Cette thèse propose trois nouveaux développements de la méthode des bases réduites (RB) et de la méthode d'interpolation empirique (EIM) pour des problèmes non-linéaires. La première contribution est une nouvelle méthodologie, la méthode progressive RB-EIM (PREIM) dont l'objectif est de réduire le coût de la phase de construction du modèle réduit tout en maintenant une bonne approximation RB finale. L'idée est d'enrichir progressivement l'approximation EIM et l'espace RB, contrairement à l'approche standard où leurs constructions sont disjointes. La deuxième contribution concerne la RB pour les inéquations variationnelles avec contraintes non-linéaires. Nous proposons une combinaison RB-EIM pour traiter la contrainte. En outre, nous construisons une base réduite pour les multiplicateurs de Lagrange via un algorithme hiérarchique qui conserve la positivité des vecteurs cette base. Nous appliquons cette stratégie aux problèmes de contact élastique sans frottement pour les maillages non-coïncidents. La troisième contribution concerne la réduction de modèles avec assimilation de données. Une méthode dédiée a été introduite dans la littérature pour combiner un modèle numérique avec des mesures expérimentales. Nous élargissons son cadre d'application aux problèmes instationnaires en exploitant la méthode POD-greedy afin de construire des espaces réduits pour tout le transitoire temporel. Enfin, nous proposons un nouvel algorithme qui produit des espaces réduits plus représentatifs de la solution recherchée tout en minimisant le nombre de mesures nécessaires pour le problème réduit final / This thesis introduces three new developments of the reduced basis method (RB) and the empirical interpolation method (EIM) for nonlinear problems. The first contribution is a new methodology, the Progressive RB-EIM (PREIM) which aims at reducing the cost of the phase during which the reduced model is constructed without compromising the accuracy of the final RB approximation. The idea is to gradually enrich the EIM approximation and the RB space, in contrast to the standard approach where both constructions are separate. The second contribution is related to the RB for variational inequalities with nonlinear constraints. We employ an RB-EIM combination to treat the nonlinear constraint. Also, we build a reduced basis for the Lagrange multipliers via a hierarchical algorithm that preserves the non-negativity of the basis vectors. We apply this strategy to elastic frictionless contact for non-matching meshes. Finally, the third contribution focuses on model reduction with data assimilation. A dedicated method has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the method to a time-dependent framework using a POD-greedy algorithm in order to build accurate reduced spaces for all the time steps. Besides, we devise a new algorithm that produces better reduced spaces while minimizing the number of measurements required for the final reduced problem

Silicon-on-Insulator (SOI) micro-ring resonators (MRRs) are versatile elements in high-density integrated optics telecommunication systems. However, small inaccuracies in the fabrication process intensely deteriorate the response of SOI MRRs. By utilizing the racetrack resonator structures with strong coupling abilities, one can improve the fabrication tolerance. For the SOI racetrack resonators, the FSR is usually large. By introducing gratings into SOI racetrack resonators, the mutual mode coupling between the clockwise and counterclockwise modes can be induced and result in the resonance splitting. The grating-assisted SOI racetrack resonators can increase the operation wavelength and open up the possibility to overcome this limitation. In this thesis, we first use the 2-D FDTD method with the effective index method (EIM) to obtain the transmission spectra of the SOI racetrack resonators. The transmission spectra are then fitted by using the time-domain coupled mode theory (CMT) to obtain the quality factor and optical parameters of the SOI racetrack resonators. Next, we demonstrate the characteristics of mode splitting resulted from the mutual mode coupling between the clockwise and counterclockwise modes in the grating-assisted racetrack resonators by utilizing both the CMT and the 2-D FDTD method with the EIM. By tuning the grating configurations, such as the length or the structure of sidewall gratings, one can obtain the desired transmission spectrum of the grating-assisted racetrack resonators. Finally, we numerically investigate the temperature-dependent spectral characterics of the grating-assisted SOI racetrack resonator by taking the thermal-optic responce of the SOI materials into account. The thermal sensitivity of this device is 95.38 pm/¢XC, and the calculted properties can help the further designs based on the grating-assisted SOI racetrack resonators.

The vertical-cavity surface-emitting laser or VCSEL is a special type of diode laser, which has established itself in optoelectronic applications asa low-cost, high-quality miniaturized light source. The development of VCSELs can be largely promoted with support from computer simulations. In this study, we have used such simulations, on one hand to understand and improve the VCSEL performance, and on the other hand to prepare for analyzing new device concepts such as transistor-VCSELs. This thesis starts with a background introduction to the principle idea of VCSELs and then states the significance of this simulation work.Then it briefly introduces the previously used simulation workbench Sentaurus and explains the mathematical approach and the computation methods of the finally chosen simulator PICS3D. The case study of a fabricated and characterized epitaxially regrown VCSEL is the major component of this work. First the device configuration is demonstrated with detailed discussion on several design features. Second the physical models of electrical, optical and thermal phenomena along with their key parameters are presented and so are the advanced models for the active region. The main results of simulation, including steady-state characteristics and small-signal modulation, show good agreement with the experimental results and reveal some imperfections of the device design and processing, such as the overestimated stability of the regrown junction and the variation of cavity length caused by over-etch. This work is also treated as an evaluation of the simulator PICS3D, and two problems are identified: one is the troublesome way to construct a 3D device by coupling several 2D layer structures together, requiring the mesh for each layer structure to be compatible; the other would be the tricky boundary setting for the adopted method, Effective Index Method (EIM), for the transverse field calculation when only a weak index guiding effect exits in the cavity. Finally, we summarize this work and suggest some tasks for further simulations.

Cette thèse aborde deux thématiques différentes. On s’intéresse d’abord au développement et à l’analyse de méthodes pour le sens électrique appliqué à la robotique. On considère en particulier la méthode des réflexions permettant, à l’image de la méthode de Schwarz, de résoudre des problèmes linéaires à partir de sous-problèmes plus simples. Ces deniers sont obtenus par décomposition des frontières du problème de départ. Nous en présentons des preuves de convergence et des applications. Dans le but d’implémenter un simulateur du problème direct d’électrolocation dans un robot autonome, on s’intéresse également à une méthode de bases réduites pour obtenir des algorithmes peu coûteux en temps et en place mémoire. La seconde thématique traite d’un problème inverse dans le domaine de la chimie quantique. Nous cherchons ici à déterminer les caractéristiques d’un système quantique. Celui-ci est éclairé par un champ laser connu et fixé. Dans ce cadre, les données du problème inverse sont les états avant et après éclairage. Un résultat d’existence locale est présenté, ainsi que des méthodes de résolution numériques. / This thesis tackle two different topics.We first design and analyze algorithms related to the electrical sense for applications in robotics. We consider in particular the method of reflections, which allows, like the Schwartz method, to solve linear problems using simpler sub-problems. These ones are obtained by decomposing the boundaries of the original problem. We give proofs of convergence and applications. In order to implement an electrolocation simulator of the direct problem in an autonomous robot, we build a reduced basis method devoted to electrolocation problems. In this way, we obtain algorithms which satisfy the constraints of limited memory and time resources. The second topic is an inverse problem in quantum chemistry. Here, we want to determine some features of a quantum system. To this aim, the system is ligthed by a known and fixed Laser field. In this framework, the data of the inverse problem are the states before and after the Laser lighting. A local existence result is given, together with numerical methods for the solving.