L’enjeu de planification de vol à bord d’un hélicoptère en tenant compte des différents paramètres environnementaux constitue un facteur clé dans le secteur aéronautique afin d’assurer une mission en toute sécurité avec un coût réduit. Ce défi concerne à la fois la phase de préparation de la mission sur une station au sol mais aussi en cours de vol pour faire face à un évènement imprévu. Nous citons un premier exemple de mission de type recherche et sauvetage qui dispose d’un temps limité pour localiser et rechercher des personnes en danger. Pour ce faire, le plan de vol généré doit suivre le relief du terrain à des altitudes relativement basses entre des points de passage désignés. L’objectif est de permettre au pilote de localiser une victime dans une durée bornée. Un deuxième type de mission comme l’assistance médicale a la particularité d’assurer un vol qui favorise le confort du passager ainsi qu’une route qui minimise le temps de vol selon la criticité de la situation. Pendant la phase dynamique, lorsqu’il s’agit d’un événement complexe telle qu’une panne moteur, une replanification de la mission devient nécessaire afin de trouver un chemin aérien qui permet d’atterrir en toute sécurité dans les plus brefs délais. Dans un autre exemple comme l’évitement d’obstacle dynamique ou de zone dangereuse, il s’agit de calculer un autre plan de vol pour atteindre la destination. Cependant à ce jour, les pilotes ne bénéficient pas de système d’autoroutage 3D permettant la replanification dynamique de mission face à une situation d’urgence. Face au défi de génération d’un plan de vol optimal, nos réflexions profondes ont abouti à la proposition d’un nouveau système de planification de chemin, pour des aéronefs, dédié à la préparation de mission avant le vol ainsi qu’à la replanification dynamique en cours de vol face à une situation d’urgence. Le système de planification peut être déployé sur une station au sol ou bien intégré comme fonction avionique au sein de l’aéronef. Les résultats obtenus ont abouti à un brevet déposé devant l’INPI et à plusieurs actions de transfert technologique au sein d’Airbus Hélicoptères. Le système s’appuie sur des techniques de discrétisation de l’espace et de calcul du plus court chemin afin de générer automatiquement des solutions flexibles en terme de profil de chemin en respectant plusieurs contraintes liées à l’appareil, le terrain et l’environnement. La solution proposée est générique et capable de s’adapter selon le type de l’aéronef, le type de la mission et la fonction objectif à minimiser. Le système de planification peut offrir des solutions avec différents compromis entre le temps d’exécution et la qualité du chemin selon le temps disponible qui peut être corrélé avec la criticité de la situation. Le fonctionnement du système de planification de chemin se compose principalement de deux phases : le prétraitement et le routage. La phase de prétraitement permet une discrétisation multi-altitude de l’espace 3D selon une précision donnée et la génération automatique d’un graphe de navigation avec une connexité paramétrable. La phase de routage 3D calcule le chemin en prenant en considération un ensemble de contraintes telles que les limitations angulaires en horizontal et en vertical et une fonction objectif à minimiser tels que la distance, le carburant, le temps, etc. Lors de la phase d’exploration du graphe, le système de planification peut communiquer avec le modèle de performance de l’appareil afin de minimiser une fonction coût liée à la performance et de s’assurer au fur et à mesure de la faisabilité de la mission. La génération automatique des plans de vol et la replanification dynamique représentent une brique essentielle pour concevoir des systèmes d’assistance au pilote ainsi que des aéronefs autonomes. / Helicopter flight planning is a key factor in the aeronautics domain in order to ensure a safe mission at a reduced cost taking into consideration different environmental parameters like terrain, weather, emergency situations, etc. This challenge concerns both the mission preparation phase using a ground station and also during the flight to cope with a complex event (mechanical failure, dynamic obstacle, bad weather conditions, etc.). We quote as a first example a Search and Rescue mission that should be performed in limited time to locate persons in danger. To achieve that, the generated flight path must follow the terrain profile at relatively low altitudes crossing the predefined waypoints. The objective is to allow the pilot to locate a victim in a short time. Another example for a mission of medical assistance where the helicopter should ensure the comfort of patients as well as minimizing the flight time to respect critical situations. During the flight, when a complex event occurs such as engine failure, re-planning the mission becomes necessary in order to find a new path that could guarantee safe landing. Unfortunately, pilots do not benefit from an embedded 3D path planning system that enables dynamic mission re-planning in case of emergency. To tackle the challenge of generating an optimal flight plan, we proposed new path planning system dedicated to mission preparation and dynamic path re-planning during critical situations. The planning system can be deployed on a ground station or embedded as an avionic function in the aircraft. The achieved results are registered as a patent at National Institute of Intellectual Property and deployed inside Airbus Helicopters through several technology transfers. The system relies on 3D space discretization and shortest path planning techniques to generate automatically flexible path profiles that respect several. The proposed solution is generic and is able to adapt to the aircraft model, mission and the objective function to be minimized. The path planning system can offer solutions with different tradeoffs between timing and path quality within the available runtime depending on the criticality of the situation. The functioning of the path planning system consists mainly of two phases : preprocessing and routing phase. The preprocessing allows a multi-altitude discretization of the 3D space according to a given precision and generate automatically a navigation graph with a configurable density. The 3D routing phase calculates the path by considering a set of constraints such as horizontal and vertical angular limitations and the objective function such as distance, fuel, time, etc. During the graph exploration phase, the path planning system can communicate with the aircraft performance model to evaluate a given criterion. Automatic flight path generation is an essential building block for designing pilot assistance systems and autonomous aircraft. In summary, we succeeded to reach the industrial expectations namely : the evaluation of mission feasibility, performances improvement, navigation workload reduction, as well as improving the flight safety. In order to realize the proposed solution, we designed a new tool for automatic 3D flight path planning. We named our tool DTANAV : Demonstration Tool for Aircraft NAVigation. It allows to apply the proposed planning process on real scenarios. Through the tool interface, the user has the possibility to set the parameters related to the mission (starting point, end point, aircraft model, navigation ceiling, etc.). Other algorithmic parameters are defined in order to control the quality and the profile of the generated solution.

This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y.

Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es- timates, locality of the error, adaptivity

As the fidelity of computer generated imagery has increased, the need to digitally create convincing natural phenomena like fluids has become fundamental to the entertainment production industry. Because fluids are complex, the underlying physics must be computationally simulated. However, because a strictly physics-based approach is both computationally expensive and difficult to control, it does not lend itself well to the way artists and directors like to work. Directors require control to achieve their specific artistic vision. Furthermore, artistic workflows rely on quick iteration and the ability to apply changes late in the production process. In this dissertation we present novel techniques in adaptive simulation and fluid post-processing to improve artistic workflows for fluid simulation. Our methods reduce fluid simulation iteration time and provide a new way for artists to intelligently resize a wide range of volumetric data including fluid simulations. To reduce iteration time, we present a more cache-friendly linear octree structure for adaptive fluid simulation that reduces the overhead of previous octree-based methods. To increase the viability of reusable effects libraries, and to give artists intuitive control over simulations late in the production process we present a ``fluid carving" technique. Fluid carving uses seam carving methods to allow intelligent resizing on a variety of fluid phenomena without the need for costly re-simulation. We present methods that improve upon traditional seam carving approaches to address issues with scalability, non-rectangular boundaries, and that generalize to a variety of different visual effects data like particles, polygonal meshes, liquids, smoke, and fire. We achieve these improvements by guiding seams along user-defined lattices that can enclose regions of interest defined as OpenVDB grids with a wide range of shapes. These techniques significantly improve artist workflows for fluid simulation and allow visual entertainment to be produced in a more intuitive, cost-effective manner.

Cette thèse contient deux parties qui étudient deux sujets différents. Les Chapitres 1-4 sont consacrés aux problèmes de discrétisation de processus à des temps d’arrêt. Dans le Chapitre 1 on étudie l'erreur de discrétisation optimale pour des intégrales stochastiques par rapport à une semimartingale brownienne multidimensionnelle continue. Dans ce cadre on établit une borne inférieure trajectorielle pour la variation quadratique renormalisée de l'erreur. On fournit une suite de temps d’arrêt qui donne une discrétisation asymptotiquement optimale. Cette suite est définie comme temps de sortie d'ellipsoïdes aléatoires par la semimartingale. Par rapport aux résultats précédents on permet une classe de semimartingales assez large. On démontre qui la borne inférieure est exacte. Dans le Chapitre 2 on étudie la version adaptative au modèle de la discrétisation optimale d’intégrales stochastique. Dans le Chapitre 1 la construction de la stratégie optimale utilise la connaissance du coefficient de diffusion de la semimartingale considérée. Dans ce travail on établit une stratégie de discrétisation asymptotiquement optimale qui est adaptative au modèle et n'utilise pas aucune information sur le modèle. On démontre l'optimalité pour une classe de grilles de discrétisation assez générale basée sur les technique de noyau pour l'estimation adaptative. Dans le Chapitre 3 on étudie la convergence en loi des erreurs de discrétisation renormalisées de processus d’Itô pour une classe concrète et assez générale de grilles de discrétisation données par des temps d’arrêt. Les travaux précédents sur le sujet considèrent seulement le cas de dimension 1. En plus ils concentrent sur des cas particuliers des grilles, ou démontrent des résultats sous des hypothèses abstraites. Dans notre travail on donne explicitement la distribution limite sous une forme claire et simple, les résultats sont démontré dans le cas multidimensionnel pour le processus et pour l'erreur de discrétisation. Dans le Chapitre 4 on étudie le problème d'estimation paramétrique pour des processus de diffusion basée sur des observations à temps d’arrêt. Les travaux précédents sur le sujet considèrent que des temps d'observation déterministes, fortement prévisibles ou aléatoires indépendants du processus. Sous des hypothèses faibles on construit une suite d'estimateurs consistante pour une classe large de grilles d'observation données par des temps d’arrêt. On effectue une analyse asymptotique de l'erreur d'estimation. En outre, dans le cas du paramètre de dimension 1, pour toute suite d'estimateurs qui vérifie un TCL sans biais, on démontre une borne inférieure uniforme pour la variance asymptotique; on montre que cette borne est exacte. Les Chapitres 5-6 sont consacrés au problème de quantification d'incertitude pour des limites d'approximation stochastique. Dans le Chapitre 5 on analyse la quantification d'incertitude pour des limites d'approximation stochastique (SA). Dans notre cadre la limite est définie comme un zéro d'une fonction donnée par une espérance. Cette espérance est prise par rapport à une variable aléatoire pour laquelle le modèle est supposé de dépendre d'un paramètre incertain. On considère la limite de SA comme une fonction de cette paramètre. On introduit un algorithme qui s'appelle USA (Uncertainty for SA). C'est une procédure en dimension croissante pour calculer les coefficients de base d'expansion de chaos de cette fonction dans une base d'un espace de Hilbert bien choisi. La convergence de USA dans cet espace de Hilbert est démontré. Dans le Chapitre 6 on analyse le taux de convergence dans L2 de l'algorithme USA développé dans le Chapitre 5. L'analyse est non trivial à cause de la dimension infinie de la procédure. Le taux obtenu dépend du modèle et des paramètres utilisés dans l'algorithme USA. Sa connaissance permet d'optimiser la vitesse de croissance de la dimension dans USA. / This thesis consists of two parts which study two separate subjects. Chapters 1-4 are devoted to the problem of processes discretization at stopping times. In Chapter 1 we study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a path wise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales and we prove that the asymptotic lower bound is attainable. In Chapter 2 we study the model-adaptive optimal discretization error of stochastic integrals. In Chapter 1 the construction of the optimal strategy involved the knowledge about the diffusion coefficient of the semimartingale under study. In this work we provide a model-adaptive asymptotically optimal discretization strategy that does not require any prior knowledge about the model. In Chapter 3 we study the convergence in distribution of renormalized discretization errors of Ito processes for a concrete general class of random discretization grids given by stopping times. Previous works on the subject only treat the case of dimension 1. Moreover they either focus on particular cases of grids, or provide results under quite abstract assumptions with implicitly specified limit distribution. At the contrast we provide explicitly the limit distribution in a tractable form in terms of the underlying model. The results hold both for multidimensional processes and general multidimensional error terms. In Chapter 4 we study the problem of parametric inference for diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids. Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp. In Chapters 5-6 we study the problem of uncertainty quantification for stochastic approximation limits. In Chapter 5 we analyze the uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm. In our setup, this limit is defined as the zero of a function given by an expectation. The expectation is taken w.r.t. a random variable for which the model is assumed to depend on an uncertain parameter. We consider the SA limit as a function of this parameter. We introduce the so-called Uncertainty for SA (USA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of this function on an orthogonal basis of a suitable Hilbert space. The almost-sure and Lp convergences of USA, in the Hilbert space, are established under mild, tractable conditions. In Chapter 6 we analyse the L2-convergence rate of the USA algorithm designed in Chapter 5.The analysis is non-trivial due to infinite dimensionality of the procedure. Moreover, our setting is not covered by the previous works on infinite dimensional SA. The obtained rate depends non-trivially on the model and the design parameters of the algorithm. Its knowledge enables optimization of the dimension growth speed in the USA algorithm, which is the key factor of its efficient performance.

The first objective of this Dissertation is to present a methodology to calculate analytically the mode shapes and corresponding natural frequencies and determine critical buckling loads of mechanically coupled microbeam resonators with a focus on micromechanical filters. The second objective is to adopt a nonlinear approach to build a reduced-order model and obtain closed-form expressions for the response of the filter to a primary resonance. The third objective is to investigate the feasibility of employing subharmonic excitation to build bandpass filters consisting of either two sets of two beams coupled mechanically or two sets of clamped-clamped beams. Throughout this Dissertation, we treat filters as distributed-parameter systems. In the first part of the Dissertation, we demonstrate the methodology by considering a mechanical filter composed of two beams coupled by a weak beam. We solve a boundary-value problem (BVP) composed of five equations and twenty boundary conditions for the natural frequencies and mode shapes. We reduce the problem to a set of three linear homogeneous algebraic equations for three constants and the frequencies in order to obtain a deeper insight into the relation between the design parameters and the performance metrics. In an approach similar to the vibration problem, we solve the buckling problem to study the effect of the residual stress on the static stability of the structure. To achieve the second objective, we develop a reduced-order model for the filter by writing the Lagrangian and applying the Galerkin procedure using its analytically calculated linear global mode shapes as basis functions. The resulting model accounts for the geometric and electric nonlinearities and the coupling between them. Using the method of multiple scales, we obtain closed-form expressions for the deflection and the electric current in the case of one-to-one internal and primary resonances. The closed-form solution shows that there are three possible operating ranges, depending on the DC voltage. For low DC voltages, the effective nonlinearity is positive and the filter behavior is hardening, whereas for large DC voltages, the effective nonlinearity is negative and the filter behavior is softening. We found that, when mismatched DC voltages are applied to the primary resonators, the first mode is localized in the softer resonator and the second mode is localized in the stiffer resonator. We note that the excitation amplitude can be increased without worrying about the appearance of multivaluedness when operating the filter in the near-linear range. The upper bound in this case is the occurrence of the dynamic pull-in instability. In the softening and hardening operating ranges, the adverse effects of the multi-valued response, such as hysteresis and jumps, limit the range of the input signal. To achieve the third objective, we propose a filtration technique based on subharmonic resonance excitation to attain bandpass filters with ideal stopband rejection and sharp rolloff. The filtration mechanism depends on tuning two oscillators such that one operates in the softening range and the other operates in the hardening range. Hardware and logic schemes are necessary to realize the proposed filter. We derive a reduced-order model using a methodology similar to that used in the primary excitation case, but with all necessary changes to account for the subharmonic resonance of order one-half. We observe that some manipulations are essential for a structure of two beams coupled by a weak spring to be suitable for filtration. To avoid these complications, we use a pair of single clamped-clamped beams to achieve our goal. Using a model derived by attacking directly the distributed-parameters problem, we suggest design guidelines to select beams that are potential candidates for building a bandpass filter. We demonstrate the proposed mechanism using an example. / Ph. D.

Systems biology is an emergent field focused on developing a system-level understanding of biological systems. In the last decade advances in genomics, transcriptomics and proteomics have gathered a remarkable amount data enabling the possibility of a system-level analysis to be grounded at a molecular level. The reverse-engineering of biochemical networks from experimental data has become a central focus in systems biology. A variety of methods have been proposed for the study and identification of the system's structure and/or dynamics. The objective of this dissertation is to introduce and propose solutions to some of the challenges inherent in reverse-engineering of biological systems. First, previously developed reverse engineering algorithms are studied and compared using data from a simulated network. This study draws attention to the necessity for a uniform benchmark that enables an ob jective comparison and performance evaluation of reverse engineering methods. Since several reverse-engineering algorithms require discrete data as input (e.g. dynamic Bayesian network methods, Boolean networks), discretization methods are being used for this purpose. Through a comparison of the performance of two network inference algorithms that use discrete data (from several different discretization methods) in this work, it has been shown that data discretization is an important step in applying network inference methods to experimental data. Next, a reverse-engineering algorithm is proposed within the framework of polynomial dynamical systems over finite fields. This algorithm is built for the identification of the underlying network structure and dynamics; it uses as input gene expression data and, when available, a priori knowledge of the system. An evolutionary algorithm is used as the heuristic search method for an exploration of the solution space. Computational algebra tools delimit the search space, enabling also a description of model complexity. The performance and robustness of the algorithm are explored via an artificial network of the segment polarity genes in the D. melanogaster. Once a mathematical model has been built, it can be used to run simulations of the biological system under study. Comparison of simulated dynamics with experimental measurements can help refine the model or provide insight into qualitative properties of the systems dynamical behavior. Within this work, we propose an efficient algorithm to describe the phase space, in particular to compute the number and length of all limit cycles of linear systems over a general finite field. This research has been partially supported by NIH Grant Nr. RO1GM068947-01. / Ph. D.

<p>Disertacija se bavi analizom uticaja raspodela podataka na rezultate algoritama diskretizacije u okviru procesa ma&scaron;inskog učenja. Na osnovu izabranih baza i algoritama diskretizacije teorije grubih skupova i stabala odlučivanja, istražen je uticaj odnosa raspodela podataka i tačaka reza određene diskretizacije.<br />Praćena je promena konzistentnosti diskretizovane tabele u zavisnosti od položaja redukovane tačke reza na histogramu. Definisane su fiksne tačke reza u zavisnosti od segmentacije multimodal raspodele, na osnovu kojih je moguće raditi redukciju preostalih tačaka reza. Za određivanje fiksnih tačaka konstruisan je algoritam FixedPoints koji ih određuje u skladu sa grubom segmentacijom multimodal raspodele.<br />Konstruisan je algoritam aproksimativne diskretizacije APPROX MD za redukciju tačaka reza, koji koristi tačke reza dobijene algoritmom maksimalne razberivosti i parametre vezane za procenat nepreciznih pravila, ukupni procenat klasifikacije i broj tačaka redukcije. Algoritam je kompariran u odnosu na algoritam maksimalne razberivosti i u odnosu na algoritam maksimalne razberivosti sa aproksimativnim re&scaron;enjima za &alpha;=0,95.</p> / <p>This dissertation analyses the influence of data distribution on the results of discretization algorithms within the process of machine learning. Based on the chosen databases and the discretization algorithms within the rough set theory and decision trees, the influence of the data distribution-cuts relation within certain discretization has been researched.<br />Changes in consistency of a discretized table, as dependent on the position of the reduced cut on the histogram, has been monitored. Fixed cuts have been defined, as dependent on the multimodal segmentation, on basis of which it is possible to do the reduction of the remaining cuts. To determine the fixed cuts, an algorithm FixedPoints has been constructed, determining these points in accordance with the rough segmentation of multimodal distribution.<br />An algorithm for approximate discretization, APPROX MD, has been constructed for cuts reduction, using cuts obtained through the maximum discernibility (MD-Heuristic) algorithm and the parametres related to the percent of imprecise rules, the total classification percent and the number of reduction cuts. The algorithm has been compared to the MD algorithm and to the MD algorithm with approximate solutions for &alpha;=0,95.</p>

Práce je zaměřená na numerickou simulaci dvoufázového proudění. Je studován matematický model a numerická aproximace toku dvou nemísitelných nestlačitelných tekutin. Rozhraní mezi tekutinami je popsáno pomocí pomocí tzv. level set metody. Představena je diskretizace problému v prostoru a v čase. Metoda konečných prvk· se zpětnou Eulerovou metodou je aplikována na Navierovy-Stokesovy rovnice a časoprostorová nespojitá Galerkinova metoda je použita k řešení transportního problému. D·raz je kladen na analýzu chyby nespojité Galerkinovy metody přímek a časoprostorové nespojité Galerkinovy metody pro transportní problém. Jsou prezentovány numerické výsledky. 1

This thesis describes the theory of model predictive control and application of the theory to synchronous drives. It shows explicit and on-line solutions and compares the results with classical vector control structure.