Publications

2023

• Properties and Proximal Point Type Methods for Strongly Quasiconvex Functions in Hilbert Spaces
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2023. We study strongly quasiconvex functions defined on Hilbert spaces and taking extended real values from both theoretical and algorithmic points of view. To this end, we verify the existence of (global) minimizers and the properties of the proximal operator for such functions, previously investigated merely on finitely dimensional spaces. Subsequently, we discuss a relaxed-inertial proximal point-type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions over closed convex sets in Hilbert spaces. We obtained a Q-linear convergence rate for the relaxed proximal point-type method and, for the corresponding function values, an asymptotic rate of O(1/k) to the optimal value of the considered optimization problem.
• Anomalous diffusion limit for a kinetic equation with a thermostatted interface
  
  • Bogdan Krzysztof
  • Komorowski Tomasz
  • Marino Lorenzo
  Probability Theory and Related Fields, Springer Verlag, 2023, 189 (1-2), pp.721-769. We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-killing condition at the interface. Both the coefficient describing the probability of killing and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in Komorowski et al. (Ann Prob 48:2290–2322, 2020), where the case of a non-degenerate probability of killing has been considered. (10.1007/s00440-023-01251-3)
  DOI : 10.1007/s00440-023-01251-3
• Inverse scattering problems for the time harmonic magnetic Schrödinger operator
  
  • LABIDI Amal
  , 2023. This thesis focuses on analyzing the inverse scattering problems by inhomogeneous medium for the time harmonic magnetic Schrödinger operator. The first inverse problem deals with the stability of the problem of identifying the magnetic and electric potentials from near field and far field patterns using geometrical optics solutions. However, we must first show that the direct scattering problem is well-posed. To do this, we used two different approaches: one is the variational approach and the other is a volume integral equation known as the Lippmann-Schwinger integral equation. Second, we consider the inverse medium scattering problem, we review the sampling methods used to determine the shape of a perturbation from measurements of scattered waves at a fixed frequency, where our focus is on the Linear Sampling Method (LSM) and the Factorization Method (FM). Several validating results are presented in 2D. In addition, it is shown that the shape of a perturbation is uniquely determined from the far field pattern for all incident plane waves. Finally, we investigate the well-posedness of the interior transmission problem and the discreteness of the set of transmission eigenvalues by applying Fredholm theory and the upper triangular Fredholm theory.
• Generalized linear sampling methods for locally perturbed periodic layers
  
  • Jenhani Nouha
  , 2023. We are interested in this PhD thesis in studying the inverse scattering problem for the reconstruction of defects in unbounded periodic structures, without a knowledge of the periodic background, and without assuming that the defect is also periodic. We first study the inverse problem using quasi-periodic incident waves. More precisely we employ the GLSM to solve the inverse problem for a single Floquet-Bloch mode. We then consider the case where non-quasi periodic incident sources are applied, and we employ the GLSM to define an indicator function for the full domain. Based on these two results, we apply the differential imaging to construct an indicator function allowing us to directly construct the defect. We give some numerical examples for the reconstruction obtained using the indicator function associated with the single Floquet-Bloch mode. We finally give a study for the well-posedness of the interior transmission problem by considering the cases when the perturbation is included in the periodic background. The study is based on the application of the Floquet-Bloch transform, a discretization with respect to the Floquet-Bloch variable and a convergence analysis to construct a solution for the Interior Transmission Problem.
• Numerical methods to estimate brain micro-structure from diffusion MRI data
  
  • Yang Zheyi
  , 2023. Diffusion magnetic resonance imaging (diffusion MRI) is a widely used non-invasive imaging modality to probe the micro-structural properties of biological tissues below the spatial resolution, by indirectly measuring the diffusion displacement of water molecules. Due to the geometrical complexity of the brain and intricate diffusion MRI mechanism, it is challenging to directly link the received signals to meaningful biophysical parameters, such as axon radii or volume fraction.In recent years, several biophysical models have been introduced to address the issue of weak interpretability. These models represent the diffusion MRI signals as a mixture of analytical signals under certain assumptions, e.g. impermeable membranes, of various disconnected simple geometries, such as spheres and sticks. Subsequently, they aim to extract the parameters of these geometries, which correlate with biophysical parameters, by inverting the analytical expression.However, the validity of these assumptions remains undetermined in actual experiments.The objective of this thesis is to improve the microstructure estimation reliability and efficiency from two perspectives. First, to facilitate the quantitative study of the valid range of biophysical models and the effect of geometrical deformation and cell membrane permeability via simulation, we proposed two reduced models derived from the Bloch-Torrey equation, respectively. For the case of the presence of permeable membranes, a new simulation approach using impermeable Laplace eigenbasis is proposed. As for the geometrical deformation, we use an asymptotic expansion with respect to the deformation angles to approximate the signal. These two reduced models enable efficient computation of signals for various values of deformation/permeability. Numerical simulations reveal that these two models can fast compute the signals within a reasonable error level compared to existing methods. Several studies have been conducted about the effects of permeability and deformation on the signals or the apparent diffusion coefficient (ADC), using the proposed models.Second, instead of inverting a simplified geometries model, we present a novel approach to associate soma size in gray matter by intermediary biomarkers. Numerical simulations identify a correlation between the volume-weighted soma radius/volume fraction and the inflection point of direction-averaged signals at high b-values (b>2500s/mm^2), offering insights for microstructure estimation. We fit a fully connected neural network using these biomarkers and compared to biophysical models, this approach offers comparable results on both synthetic and in vivo data and fast estimation since no inversion is involved.
• Méthode de décomposition de domaine pour les problèmes couplés acoustique-élastique, dans le domaine temporel. Application aux explosions sous-marines.
  
  • Nassor Alice
  , 2023. Ce travail étudie les approches globales en temps de décomposition de domaine pour résoudre des problèmes transitoires d'interaction fluide-structure. Afin de déterminer un algorithme optimal, nous étudions dans un premier temps la solvabilité des problèmes élastodynamiques et acoustiques transitoires avec des conditions aux frontières de type Robin et de Neumann. Nous énonçons des résultats de solvabilité, en soulignant les différentes régularités espace-temps des solutions. Nous étudions également la solvabilité du problème couplé élastodynamique-acoustique transitoire. Puis en nous basant sur ces résultats mathématiques, nous proposons ensuite un algorithme itératif global en temps basé sur les conditions aux limites de type Robin pour le problème couplé et prouvons sa convergence.Ces résultats sont ensuite mis en oeuvre pour coupler deux méthodes numériques efficaces. La réponse du fluide en temps discret est obtenue à l'aide d'une approche Z-BEM qui combine (i) une méthode d'éléments de frontière (BEM) accélérée par la méthode des matrices hiérarchiques dans le domaine de Laplace et (ii) une quadrature de convolution. La réponse de la structure est modélisée à l'aide de la méthode des éléments finis. Nous développons de cette manière une méthode numérique de couplage itérative globale en temps à convergence garantie, permettant en outre d'utiliser deux méthodes numériques distinctes de manière non intrusive.Plusieurs améliorations sont ensuite proposées: une méthode d'accélération de convergence est mise en œuvre et une approximation à haute fréquence est proposée pour améliorer l'efficacité de la Z-BEM. On propose ensuite un deuxième couplage itératif global-en-temps basé sur une interface acoustique-acoustique, dont la convergence est également démontrée. Ce couplage permet ensuite d'introduire des effets non linéaires dus au phénomène de cavitation pour préciser le modèle fluide. La Z-BEM est enfin adaptée en utilisant la méthode des images pour permettre la prise en compte d'une surface libre.Cette méthode est appliquées à des problèmes à dynamique rapide de dispersion d'ondes de choc acoustiques par des structures élastiques immergées et permet de simuler des configurations réalistes rencontrées dans l'industrie navale.
• Wave propagation in quasi-periodic media
  
  • Amenoagbadji Pierre
  , 2023. The goal of this thesis is to develop efficient numerical methods for the solution of the time-harmonic wave equation in quasiperiodic media, in the spirit of methods previously developed for periodic media. The goal is to use as in quasiperiodic homogenization the idea that an elliptic PDE with quasiperiodic coefficients can be interpreted as the cut of a higher-dimensional PDE which is elliptically degenerate, but with periodic coefficients. The periodicity property allows to use adapted tools, but the non-elliptic aspect makes the mathematical and numerical analysis of the PDE delicate. One application concerns transmission problems between periodic half-spaces (typically photonic crystals) when (1) the interface does not cut the periodic half-spaces in a direction of periodicity, or (2) when the periodic media have noncommensurate periods along the interface.
• A simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI
  
  • Fang Chengran
  • Yang Zheyi
  • Wassermann Demian
  • Li Jing-Rebecca
  Medical Image Analysis, Elsevier, 2023, 90, pp.102979. We propose a framework to train supervised learning models on synthetic data to estimate brain microstructure parameters using diffusion magnetic resonance imaging (dMRI). Although further validation is necessary, the proposed framework aims to seamlessly incorporate realistic simulations into dMRI microstructure estimation. Synthetic data were generated from over 1,000 neuron meshes converted from digital neuronal reconstructions and linked to their neuroanatomical parameters (such as soma volume and neurite length) using an optimized diffusion MRI simulator that produces intracellular dMRI signals from the solution of the Bloch–Torrey partial differential equation. By combining random subsets of simulated neuron signals with a free diffusion compartment signal, we constructed a synthetic dataset containing dMRI signals and 40 tissue microstructure parameters of 1.45 million artificial brain voxels. To implement supervised learning models we chose multilayer perceptrons (MLPs) and trained them on a subset of the synthetic dataset to estimate some microstructure parameters, namely, the volume fractions of soma, neurites, and the free diffusion compartment, as well as the area fractions of soma and neurites. The trained MLPs perform satisfactorily on the synthetic test sets and give promising in-vivo parameter maps on the MGH Connectome Diffusion Microstructure Dataset (CDMD). Most importantly, the estimated volume fractions showed low dependence on the diffusion time, the diffusion time independence of the estimated parameters being a desired property of quantitative microstructure imaging. The synthetic dataset we generated will be valuable for the validation of models that map between the dMRI signals and microstructure parameters. The surface meshes and microstructures parameters of the aforementioned neurons have been made publicly available. (10.1016/j.media.2023.102979)
  DOI : 10.1016/j.media.2023.102979
• Personalized incentives as feedback design in generalized Nash equilibrium problems
  
  • Fabiani Filippo
  • Simonetto Andrea
  • Goulart Paul J.
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2023, 68 (12), pp.7724-7739. We investigate both stationary and time-varying, nonmonotone generalized Nash equilibrium problems that exhibit symmetric interactions among the agents, which are known to be potential. As may happen in practical cases, however, we envision a scenario in which the formal expression of the underlying potential function is not available, and we design a semi-decentralized Nash equilibrium seeking algorithm. In the proposed two-layer scheme, a coordinator iteratively integrates the (possibly noisy and sporadic) agents' feedback to learn the pseudo-gradients of the agents, and then design personalized incentives for them. On their side, the agents receive those personalized incentives, compute a solution to an extended game, and then return feedback measurements to the coordinator. In the stationary setting, our algorithm returns a Nash equilibrium in case the coordinator is endowed with standard learning policies, while it returns a Nash equilibrium up to a constant, yet adjustable, error in the time-varying case. As a motivating application, we consider the ridehailing service provided by several companies with mobility as a service orchestration, necessary to both handle competition among firms and avoid traffic congestion, which is also adopted to run numerical experiments verifying our results. (10.1109/TAC.2023.3287218)
  DOI : 10.1109/TAC.2023.3287218
• On the use of a tailored fluid-fluid Green's function to predict scattering from two-phase fluid interfaces
  
  • Pacaut Louise
  • Serre Gilles
  • Mercier Jean-François
  • Chaillat Stéphanie
  , 2023, 268 (8), pp.434-443. In naval applications, the precise knowledge of the acoustical behaviour of two-phase fluids is of interest. Indeed, they appear in many configurations as bubbles curtains, cavitation (along propellers and pipes) or two-phase turbulent boundary layers and wakes (gas exhaust). To model this problem, we focus on a Boundary Element/ Boundary Element coupling for low to moderate frequencies. We propose two approaches: (a) solving the Helmholtz equations in each phase by introducing the free field Green's function and (b) determining a tailored Green's function, taking into account the presence of the two-phase fluid. The determination of a tailored Green's function has two main advantages: (i) it allows a reduction of the numerical model, since it contains all the information on the fluid-fluid coupling, in particular the transmission conditions and resonances and (ii) it is sourceindependent and thus it gives directly the answer to any source. Numerical tests on the evaluation of radiated noise are performed to determine the efficiency of the approach based on a tailored Green's function compared to the free-field based one. (10.3397/IN_2023_0076)
  DOI : 10.3397/IN_2023_0076
• Beyond the Fermat Optimality Rules
  
  • Grad Sorin-Mihai
  • Abbasi Malek
  • Théra Michel
  , 2023.
• Contributions à la résolution de problèmes d'optimisation combinatoire difficiles
  
  • Ales Zacharie
  , 2023.
• Étude de deux problèmes de propagation d’ondes en milieu électromagnétique dispersif : 1) Stabilité en temps long dans un milieu de Drude-Lorentz; 2) Transmission entre une couche de metamateriau et un diélectrique.
  
  • Rosas Martinez Luis Alejandro
  , 2023. Cette thèse traite de deux problèmes indépendants liés aux phénomènes de propagation des ondes dans les milieux dispersifs. Dans la première partie, nous étudions le comportement en temps long des solutions des équations de Maxwell dans des milieux dissipatifs généralisés de Drude-Lorentz. Plus précisément, nous souhaitons quantifier les pertes dans de tels milieux à l'aide du taux de décroissance de l'énergie électromagnétique pour le problème de Cauchy correspondant. Cette première partie est elle-même composée de deux approches. La première, l'approche par fonctions de Lyapunov en fréquence, consiste à obtenir une inégalité différentielle (en temps) pour certaines fonctionnelles de la solution, les fonctions de Lyapunov L(k) où k désigne la fréquence spatiale. Les estimations de stabilité sont ensuite obtenues par l'intégration en temps de l'inégalité différentielle. En développant cette méthode, nous obtenons un résultat de stabilité polynomiale sous des hypothèses de dissipation fortes. La deuxième approche, l'approche modale, exploite les propriétés spectrales de l'opérateur hamiltonien apparaissant dans le problème de Cauchy. Cette dernière approche améliore la première en autorisant des hypothèses de dissipation faibles. Dans la deuxième partie du travail, nous nous intéressons au problème de transmission d'une couche de métamatériau de Drude non dissipatif dans un milieu diélectrique. Dans ce contexte, nous considérons les équations de Maxwell temporelles bidimensionnel en polarisation TM et nous les reformulons en une équation de Schrödinger dont le Hamiltonien, A, est un opérateur autoadjoint non borné. La transformation de Fourier nous permet de travailler avec des Hamiltoniens réduits A(k), k ∈ R. Enfin, nous nous intéressons au spectre ponctuel du Hamiltonien réduit qui est lié aux modes guidés du problème original. Cette étude débouche sur une relation de dispersion dont la difficulté réside dans son caractère hautement non linéaire par rapport au paramètre spectral. Nous prouvons l'existence d'une infinité dénombrable de branches de solutions pour la relation de dispersion : les courbes de dispersion. Nous donnons une analyse précise de ces courbes et mettons en lumière, notamment, l'existence d'ondes guidées correspondant à des palsmons surface.
• A hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems
  
  • Modave Axel
  • Chaumont-Frelet Théophile
  Journal of Computational Physics, Elsevier, 2023, 493, pp.112459. A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns corresponding to characteristic variables are defined at the interface between the elements, and the physical fields are eliminated to obtain a reduced system. The reduced system can be written as a fixed-point problem that can be solved with stationary iterative schemes. Numerical results with 2D benchmarks are presented to study the performance of the approach. Compared to the standard HDG approach, the properties of the reduced system are improved with CHDG, which is more suited for iterative solution procedures. The condition number of the reduced system is smaller with CHDG than with the standard HDG method. Iterative solution procedures with CGN or GMRES required smaller numbers of iterations with CHDG. (10.1016/j.jcp.2023.112459)
  DOI : 10.1016/j.jcp.2023.112459
• Beyond the Fermat Optimality Rules
  
  • Grad Sorin-Mihai
  • Abbasi Malek
  • Théra Michel
  , 2023.
• Acoustic waveguide with a dissipative inclusion
  
  • Chesnel Lucas
  • Heleine Jérémy
  • Nazarov Sergei A.
  • Taskinen Jari
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57 (6), pp.3585-3613. We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient η, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when η → 0+ (small dissipation) and when η → +∞ (large dissipation). Surprisingly, at the limit η → +∞, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to η. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations. (10.1051/m2an/2023070)
  DOI : 10.1051/m2an/2023070
• Spectrum of the Dirichlet Laplacian in a thin cubic lattice
  
  • Chesnel Lucas
  • Nazarov Sergei A.
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57 (6), pp.3251-3273. We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width ε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ as ε tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length O(e−δ/ε), δ > 0, while the length of the next spectral segments is O(ε). To establish these results, we need to study in detail the properties of the Dirichlet Laplacian AΩ in the geometry Ω obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max–min arguments as well as a well-chosen Poincaré–Friedrichs inequality, we prove that AΩ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for AΩ, that is no non trivial bounded solution at the threshold frequency for AΩ. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis. (10.1051/m2an/2023082)
  DOI : 10.1051/m2an/2023082
• Constrained Hierarchical Clustering via Graph Coarsening and Optimal Cuts
  
  • Mauduit Eliabelle
  • Simonetto Andrea
  , 2023. Motivated by extracting and summarizing relevant information in short sentence settings, such as satisfaction questionnaires, hotel reviews, and X/Twitter, we study the problem of clustering words in a hierarchical fashion. In particular, we focus on the problem of clustering with horizontal and vertical structural constraints. Horizontal constraints are typically cannot-link and must-link among words, while vertical constraints are precedence constraints among cluster levels. We overcome state-of-the-art bottlenecks by formulating the problem in two steps: first, as a soft-constrained regularized least-squares which guides the result of a sequential graph coarsening algorithm towards the horizontal feasible set. Then, flat clusters are extracted from the resulting hierarchical tree by computing optimal cut heights based on the available constraints. We show that the resulting approach compares very well with respect to existing algorithms and is computationally light.
• Combined inversion methods for inverse conductivity problems
  
  • Jerbi Rahma
  , 2023. In this thesis, we develop an inversion method that combines the use of a Kohn-Vogelius type cost functional with a non-overlapping domain decomposition method as an iterative solver. The idea behind this method is to iterate simultaneously on the solution of the direct problem using the domain decomposition method and on the unknown of the inverse problem using gradient descent on the Kohn-Vogelius cost functional. This type of approach falls into the category of ”one-shot inversion methods,” and its use has the potential to significantly reduce the cost of inversion when the numerical solution of the direct problem is costly. We are particularly interested in the case of geometric inverse problems where the unknown of the inverse problem is the support of a physical parameter’s discontinuity. The developments made in this area were modeled on the inverse electrical conductivity problem, where the goal is to reconstruct the conductivity discontinuity interface from Cauchy data on the domain boundary. We prove the local convergence of the method in simplified cases and numerically show its efficiency for some two dimensional experiments with synthetic data. Additionally, we extend our approach to the more complex case where we also iterate on the value of con- ductivity. In this context, we have also developed an alternating inversion algorithm between the geometry and the inner value of the conductivity, with an adaptive descent step.
• THE L 2 -NORM OF THE FORWARD STOCHASTIC INTEGRAL W.R.T. FRACTIONAL BROWNIAN MOTION H > 1 2
  
  • Ohashi Alberto
  • Russo Francesco
  , 2023. In this article, we present the exact expression of the L 2-norm of the forward stochastic integral driven by the multi-dimensional fractional Brownian motion with parameter 1 2 < H < 1. The class of integrands only requires rather weak integrability conditions compatible w.r.t. a random finite measure whose density is expressed as a second-order polynomial of the underlying driving Gaussian noise. A simple consequence of our results is the exact expression of the L 2-norm for the pathwise Young integral.
• The Half-Space Matching method for elastodynamic scattering problems in unbounded domains
  
  • Bécache Éliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Journal of Computational Physics, Elsevier, 2023, pp.112320. In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters. (10.1016/j.jcp.2023.112320)
  DOI : 10.1016/j.jcp.2023.112320
• Scattered wavefield in the stochastic homogenization regime
  
  • Garnier Josselin
  • Giovangigli Laure
  • Goepfert Quentin
  • Millien Pierre
  , 2023. In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments.
• Optimal control under uncertainties for the vertical landing of the first stage of a reusable launch vehicle
  
  • Leparoux Clara
  , 2023. The work in this thesis focuses on the development of a robust trajectory planning and optimal control method. It provides theoretical justifications for the method presented, proving the existence of solutions to the problem formulated. Finally, the method is applied to a trajectory planning problem for the vertical landing of a reusable launch vehicle first stage.
• The isometry of symmetric-Stratonovich integrals w.r.t. Fractional Brownian motion $H< \frac{1}{2}$
  
  • Ohashi Alberto
  • Russo Francesco
  • Viens Frederi
  , 2023. In this work, we present a detailed analysis on the exact expression of the $L^2$-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion $B$ with parameter $\frac{1}{4} < H < \frac{1}{2}$. Our main result is a complete description of a Hilbert space of integrand processes which realizes the $L^2$-isometry where none regularity condition in the sense of Malliavin calculus is imposed. The main idea is to exploit the regularity of the conditional expectation of the tensor product of the increments $B_{t-\delta,t+\delta}\otimes B_{s-\epsilon,s+\epsilon}$ onto the Gaussian space generated by $(B_s,B_t)$ as $(\delta,\epsilon)\downarrow 0$. The Hilbert space is characterized in terms of a random Radon $\sigma$-finite measure on $[0,T]^2$ off diagonal which can be characterized as a product of a non-Markovian version of the stochastic Nelson derivatives. As a by-product, we present the exact explicit expression of the $L^2$-norm of the pathwise rough integral in the sense of Gubinelli.
• Optimizing Variational Circuits for Higher-Order Binary Optimization
  
  • Verchère Zoé
  • Elloumi Sourour
  • Simonetto Andrea
  , 2023. Variational quantum algorithms have been advocated as promising candidates to solve combinatorial optimization problems on near-term quantum computers. Their methodology involves transforming the optimization problem into a quadratic unconstrained binary optimization (QUBO) problem. While this transformation offers flexibility and a ready-to-implement circuit involving only two-qubit gates, it has been shown to be less than optimal in the number of employed qubits and circuit depth, especially for polynomial optimization. On the other hand, strategies based on higher-order binary optimization (HOBO) could save qubits, but they would introduce additional circuit layers, given the presence of higher-than-two-qubit gates. In this paper, we study HOBO problems and propose new approaches to encode their Hamiltonian into a ready-to-implement circuit involving only two-qubit gates. Our methodology relies on formulating the circuit design as a combinatorial optimization problem, in which we seek to minimize circuit depth. We also propose handy simplifications and heuristics that can solve the circuit design problem in polynomial time. We evaluate our approaches by comparing them with the state of the art, showcasing clear gains in terms of circuit depth.