It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas. In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM). Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula. It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using only a single variational step. Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations too / indeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented. Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green&rsquo / s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well. In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.

<p>Det här arbetet preciserar ett problem utifrån ett problemområde som faller inom ramen för användarcentrerad design. Den preciserade problemformuleringen är: I vilken utsträckning kan ett gränssnitt göras användbart genom prototyping, aktiv användarmedverkan och iterativ utveckling genom användningen av en modifierad version av ISO 13407 standarden som ett utvecklingsramverk? Den här frågeställningen har tillämpats och avgränsats till ett pekskärmsgränssnitt vid namn Navigator som är utvecklat av Asko. Resultaten visar att de gränssnittsprototyper som har skapats är användbara. Vidare visar resultaten även att det inte är tillräckligt att enbart använda tre nyckelprinciper för användarcentrerad design i syfte att uppnå användbarhet.</p>

The numerical treatment of linear-quadratic regulator problems for parabolic partial differential equations (PDEs) on infinite time horizons requires the solution of large scale algebraic Riccati equations (ARE). The Newton-ADI iteration is an efficient numerical method for this task. It includes the solution of a Lyapunov equation by the alternating directions implicit (ADI) algorithm in each iteration step. On finite time intervals the solution of a large scale differential Riccati equation is required. This can be solved by a backward differentiation formula (BDF) method, which needs to solve an ARE in each time step. Here, we study the selection of shift parameters for the ADI method. This leads to a rational min-max-problem which has been considered by many authors. Since knowledge about the complete complex spectrum is crucial for computing the optimal solution, this is infeasible for the large scale systems arising from finite element discretization of PDEs. Therefore several alternatives for computing suboptimal parameters are discussed and compared for numerical examples.

We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss.

We present a novel approach for deriving analytical solutions to transport equations expressedin similarity variables. We apply a fixed-point iteration procedure to these transformedequations by formally solving for the highest derivative term and then integrating to obtainan expression for the solution in terms of a previous estimate. We are able to analyticallyobtain the Lipschitz condition for this iteration procedure and, from this (via requirements forconvergence given by the contraction mapping principle), deduce a range of values for the outerlimit of the solution domain, for which the fixed-point iteration is guaranteed to converge.

Design continuously re-defines its meaning. Over the last years, the way designers interpret information, solve problems as well as prototype and express ideas has received increased attention from domains outside of traditional design, especially from the business world. However, much of the design thinking discourse outside of traditional design centres around a few widely-read practitioner books and only builds on a rudimentary understanding of its principles. Likewise, the academic literature only offers a few rigorous investigations of the application of design thinking in the management and innovation domain, especially when it comes to thedevelopment of novice multidisciplinary teams. Therefore, this thesis provides an evaluation of the influence of the following five key themes discussed in the design thinking literature: Team diversity, iteration, learning styles, creative confidence, and team communication. These themes were explored during a quantitative quasi-experimental research study, which was built on a novel research framework. Data wascollected from 42 German research participants over a period of 10 month. The longitudinal perspective enabled the researcher to illustrate how novices develop design thinking competencies in projects over time. While investigating team diversity, multidisciplinary teams were found to produce significantly better project outcomes than single-discipline teams. On the other hand, diversity of personality traits was not found to have a significant effect on the final performance of teams. The exploration of iteration behaviour revealed that multidisciplinary teams did not iterate significantly more than single-discipline teams. In addition, more experienced participants approached design thinking projects slightly less iteratively thannovices. Overall, the degree of iteration was not found to have a significant effect on the final team performance. Regarding the use of different learning styles, it was discovered that, teams with a balance of learning styles achieved significantly better project outcomes than less-balanced teams. In terms of learning styles, participants approached design thinking tasks mainly through rational conceptualisation rather than concrete experience. Theanalysis of individual and team confidence showed that creative confidence developed slowly and linearly over the course of a project, but only partly carried over to new project and team settings. Furthermore, no evidence was found that higher levels of creative confidence directly influenced the quality of the project outcomes. The investigation of team communication revealed that the importance of individuals in design thinking teams significantlychanged over the course of a project. Contrary to previous assumptions, high degrees of internal team cohesion were found to have a significant negative effect on project outcomes. While several of these findings clarify and reiterate existing design thinking theory, others call for an adjustment of theory and highlight the need for more rigorous research. Several recommendations are offered for practitioners, educators, and researchers on how to incorporate the presented findings into practice and future research.

L'objectif de cette recherche est de définir les significations du N (nom) fois au travers des principales constructions qui l'utilisent en français contemporain, et d‟expliquer la formation des expressions dérivées autrefois, des fois (que), parfois, quelquefois, toutefois, une fois (que). Examiné sous l'angle du lexique, le mot fois est un substantif féminin, qui, précédé d'un déterminant, se constitue en groupe nominal (GN). Le N fois dénote un comptage d'occurrences, c'est-à-dire d'unités de discours, qui sont le plus souvent des processus verbaux ou des prédicats. Il s'agit donc d‟unités linguistiques ce qui explique que fois apparaît dans des contextes divers. Nous avons classé les principales constructions en fonction des déterminants relevés dans le GN fois. D'une part, les GN composés des adjectifs cardinaux ou quantificateurs + fois permettent de compter des occurrences du langage et ont une fonction adverbiale de quantification ; si les occurrences comptées sont des processus verbaux, ils portent en sus une notion temporelle à valeur itérative ou fréquentielle. Avec les autres déterminants (article défini et indéfini, adjectif démonstratif, quantificateurs universels tous les et chaque), fois est un nom temporel qui tend à se standardiser. Les expressions dérivées évoluent sémantiquement et produisent une forme adverbiale préalable à d'autres transformations. Nous pouvons ainsi identifier le groupe des adverbiaux de quantification, le plus ancien, qui a facilité l'apparition des expressions dérivées, et un groupe plus récent où fois apparaît comme un nom temporel au même titre que jour, instant ou moment. / We aim in this search to define the significations of N (name) fois through the main constructions using it in modern French, and to explain how it helped in forming the derived expressions autrefois, des fois (que), parfois, quelquefois, toutefois, une fois (que). From a lexical point of view, the word fois is a feminine substantive, and, built with a determiner, it forms a nominal group (NG). The N fois denotes an occurrences counting, that means discurse units counting, mostly verbal process or predicates. Therefore it concerns liunguistic units and that's why fois is used in several contexts. We organized the main constructions according to the determiner used in NG fois. Firstly, NG built with cardinal adjectives or quantifiers + fois allow counting language occurrences and function as a quantification adverb ; if the occurrences are verbal process, they bring besides a temporal notion with iterative or frequential value. With other déterminers (definite and indefinite article, demonstrative adjective, universal quantifiers tous les and chaque), fois is a temporal name on the way to standardisation. The derived expressions change semantically and come to an adverbial form prior other transformations. This concludes to identify the group of quantification adverbs, the former one, which helped the formation of derived expressions, and a newer group in which fois is a temporal name as jour, instant or moment.

Les processus décisionnels de Markov (MDPs) modélisent des problèmes de décisionsséquentielles dans lesquels un utilisateur interagit avec l’environnement et adapte soncomportement en prenant en compte les signaux de récompense numérique reçus. La solutiond’unMDP se ramène à formuler le comportement de l’utilisateur dans l’environnementà l’aide d’une fonction de politique qui spécifie quelle action choisir dans chaque situation.Dans de nombreux problèmes de décision du monde réel, les utilisateurs ont despréférences différentes, donc, les gains de leurs actions sur les états sont différents et devraientêtre re-décodés pour chaque utilisateur. Dans cette thèse, nous nous intéressonsà la résolution des MDPs pour les utilisateurs ayant des préférences différentes.Nous utilisons un modèle nommé MDP à Valeur vectorielle (VMDP) avec des récompensesvectorielles. Nous proposons un algorithme de recherche-propagation qui permetd’attribuer une fonction de valeur vectorielle à chaque politique et de caractériser chaqueutilisateur par un vecteur de préférences sur l’ensemble des fonctions de valeur, où levecteur de préférence satisfait les priorités de l’utilisateur. Etant donné que le vecteurde préférences d’utilisateur n’est pas connu, nous présentons plusieurs méthodes pourrésoudre des MDP tout en approximant le vecteur de préférence de l’utilisateur.Nous introduisons deux algorithmes qui réduisent le nombre de requêtes nécessairespour trouver la politique optimale d’un utilisateur: 1) Un algorithme de recherchepropagation,où nous propageons un ensemble de politiques optimales possibles pourle MDP donné sans connaître les préférences de l’utilisateur. 2) Un algorithme interactifd’itération de la valeur (IVI) sur les MDPs, nommé algorithme d’itération de la valeurbasé sur les avantages (ABVI) qui utilise le clustering et le regroupement des avantages.Nous montrons également comment l’algorithme ABVI fonctionne correctement pourdeux types d’utilisateurs différents: confiant et incertain.Nous travaillons finalement sur une méthode d’approximation par critére de regret minimaxcomme méthode pour trouver la politique optimale tenant compte des informationslimitées sur les préférences de l’utilisateur. Dans ce système, tous les objectifs possiblessont simplement bornés entre deux limites supérieure et inférieure tandis que le systèmeine connaît pas les préférences de l’utilisateur parmi ceux-ci. Nous proposons une méthodeheuristique d’approximation par critère de regret minimax pour résoudre des MDPsavec des récompenses inconnues. Cette méthode est plus rapide et moins complexe queles méthodes existantes dans la littérature. / Markov decision processes (MDPs) are models for solving sequential decision problemswhere a user interacts with the environment and adapts her policy by taking numericalreward signals into account. The solution of an MDP reduces to formulate the userbehavior in the environment with a policy function that specifies which action to choose ineach situation. In many real world decision problems, the users have various preferences,and therefore, the gain of actions on states are different and should be re-decoded foreach user. In this dissertation, we are interested in solving MDPs for users with differentpreferences.We use a model named Vector-valued MDP (VMDP) with vector rewards. We propose apropagation-search algorithm that allows to assign a vector-value function to each policyand identify each user with a preference vector on the existing set of preferences wherethe preference vector satisfies the user priorities. Since the user preference vector is notknown we present several methods for solving VMDPs while approximating the user’spreference vector.We introduce two algorithms that reduce the number of queries needed to find the optimalpolicy of a user: 1) A propagation-search algorithm, where we propagate a setof possible optimal policies for the given MDP without knowing the user’s preferences.2) An interactive value iteration algorithm (IVI) on VMDPs, namely Advantage-basedValue Iteration (ABVI) algorithm that uses clustering and regrouping advantages. Wealso demonstrate how ABVI algorithm works properly for two different types of users:confident and uncertain.We finally work on a minimax regret approximation method as a method for findingthe optimal policy w.r.t the limited information about user’s preferences. All possibleobjectives in the system are just bounded between two higher and lower bounds while thesystem is not aware of user’s preferences among them. We propose an heuristic minimaxregret approximation method for solving MDPs with unknown rewards that is faster andless complex than the existing methods in the literature.

Orientador: Masayoshi Tsuchida / Banca: José Manoel Balthazar / Banca: Elso Drigo Filho / Resumo: Neste trabalho estuda-se alguns sistemas dinâmicos utilizando um novo método para aproximar a matriz de transição de estados (STM) para sistemas periódicos no tempo. Este método é baseado na transformação de Lyapunov-Floquet (L-F), e utiliza a expansão polinomial de Chebyshev para aproximar o termo periódico. O método iterativo de Picard é usado para aproximar a STM. Os multiplicadores de Floquet, determinados através deste método, permitem construir o diagrama de estabilidade do sistema dinâmico. Esta técnica é aplicada para analisar a estabilidade e os pontos de bifurcação do sistema dinâmico formado por um pêndulo elástico com excitação vertical periódica no suporte. Além dessa aplicação, é analisada também a equação de Mathieu e a estabilidade do sistema dinâmico constituído por partículas carregadas e imersas em um campo magnético perturbado. / Abstract: In this work some dynamic systems are studied using a new method to approach state transition matrix (STM) for time-periodic systems. This method is based on Lyapunov- Floquet transformation (transformation L-F) and uses the Chebyshev polynomial expansion to approach the periodical term. The Picard iterative method is used to approach the STM. The Floquet multipliers determined through this method, allow to draw the stability diagram of the dynamic system. This technique is applied to analyze the stability and bifurcation points of the dynamic system formed by an elastic pendulum with periodic vertical excitation on support. Besides this application, the Mathieu equation is analyzed and also the stability of the dynamical system constituted by charged particle in a perturbed magnetic field is discussed. / Mestre

The preprint delivers an efficient solution technique for the numerical simulation of crack propagation of 2D linear elastic formulations based on finite elements together with the conjugate gradient method in order to solve the corresponding linear equation systems. The developed iterative numerical approach using hierarchical preconditioners comprehends the interesting feature that the hierarchical data structure will not be destroyed during crack propagation. Thus, one gets the possibility to simulate crack advance in a very effective numerical manner including adaptive mesh refinement and mesh coarsening. Test examples are presented to illustrate the efficiency of the given approach. Numerical simulations of crack propagation are compared with experimental data.