Yannick Privat

Dans cette thèse, nous nous demandons si certaines formes présentes dans la nature résultent de l'optimisation d'un critère. Plus précisément, nous considérons un organe ou une partie du corps humain et tentons de deviner un critère que la nature aurait pu chercher à optimiser. Nous résolvons alors le problème d'optimisation de formes résultant afin de comparer la forme obtenue, théoriquement ou numériquement, avec la forme réelle de l'organe. Si ces deux formes sont proches, on pourra en déduire que le critère est convaincant. Dans la première partie de cette thèse, nous considérons l'exemple d'une fibre nerveuse de type axone ou dendrite. Nous proposons deux critères pour expliquer sa forme. Le premier traduit l'atténuation dans le temps du message électrique traversant la fibre et le second l'atténuation dans l'espace de ce message. Dans notre choix de modélisation, nous distinguons deux types de fibres nerveuses : celles qui sont connectées au noyau de la cellule et celles qui sont connectées entre elles. Les problèmes correspondants se ramènent à la minimisation par rapport au domaine des valeurs propres d'un opérateur elliptique et d'une fonction de transfert faisant intervenir la trace sur le bord du domaine du potentiel électrique au sein de la fibre. La seconde partie de cette thèse est dédiée à l'optimisation de la forme d'un arbre bronchique ou d'une partie de cet arbre. Nous considérons un critère de type << énergie dissipée >>. Dans une étude théorique, nous prouvons tout d'abord que le cylindre n'est pas une conduite optimale pour minimiser l'énergie dissipée par un fluide newtonien incompressible satisfaisant aux équations de Navier-Stokes. Nous effectuons ensuite des simulations en deux et trois dimensions afin de tester numériquement si l'arbre bronchique est ou non optimal.

In this article, we investigate first-order parameter insensitization for linear controlled evolutions: given a family A(σ) of generators on a Hilbert space H and a control operator B, does there exist a single control, independent of σ, that both steers the nominal system to a prescribed final state and cancels the first order sensitivity at the nominal system of the endpoint σ → yσ(T ), where yσ is the state of the system? We start from an abstract framework that does not depend on the dimension or on the type of system (ODE or PDE), and we derive the associated sensitivity cascade system. Insensitization is then recast as a controllability problem for an extended state (y, z). We illustrate this approach with two significant examples: in finite dimension, we prove the equivalence between four natural formulations and provide a necessary and sufficient algebraic condition of Kalman-type rank. We also treat infinite-dimensional examples, namely: the heat equation with uncertain diffusion coefficient, and show that the natural null and mixed approximate/exact insensitization problems are solvable; the wave equation with uncertain potential under appropriate geometric conditions on the control region allowing to derive approximate insensitization properties.

This article presents geometric optimal control techniques for analyzing geodesics in time-optimal Zermelo navigation problems on 2-spheres of revolution. We classify the problem by analyzing the pair ( F 0 , g ) , which represents the current (or wind) and the Riemannian metric. Using the maximum principle, the dynamics of geodesics are described by a Hamiltonian vector field on the cotangent bundle T ∗ S 2 . Our primary motivation is the application to micromagnetism, specifically spin magnetization reversal in ferromagnetic ellipsoidal samples. This model depends on four parameters and the amplitude of the applied magnetic field. The problem is formulated as a Zermelo navigation on the 2-sphere, where geodesics are classified as elliptic, hyperbolic, or abnormal. We demonstrate that the transition set | F 0 | g = 1 , which separates weak and strong current domains, is critical for understanding optimality. A key result shows that abnormal geodesics intersect this set with semi-cubical cusp singularities, a phenomenon we term the Landau–Lifshitz billiard. The analysis of the transition set’s connected components is complex and complemented by algebraic geometry and symbolic computations. We further reveal that hyperbolic geodesics lose optimality at their second intersection with the abnormal arc. Our numerical simulations complement this analysis by computing conjugate and cut loci, wavefronts, and accessibility sets, providing new insights into optimal magnetization switching under bounded control.

Analysing reachability associated to a control system is a subtle issue, especially for infinitedimensional dynamics, and when controls are subject to bounded constraints. We develop a computer-assisted framework for establishing non-reachability in linear parabolic PDEs governed by strongly elliptic operators, extending recent finite-dimensional techniques introduced in to the PDE setting. The non-reachability of a given target is shown to be equivalent to proving that a properly defined dual functional takes negative values. Our approach combines rigorous numerics with explicit convergence estimates for discretisations of the adjoint equation, ensuring mathematically certified results with tight error bounds. We demonstrate the wide applicability of our framework on Laplacian-driven control systems, showcasing its accuracy and reliability under various types of control constraints.

This article explores a variant of Kac's famous problem, "Can one hear the shape of a drum?", by addressing a geometric inverse problem in acoustics. Our objective is to reconstruct the shape of a cuboid room using acoustic signals measured by microphones placed within the room. By examining this straightforward configuration, we aim to understand the relationship between the acoustic signals propagating in a room and its geometry. This geometric problem can be reduced to locating a finite set of acoustic point sources, known as image sources. We model this issue as a finite-dimensional optimization problem and propose a solution algorithm inspired by super-resolution techniques. This involves a convex relaxation of the finite-dimensional problem to an infinite-dimensional subspace of Radon measures. We provide analytical insights into this problem and demonstrate the efficiency of the algorithm through multiple numerical examples.

This article explores a variant of Kac’s famous problem, ‘Can one hear the shape of a drum?’, by addressing a geometric inverse problem in acoustics. Our objective is to reconstruct the shape of a cuboid room using acoustic signals measured by microphones placed within the room. By examining this straightforward configuration, we aim to understand the relationship between the acoustic signals propagating in a room and its geometry. This geometric problem can be reduced to locating a finite set of acoustic point sources, known as image sources. We model this issue as a finite-dimensional optimization problem and propose a solution algorithm inspired by super-resolution techniques. This involves a convex relaxation of the finite-dimensional problem to an infinite-dimensional subspace of Radon measures. We provide analytical insights into this problem and demonstrate the efficiency of the algorithm through multiple numerical examples.

This article explores a variant of Kac's famous problem, ‘‘Can one hear the shape of a drum?'', by addressing a geometric inverse problem in acoustics. Our objective is to reconstruct the shape of a cuboid room using acoustic signals measured by microphones placed within the room. By examining this straightforward configuration, we aim to understand the relationship between the acoustic signals propagating in a room and its geometry. This geometric problem can be reduced to locating a finite set of acoustic point sources, known as image sources. We model this issue as a finite-dimensional optimization problem and propose a solution algorithm inspired by super-resolution techniques. This involves a convex relaxation of the finite-dimensional problem to an infinite-dimensional subspace of Radon measures. We provide analytical insights into this problem and demonstrate the efficiency of the algorithm through multiple numerical examples.

We present an algorithm that fully reverses the shoebox image source method (ISM), a popular and widely used room impulse response (RIR) simulator for cuboid rooms introduced by Allen and Berkley in 1979. More precisely, given a discrete multichannel RIR generated by the shoebox ISM for a microphone array of known geometry, the algorithm reliably recovers the 18 input parameters. These are the 3D source position, the 3 dimensions of the room, the 6-degrees-of-freedom room translation and orientation, and an absorption coefficient for each of the 6 room boundaries. The approach builds on a recently proposed gridless image source localization technique combined with new procedures for room axes recovery and first-order-reflection identification. Extensive simulated experiments reveal that near-exact recovery of all parameters is achieved for a 32-element, 8.4-cm-wide spherical microphone array and a sampling rate of 16 kHz using fully randomized input parameters within rooms of size 2$\times 2\times$2 to 10$\times 10\times$5 meters. Estimation errors decay towards zero when increasing the array size and sampling rate. The method is also shown to strongly outperform a known baseline, and its ability to extrapolate RIRs at new positions is demonstrated. Crucially, the approach is strictly limited to low-passed discrete RIRs simulated using the vanilla shoebox ISM. Nonetheless, it represents to our knowledge the first algorithmic demonstration that this difficult inverse problem is in-principle fully solvable over a wide range of configurations.