Yang Zheyi
  • Mekkaoui Imen
  • Hesthaven Jan
  • Li Jing-Rebecca
  MathematicS In Action, Société de Mathématiques Appliquées et Industrielles (SMAI), 2023, 12 (1), pp.65-85. The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses can be modeled by the Bloch-Torrey partial differential equation (PDE). The associated diffusion MRI signal is the spatial integral of the solution of the Bloch-Torrey PDE. In addition to the signal, the time-dependent apparent diffusion coefficient (ADC) can be obtained from the solution of another partial differential equation, called the HADC model, which was obtained using homogenization techniques. In this paper, we analyze the Bloch-Torrey PDE and the HADC model in the context of geometrical deformations starting from a canonical configuration. To be more concrete, we focused on two analytically defined deformations: bending and twisting. We derived asymptotic models of the diffusion MRI signal and the ADC where the asymptotic parameter indicates the extent of the geometrical deformation. We compute numerically the first three terms of the asymptotic models and illustrate the effects of the deformations by comparing the diffusion MRI signal and the ADC from the canonical configuration with those of the deformed configuration. The purpose of this work is to relate the diffusion MRI signal more directly with tissue geometrical parameters. (10.5802/msia.32)
  DOI : 10.5802/msia.32
• 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, 2023, 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
• A General Comparison Principle for Hamilton Jacobi Bellman Equations on Stratified Domains
  
  • Jerhaoui Othmane
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2023, 29. This manuscript aims to study finite horizon, first order Hamilton Jacobi Bellman (HJB) equations on stratified domains. This problem is related to optimal control problems with discontinuous dynamics. We use nonsmooth analysis techniques to derive a strong comparison principle as in the classical theory and deduce that the value function is the unique viscosity solution. Furthermore, we prove some stability results of the Hamilton Jacobi Bellman equation. Finally, we establish a general convergence result for monotone numerical schemes in the stratified case. (10.1051/cocv/2022089)
  DOI : 10.1051/cocv/2022089
• 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, 2023, 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. (10.48550/arXiv.2310.04127)
  DOI : 10.48550/arXiv.2310.04127
• Wave propagation in one-dimensional quasiperiodic media
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  Communications in Optimization Theory, Mathematical Research Press, 2023. This work is devoted to the resolution of the Helmholtz equation −(µ u) − ρ ω 2 u = f in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed in [10, 19] for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation. (10.48550/arXiv.2301.01159)
  DOI : 10.48550/arXiv.2301.01159
• Variational methods for solving numerically magnetostatic systems
  
  • Ciarlet Patrick
  • Jamelot E.
  Advances in Computational Mathematics, Springer Verlag, 2023. In this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements, or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of $T$-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on an augmented variational formulation obtained with the help of the Weighted Regularization Method.
• An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Lassounon David
  • Rihani Mahran
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2023, 61 (3), pp.1316-1339. In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments. (10.1137/22M1495998)
  DOI : 10.1137/22M1495998
• ON SDEs FOR BESSEL PROCESSES IN LOW DIMENSION AND PATH-DEPENDENT EXTENSIONS
  
  • Ohashi Alberto
  • Russo Francesco
  • Teixeira Alan
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2023. The Bessel process in low dimension (0 ≤ δ ≤ 1) is not an Itô process and it is a semimartingale only in the cases δ = 1 and δ = 0. In this paper we first characterize it as the unique solution of an SDE with distributional drift or more precisely its related martingale problem. In a second part, we introduce a suitable notion of path-dependent Bessel processes and we characterize them as solutions of path-dependent SDEs with distributional drift.
• H-matrix accelerated FEM-BEM coupling for dynamic analysis of naval structures in pulsating potential fluids
  
  • Mavaleix-Marchessoux Damien
  • Bonnet Marc
  • Chaillat Stéphanie
  • Leblé Bruno
  , 2021. This article addresses one of the components of our ongoing work towards an efficient computational modeling methodology for evaluating all effects on a submerged structure of a remote underwater explosion. Following up on a previous study devoted to computing the transient acoustic fields induced by the shock wave initially sent by the blast on a rigid submarine, we focus here on the second stage of the underwater event, namely solving the transient fluid-structure interaction (FSI) between the structure and the incompressible potential flow induced by the delayed, and slower, oscillations of the gas bubble created by the remote blast. The boundary element method (BEM) is the best-suited approach for handling potential flow problems in large fluid domains (idealized as unbounded), whereas the finite element method (FEM) naturally applies to the transient structure analyses. To perform the FEM-BEM coupling we use a sub-cycling approach that alternates fluid and solid analyses with Neumann boundary conditions. The transient nature of the coupled analysis and the recourse to sub-cycling together make the overall procedure rely on a large number of BEM potential flow solutions, while the complexities of the wet surface and of the solid transient response imply a need for large BE models for the flow potential. This combination of reasons mandates accelerating the BE component. Accordingly, our main contribution is to study the feasibility and effectiveness of coupling the Hierarchical-matrix accelerated BEM (H-BEM) and the FEM for the FSI problems of interest. In particular, we show that the same integral operators can be used at all time instants in spite of the expected global motion of the submerged structure, a feature that the H-BEM can exploit to full advantage. The proposed original treatment is validated against analytical solutions for the case of a motionless or mobile rigid spherical immersed object, and then tested on a complex configuration representative of target applications.
• Time-vs. frequency-domain inverse elastic scattering: Theory and experiment
  
  • Liu Xiaoli
  • Song J
  • Pourahmadian Fatmeh
  • Haddar Houssem
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023, 83 (3). This study formally adapts the time-domain linear sampling method (TLSM) for ultrasonic imaging of stationary and evolving fractures in safety-critical components. The TLSM indicator is then applied to the laboratory test data of [22, 18] and the obtained reconstructions are compared to their frequency-domain counterparts. The results highlight the unique capability of the time-domain imaging functional for high-fidelity tracking of evolving damage, and its relative robustness to sparse and reduced-aperture data at moderate noise levels. A comparative analysis of the TLSM images against the multifrequency LSM maps of [22] further reveals that thanks to the full-waveform inversion in time and space, the TLSM generates images of remarkably higher quality with the same dataset. (10.1137/22M1522437)
  DOI : 10.1137/22M1522437
• Shape optimization of peristaltic pumps transporting rigid particles in Stokes flow
  
  • Bonnet Marc
  • Liu Ruowen
  • Veerapaneni Shravan
  • Zhu Hai
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2023, 45 (1), pp.B78-B106. This paper presents a computational approach for finding the optimal shapes of peristaltic pumps transporting rigid particles in Stokes flow. In particular, we consider shapes that minimize the rate of energy dissipation while pumping a prescribed volume of fluid, number of particles and/or distance traversed by the particles over a set time period. Our approach relies on a recently developed fast and accurate boundary integral solver for simulating multiphase flows through periodic geometries of arbitrary shapes. In order to fully capitalize on the dimensionality reduction feature of the boundary integral methods, shape sensitivities must ideally involve evaluating the physical variables on the particle or pump boundaries only. We show that this can indeed be accomplished owing to the linearity of Stokes flow. The forward problem solves for the particle motion in a slip-driven pipe flow while the adjoint problems in our construction solve quasi-static Dirichlet boundary value problems backwards in time, retracing the particle evolution. The shape sensitivities simply depend on the solution of one forward and one adjoint (for each shape functional) problems. We validate these analytic shape derivative formulas by comparing against finite-difference based gradients and present several examples showcasing optimal pump shapes under various constraints. (10.1137/21M144863X)
  DOI : 10.1137/21M144863X
• A Quantum Algorithm for the Sub-Graph Isomorphism Problem
  
  • Mariella Nicola
  • Simonetto Andrea
  ACM Transactions on Quantum Computing, ACM, 2023, 4 (2). We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number of qubits that scales logarithmically with the number of vertices of the graphs; and (2) on a new Ansatz that can efficiently probe the permutation space. Simulations are then presented to showcase the approach on graphs up to 16 vertices, whereas, given the logarithmic scaling, the approach could be applied to realistic sub-graph isomorphism problem instances in the medium term. (10.1145/3569095)
  DOI : 10.1145/3569095
• Robust Motion Planning of the Powered Descent of a Space Vehicle
  
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  IFAC-PapersOnLine, Elsevier, 2023, 56 (2), pp.2001-2006. The motion planning of powered descent problems has often been treated in the deterministic optimal control framework, which provides efficient theoretical and numerical tools. However, future applications require robustness, usually obtained by introducing stochastic components in the dynamics to model uncertainties. After stating the robust motion planning problem, this paper proposes a deterministic approximation which avoids the computational difficulties of stochastic optimal control. The approach consists of guiding the mean while reducing the covariance, the dynamics of these two quantities being approximated thanks to statistical linearization. In addition, since feedback control is necessary to control covariance, two techniques are provided to deal with actuator limits when the control is stochastic. (10.1016/j.ifacol.2023.10.1095)
  DOI : 10.1016/j.ifacol.2023.10.1095
• The linear sampling method for random sources
  
  • Garnier Josselin
  • Haddar Houssem
  • Montanelli Hadrien
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2023, 16 (3), pp.1572-1593. (10.1137/22M1531336)
  DOI : 10.1137/22M1531336
• Inversion of Eddy-Current Signals Using a Level-Set Method and Block Krylov Solvers
  
  • Audibert Lorenzo
  • Girardon Hugo
  • Haddar Houssem
  • Jolivet Pierre
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2023, 45 (3), pp.B366-B389. The application motivating this work is related to the identification of deposits inside nuclear power plant steam generators using eddy-current probes. We consider a realistic experimental process that relies on the scan of a domain by sweeping along a tube axis a probe made out of coils, playing the role of the sources/receivers. Solving the inverse shape problem associated with these measurements using a least squares method requires solutions to the eddy-current and the adjoint problems for a large number of right-hand sides at each gradient-descent iteration. Additional cost in the forward solver comes from the use of a potential formulation of the problem that has the advantage of being independent from the topology of the conductive media (that may vary during iterations). We use a level-set approach to avoid remeshing and handle unknown topologies. The crucial ingredient in our algorithm is an optimized way of handling high numbers of right-hand sides for iterative solvers of large-scale problems. We first benchmark various block Krylov methods, block GMRES and block BGCRODR, to test their effectiveness compared to their standard counterpart, i.e., GMRES and GCRODR. Then, we propose for BGCRODR a new implementation for recycling information from previously generated Krylov bases that scales better than traditional approaches. This part is independent from the practical inverse problem at hand. The efficiency of the overall inversion procedure is finally demonstrated on realistic synthetic 3D examples. (10.1137/20M1382064)
  DOI : 10.1137/20M1382064
• A posteriori error estimates for mixed finite element discretizations of the Neutron Diffusion equations
  
  • Ciarlet Patrick
  • Do Minh Hieu
  • Madiot François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57 (1), pp.1-27. We analyse a posteriori error estimates for the discretization of the neutron diffusion equations with mixed finite elements. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. We pay particular attention to AMR strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications. We exhibit a robust marker strategy for this specific constraint, the direction marker strategy. (10.1051/m2an/2022078)
  DOI : 10.1051/m2an/2022078
• Modelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM
  
  • Dansou Anicet
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Road Materials and Pavement Design, Taylor & Francis, 2023, 24, pp.631-652. We present a computational modeling approach aimed at investigating the effect of fiber grid reinforcement on crack opening displacement and fatigue crack propagation. Grid reinforcements are modeled using elastic membrane finite elements, while the cracked concrete is treated using a symmetric boundary element method (BEM), which in particular allows easy geometrical modelling and meshing of cracks. The BEM is accelerated by the fast multipole method, allowing the handling of potentially large BEM models entailed by three-dimensional configurations hosting multiple cracks. Fatigue crack growth is modelled using the Paris law. The proposed computational approach is first verified on a reinforced cracked beam, and then applied to a three-dimensional configuration featuring a grid-reinforced asphalt pavement. (10.1080/14680629.2022.2029755)
  DOI : 10.1080/14680629.2022.2029755
• Stability estimate for an inverse problem for the time harmonic magnetic schrödinger operator from the near and far field pattern
  
  • Bellassoued Mourad
  • Haddar Houssem
  • Labidi Amal
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2023, 55 (4), pp.2475-2504. We derive conditional stability estimates for inverse scattering problems related to time harmonic magnetic Schrödinger equation. We prove logarithmic type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions. (10.1137/22M1481956)
  DOI : 10.1137/22M1481956