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In this thesis, we demonstrated the effectiveness and the accuracy of the FD-FD method for complex optical waveguide structures such as the micro ring resonator, micro disk resonator, tapered waveguides and waveguides terminated with tilted facets. We are able to achieve the goals by deriving the following modification/extension of the original FD-FD methods. In frequency domain, we can build an accurate frequency-domain modal absorbing boundary condition (ABC) for both the homogeneous region and for the layered background. This allows us to connect the analytical modal solutions with FD solutions and thus reduce the area of the FD domain. In addition, we adopt an effective index averaging method for representing equivalent material for grid cells containing more than one kind of materials. For the TM case, for each grid cell we need to compute effective indices for all four surrounding cells (left, right, up, and down). For the TE case, we need to compute just one effective index within each grid cell. Note that we employ two different averaging schemes for the TE and the TM cases. To solve the huge block tri-diagonal matrix equation (derived from the FD-FD approximation) we modified the Thomas method and we were able to obtain the solutions of linear equations involving more than a hundred thousand variables under a few minutes. We used our method to analyze optical micro-ring waveguides, micro-disk cavities, adiabatic tapered waveguides and waveguides terminated with tilted facets. The simulated results include the reflection coefficients, transmission coefficients and field distribution.

We say that (weak/strong) time duality holds for continuous time quasi-birth-and-death-processes if, starting from a fixed level, the first hitting time of the next upper level and the first hitting time of the next lower level have the same distribution. We present here a criterion for time duality in the case where transitions from one level to another have to pass through a given single state, the so-called bottleneck property. We also prove that a weaker form of reversibility called balanced under permutation is sufficient for the time duality to hold. We then discuss the general case.

Photonic crystals have allowed unprecedented control of light and have allowed bringing new functionalities on chip. Photonic crystal waveguides (PCWs), which are linear defects in a photonic crystal, have unique features that distinguish these waveguides from other waveguides. The unique features include very large dispersion, existence of slow light, and the possibility of tailoring the dispersion properties for guiding light. In my research, I have overcome some of the challenges in using slow light in PCWs. In this work, I have demonstrated (i) high efficiency coupling of light into slow group velocity modes of a PCW, (ii) large bandwidth high transmission and low dispersion bends in PCWs, (iii) accurate modeling of pulse propagation in PCWs, (iv) high efficiency absorbing boundary conditions for dispersive slow group velocity modes of PCWs. To demonstrate the utility of slow light in designing high efficiency devices, I have demonstrated refractive index sensors using slow light in PCWs. In the end, a few high efficiency devices based on slow light in PCWs are mentioned. The remaining issues in the widespread use of PCW are also discussed in the last chapter.

Organisms have movements that are usually modeled by particles’ random walks. Under some mathematical technical assumptions the movements are described by diffusion equations. However, empirical data often show that the movements are not simple random walks. Instead, they are correlated random walks and are described by telegraph equations. This thesis considers telegraph equations with and without bias corresponding to correlated random walks with and without bias. Analytical solutions to the equations with absorbing boundary conditions and their mean passage times are obtained. Numerical simulations of the corresponding correlated random walks are also performed. The simulation results show that the solutions are approximated very well by the corresponding correlated random walks and the mean first passage times are highly consistent with those from simulations on the corresponding random walks. This suggests that telegraph equations can be a good model for organisms with the movement pattern of correlated random walks. Furthermore, utilizing the consistency of mean first passage times, we can estimate the parameters of telegraph equations through the mean first passage time, which can be estimated through experimental observation. This provides biologists an easy way to obtain parameter values. Finally, this thesis analyzes the velocity distribution and correlations of movement steps of amoebas, leaving fitting the movement data to telegraph equations as future work.

Differential equations arise frequently in modeling of physical systems, often resulting in linear eigenvalue problems. However, when dealing with large physical domains, solving such problems can be computationally expensive. This thesis examines an alternative approach to solving these problems, which involves utilizing absorbing boundary conditions and a Dirichlet-to-Neumann maps to transform the large sparse linear eigenvalue problem into a smaller nonlinear eigenvalue problem (NEP). The NEP is then solved using augmented Newton’s method. The specific equation investigated in this thesis is the two-dimensional Helmholtz equation, defined on the interval (x, y) ∈ [0, 10] × [0, 1], with the absorbing boundary condition introduced at x = 1. The results show a significant reduction in computational time when using this method compared to the original linear problem, making it a valuable tool for solving large linear eigenvalue problems. Another result is that the NEP does not affect the computational error compared to solving the linear problem, which further supports the NEP as an attractive alternative method.

No description available.

Mes travaux de thèse portent sur la construction de conditions aux limites absorbantes (CLAs) pour des problèmes de propagation d'ondes posés dans des milieux limités par des surfaces régulières. Ces conditions sont nouvelles car elles prennent en compte non seulement les ondes proagatives (comme la plupart des CLAs existantes) mais aussi les ondes évanescentes et rampantes. Elles sont donc plus performantes que les conditions existantes. De plus, elles sont facilement implémentables dans un schéma d'éléments finis de type Galerkine Discontinu (DG) et ne modifie pas la condition de stabilité de Courant-Friedrichs-Lewy (CFL). Ces CLAs ont été implémentées dans un code simulant la propagation des ondes acoustiques ainsi que dans un code simulant la propagation des ondes en régime harmonique. Les comparaisons réalisées entre les nouvelles conditions et celles qui sont les plus utilisées dans la littérature montrent que prendre en compte les ondes évanescentes et les ondes rampantes permet de diminuer les réflexions issues de la frontière artificielle et donc de rapprocher la frontière artificielle du bord de l'obstacle. On limite ainsi les coûts de calcul, ce qui est un des avantages de mes travaux. De plus, compte tenu du fait que les nouvelles CLAs sont écrites pour des frontières quelconques, elles permettent de mieux adapter le domaine de calcul à la forme de l'obstacle et permettent ainsi de diminuer encore plus les coûts de calcul numérique. / In my PhD, I have worked on the construction of absorbing boundary conditions (ABCs) designed for wave propagation problems set in domains bounded by regular surfaces. These conditions are new since they take into account not only propagating waves (as most of the existing ABCs) but also evanescent and creeping waves. Therefore, they outperform the existing ABCs. Moreover, they can be easily implemented in a discontinuous Galerkin finite element scheme and they do not change the Courant-Friedrichs-Lewy stability condition. These ABCs have been implemented in two codes that respectively simulate the propagation of acoustic waves and harmonic waves. The comparisons performed between these ABCs and the ABCs mostly used in the litterature show that when we take into account evanescent and creeping waves, we reduce the reflections coming from the artificial boundary. Therefore, thanks to these new ABCs, the artificial boundary can get closer to the obstacle. Consequently, we reduce the computational costs which is one of the advantages of my work. Moreover, since these new ABCs are written for any kind of boundary, we can adapt the shape of the computational domain and thus we can reduce again the computational costs.

This thesis outlines an iterative approach for determining the scattered wave for two dimensional multiple acoustic scattering problems using high order local absorbing boundary conditions and second order finite difference. We seek to approximate the total wave as it is scattered off of multiple arbitrarily shaped obstacles. This is done by decomposing the scattered wave into the superposition of single scattered waves. We then repeatedly solve the single scattering system for each obstacle, while updating the boundary conditions based off the incident wave and the scattered wave off the other obstacles. We solve each single scattering by enclosing the obstacle in a circular artificial boundary and generating a curvilinear coordinate system for the computational region between the obstacle and the artificial boundary. We impose an absorbing boundary condition, specifically Karp's Farfield Expansion ABC, on the artificial boundary. We use a finite difference method to discretize the governing equations and to discretize the absorbing boundary conditions. This will create a linear system whose solution will approximate the single scattered wave. The forcing vector of the linear system is determined from the total influence on the obstacle boundary from the incident wave and the scattered waves from the other obstacles. In each iteration, we solve the singular acoustic scattering problem for each obstacle by using the scattered wave approximations from the other obstacles obtained from the previous iteration. The iterations continue until the solutions converge. This iterative method scales well to multiple scattering configurations with many obstacles, and achieves errors on the order of 1E-5 in less than five minutes. This is due to using LU factorization to solve the linear systems, paired with parallelization. I will include numerical results which demonstrate the accuracy and advantages of this iterative technique.

La méthode d’imagerie la plus répandue dans l’industrie pétrolière est la RTM (Reverse Time Migration) qui repose sur la simulation de la propagation des ondes dans le sous-sol. Nous nous sommes concentrés sur un propagateur d'ondes élastiques 3D en milieu anisotrope de type TTI (Tilted Transverse Isotropic). Nous avons directement travaillé dans le code de recherche de Total DIVA (Depth Imaging Velocity Analysis), basé sur une discrétisation par la méthode de Galerkin Discontinue et le schéma Leap-Frog, et développé pour le calcul parallèle intensif – HPC (High Performance Computing). Nous avons ciblé plus particulièrement deux contributions possibles qui, si elles supposent des compétences très différentes, ont la même finalité : réduire les coûts de calculs requis pour la simulation. D'une part, les conditions aux limites classiques de type PML (Perfectly Matched Layers) ne sont pas stables dans des milieux TTI. Nous avons proposé de formuler une CLA (Conditions aux Limites Absorbantes) stable dans des milieux anisotropes. La méthode de construction repose sur les propriétés des courbes de lenteur, ce qui donne à notre approche un caractère original. D'autre part, le parallélisme initial, basé sur une décomposition de domaine et des communications par passage de messages à l'aide de la bibliothèque MPI, conduit à un déséquilibrage de charge qui détériore son efficacité parallèle. Nous avons corrigé cela en remplaçant le paradigme parallélisme par l'utilisation de la programmation à base de tâches sur support d'exécution. Cette thèse a été réalisée dans le cadre de l'action de recherche DIP (Depth Imaging Partnership) qui lie la compagnie pétrolière Total et Inria. / The most common method of Seismic Imaging is the RTM (Reverse Time Migration) which depends on wave propagation simulations in the subsurface. We focused on a 3D elastic wave propagator in anisotropic media, more precisely TTI (Tilted Transverse Isotropic). We directly worked in the Total code DIVA (Depth Imaging Velocity Analysis) which is based on a discretization by the Discontinuous Galerkin method and the Leap-Frog scheme, and developed for intensive parallel computing – HPC (High Performance Computing). We choose to especially target two contributions. Although they required very different skills, they share the same goal: to reduce the computational cost of the simulation. On one hand, classical boundary conditions like PML (Perfectly Matched Layers) are unstable in TTI media. We have proposed a formulation of a stable ABC (Absorbing Boundary Condition) in anisotropic media. The technique is based on slowness curve properties, giving to our approach an original side. On the other hand, the initial parallelism, which is based on a domain decomposition and communications by message passing through the MPI library, leads to load-imbalance and so poor parallel efficiency. We have fixed this issue by replacing the paradigm for parallelism by the use of task-based programming through runtime system. This PhD thesis have been done in the framework of the research action DIP (Depth Imaging Partnership) between the Total oil company and Inria.