Publications

2023

• 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
• Minimizing recovery cost of network optimization problems
  
  • Alès Zacharie
  • Elloumi Sourour
  Networks, Wiley, 2023, 83 (1). We propose a two-stage recoverable robustness approach that minimizes the recovery cost. In many applications, once the uncertainty ξ is revealed, it can be more important to recover a solution x ξ which is as similar as possible to the nominal solution x nom than to minimize the nominal objective value of x ξ. This for example occurs when the nominal solution is implemented on a regular basis or when the uncertainty is revealed late. We define the proactive problem which minimizes the weighted recovery costs over a discrete set of scenarios while ensuring optimality of the nominal objective value of x nom. We model the recovery cost of a scenario by a distance between the first-stage nominal solution and the second-stage solution recovered for this scenario. We show for two different solution distances d val and dstruct that the proactive problem is N P-hard for both the integer min-cost flow problem with uncertain arc demands and for the integer max-flow problem with uncertain arc capacities. For these two problems, we prove that once uncertainty is revealed, even identifying a reactive solution x r with a minimal distance to a given solution x nom is N P-hard for dstruct, and is polynomial for d val. We highlight the benefits of the proactive approach in a case study on a railroad planning problem. First, we compare it to the anchored and the k-distance approaches. Then, we show the efficiency of the proactive solution over reactive solutions. Finally, we illustrate the recovery cost reduction when relaxing the optimality constraint on the nominal objective of the proactive solution x nom. We also consider the min-max version of the proactive problem where we minimize the maximal recovery cost over all scenarios. We show that the same complexity results hold for this version. We also exhibit a class of problems for which the set of extreme points of the convex hull of a discrete uncertainty set always contain a worst-case scenario. We show that this result does not hold for three distinct classes deduced from the first one. (10.1002/net.22121)
  DOI : 10.1002/net.22121
• Mixed integer (non)linear approaches for the satellite constellation design problem
  
  • Mencarelli Luca
  • Floquet Julien
  • Georges Frédéric
  • Grenier Dominique
  Optimization and Engineering, Springer Verlag, 2023, 24, pp.2299–2320. In this paper, we propose mathematical optimization models to solve the satellite constellation design problem for discontinuous coverage. In such a design problem, the aim is to determine the minimal number of satellites (and, incidentally, their 3D placements) in order to observe a fixed Earth region within a given revisiting time. Two Mixed Integer Nonlinear formulations are introduced. The first one is a feasibility problem based on the direct mathematical definition of pixel observability. The second one consists in introducing a set of indicator variables which specify if a satellite observes a pixel at a given time-stamp. In order to obtain a linear problem, the possible positions of the satellites are discretized. Finally, computational results show the potential and limitations of the proposed approaches. (10.1007/s11081-022-09774-9)
  DOI : 10.1007/s11081-022-09774-9
• 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, 20, pp.1111–1138. 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. (10.30757/ALEA.v20-41)
  DOI : 10.30757/ALEA.v20-41
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• Relaxed-Inertial Proximal Point Type Algorithms for Quasiconvex Minimization
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  Journal of Global Optimization, Springer Verlag, 2023, 85 (3), pp.615-635. We propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. A relaxed version of the method where the constraint set is only closed and convex is also discussed, and so is the case of a quasiconvex objective function. Numerical experiments illustrate the theoretical results. (10.1007/s10898-022-01226-z)
  DOI : 10.1007/s10898-022-01226-z
• 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
• Distributed Personalized Gradient Tracking with Convex Parametric Models
  
  • Notarnicola Ivano
  • Simonetto Andrea
  • Farina Francesco
  • Notarstefano Giuseppe
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2023, 68 (1). (10.1109/TAC.2022.3147007)
  DOI : 10.1109/TAC.2022.3147007