ED Mathématiques et Informatique
Study of particle methods in nonlinear filtering. Application to passive trajectography.
by Luc DE MONTELLA (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 14h00 - Ada Lovelace 200 Av. de la Vieille Tour, 33405 Talence, Centre Inria de l'université de Bordeaux
in front of the jury composed of
- Pierre DEL MORAL - Directeur de recherche - INRIA Bordeaux - Directeur de these
- Nicolas CHOPIN - Professeur - ENSAE - Rapporteur
- Karim DAHIA - Ingénieur de recherche - ONERA - Examinateur
- Nadia OUDJANE - Ingénieure de recherche - EDF R&D - Examinateur
- Julian TUGAUT - Maître de conférences - Université Jean Monnet - Examinateur
- Pierrick LEGRAND - Professeur des universités - Université de Bordeaux - Examinateur
This thesis presents a theoretical and applied study of particle filtering methods, with the ambition of strengthening confidence in their use for critical applications, such as in military or submarine domains, for example. First, we focus on the Diffusion Monte Carlo (DMC) method, a variant of particle methods used in physics to compute the ground state of quantum systems. We establish assumptions that guarantee the uniform-in-time convergence of this method on non-compact state spaces while ensuring that the conditions remain flexible enough to include Gaussian linear models. This work thus constitutes the first result of this kind for particle methods. To provide a concrete example that meets our assumptions and to study the implications of our theorem, we conduct a detailed analysis of the coupled harmonic oscillator. This study allows us to highlight cases where the DMC exhibits asymptotic convergence properties, even though its error diverges for any finite number of particles. This result underscores the importance of establishing uniform-in-time convergence guarantees. Furthermore, we show that this divergence is not inevitable : a modification of DMC can be sufficient to ensure its convergence, thereby opening new perspectives for its application to more complex systems. Building on our theoretical work, we explore the problem of passive tracking, aiming to develop practical solutions to enhance the capabilities of particle methods. Our first approach involves integrating acoustic pressure level measurements with the azimuth data which are traditionally used for tracking. Although pressure measurements are commonly available, they are often overlooked. By leveraging the ability of particle filters to approximate multimodal distributions, which is necessary to combine this information, we demonstrate through numerical studies that this approach significantly improves tracking performance. To more comprehensively evaluate this performance, we compare it to the Cramér-Rao bound associated with filters that use only bearing data, with this bound serving as a lower limit for the theoretical minimal error in this case. We then show that using acoustic pressure level measurements allows, in certain scenarios, for an error lower than the theoretical minimal error for filters using only bearing data. This encouraging result opens promising prospects for future applications, but the study presented here is only a first step before the method can be fully operational in real-world conditions. To propose an immediately applicable solution, we conduct an in-depth analysis of the quality of information in a passive bearing-based localization context. During this analysis, we examine the Fisher information associated with the bearing-based localization problem, as well as the convergence properties of position estimators. This work leads to the definition of a maneuvering protocol that enables an observer to localize a target with precision and efficiency.