Made available in DSpace on 2016-06-02T19:56:53Z (GMT). No. of bitstreams: 1 DissMF.pdf: 1677378 bytes, checksum: 5f2c8e8509c97ea1775dbbda2508174b (MD5) Previous issue date: 2005-03-23 / Universidade Federal de Minas Gerais / This work studied the horizontal-flow anaerobic immobilized biomass (HAIB) for the treatment of wastewater, developed in the Department of Hydraulics and Sanitation of the School of Engineering of São Carlos - USP. For the simulation and modeling of RAHLF, data of two HAIB were used, and; when the regime is the dynamic, one in bench scale treating wastewater containing poisonous substances (BTEX) and when the regime is the stationary, other in pilot scale treating sewer sanitarium. The study was divided in two parts: stationary regime and transient regime. In the first case the models were analyzed: pseudo - homogeneous with and without axial dispersion and the heterogeneous with axial dispersion; in the second case, the models were analyzed: pseudo - homogeneous and heterogeneous. The models of the first case were resolved respectively for the numeric methods of Runge - Kutta, of the finite differences and of the orthogonal collocation. In those models, when if it despised the axial dispersion the kinetic constant of first order was adjusted. Already in the other two models acted in this same case, they were adjusted the kinetic constant of first order and the coefficient of axial dispersion simultaneously. All of the methods numeric employees in the first case described the tendency of variation of the concentration well along the reactor and inside the particle, making possible the convergence of the solutions. The models without axial dispersion foresaw concentration values in the exit closest of the experimental values than the models with dispersion, and this, for his/her time foresaw closer values to the experimental ones in the intermediate points. In the second case, the models were solved through two methods, the method of the finite differences and the method of the sequence. The methods got to solve the equations that describe the behavior of the reactor satisfactorily and they show that the answer of the exit in the reactor is appropriately made calculations. / Este trabalho estudou o reator anaeróbio horizontal de leito fixo (RAHLF) para o tratamento de águas residuárias, desenvolvido no Departamento de Hidráulica e Saneamento da Escola de Engenharia de São Carlos USP. Para a simulação e modelagem do RAHLF, foram utilizados dados de dois RAHLFs, sendo que; quando o regime é o dinâmico, um em escala de bancada tratando águas residuárias contendo substâncias tóxicas (BTEX) e quando o regime é o estacionário, outro em escala piloto tratando esgoto sanitário. O estudo foi dividido em duas partes: regime estacionário e regime transiente. No primeiro caso foram analisados os modelos: pseudo homogêneo com e sem dispersão axial e o heterogêneo com dispersão axial; no segundo caso, foram analisados os modelos: pseudo homogêneo e heterogêneo. Os modelos do primeiro caso foram resolvidos respectivamente pelos métodos numéricos de Runge Kutta, das diferenças finitas e da colocação ortogonal. Nesses modelos, quando se desprezava a dispersão axial a constante cinética de primeira ordem foi ajustada. Já nos outros dois modelos representados neste mesmo caso, foram ajustados simultaneamente a constante cinética de primeira ordem e o coeficiente de dispersão axial. Todos os métodos numéricos empregados no primeiro caso descreveram bem a tendência de variação da concentração ao longo do reator e no interior da partícula, possibilitando a convergência das soluções. Os modelos sem dispersão axial previram valores de concentração na saída mais próximo dos valores experimentais do que os modelos com dispersão, e este, por sua vez previram valores mais próximos aos experimentais nos pontos intermediários. No segundo caso, os modelos foram solucionados através de dois métodos, o método das diferenças finitas e o método da seqüência. Os métodos conseguiram resolver satisfatoriamente as equações que descrevem o comportamento do reator e mostram que a resposta da saída no reator é adequadamente calculada.

This term project describes transformation of system of diferential equations into polynomial form. Such transformed systems of diferential equations can be subsequently solved using Taylor series. This method enables computing of initial problem's numeric solution using dynamical order selection in order to achieve required accuracy. The work mathematically proves, that transformed systems of diferential equations have the same solution as the original systems. This transformation can be used for all mathematic functions commonly used in technical applications. The work also focuses on optimization of given problem and implements it in programme taylor. This progamme enables user to solve given diferential equations with chosen parameters.

Made available in DSpace on 2016-06-02T19:56:42Z (GMT). No. of bitstreams: 1 DissDVRF.pdf: 670377 bytes, checksum: a7df05c407160cb1a845c224293f75b6 (MD5) Previous issue date: 2004-03-05 / Universidade Federal de Sao Carlos / This work studied the anaerobic horizontal reactor of fixed bed (RAHLF)for treatment of waste waters, developed in the Department of Hydraulics and Sanitation of the School of Engineering of São Carlos USP by investigating some mathematical models with two different RAHLFs conditions in different scales: the first in pilot scale by treating domestic and other in bench scale by treating synthetic substratum. The pseudo-homogeneous and heterogeneous models were investigated by considering or not the substratum dispersion in the axial direction. The models were resolved numerically by fourth order Runge-Kutta method, orthogonal collocation and finite differences. The values were compared to reactor experimental values. In the pseudo-homogeneous resolution model with dispersion by using the reactor data in pilot scale was adjusted the axial dispersion coefficient to 1,65.10-3 m2.s-1, after that was used in the solution of the heterogeneous model. The benches reactor values were used to obtain the dispersion coefficient as Zero. The parameter kinetic variation in the reactor was investigated by verifying that this variation is not responsible for the difference between the experimental data and those previewed by the model. / Este trabalho estudou o reator anaeróbio horizontal de leito fixo (RAHLF) para tratamento de águas residuárias, desenvolvido no Departamento de Hidráulica e Saneamento da Escola de Engenharia de São Carlos USP, investigando alguns modelos matemáticos com as condições de operação de dois RAHLFs em diferentes escalas: um em escala piloto tratando esgoto doméstico e outro em escala de bancada tratando substrato sintético. Foram investigados os modelos pseudo-homogêneos e heterogêneos, e estes por sua vez, considerando ou não a dispersão axial. Os modelos foram resolvidos numericamente utilizando os métodos de Runge-Kutta de quarta ordem, colocação ortogonal e diferenças finitas, comparando os valores obtidos pelos modelos com os valores experimentais dos reatores. Na resolução do modelo pseudo-homogêneo com dispersão utilizando os dados do reator em escala piloto foi ajustado o coeficiente de dispersão axial com valor de 1,65.10-3 m2.s-1, o qual foi posteriormente utilizado na solução do modelo heterogêneo. Utilizando os dados do reator em escala de bancada o ajuste forneceu um coeficiente de dispersão igual a zero. A variação do parâmetro cinético ao longo do reator foi investigada, verificando que esta variação não é responsável pelo desvio entre os dados experimentais e os previstos pelo modelo.

When calculating the dynamics of a quantum system, including the effect of its environment is highly relevant since virtually any real quantum system is exposed to environmental influences. It has turned out that the widely used perturbative approaches to treat such so-called open quantum systems have severe limitations. Furthermore, due to current experiments which have implemented strong system-environment interactions the non-perturbative regime is far from being academical. Therefore determining the exact dynamics of an open quantum system is of fundamental relevance. The hierarchy of pure states (HOPS) formalism poses such an exact approach. Its novel and detailed derivation, as well as several numerical aspects constitute the main methodical part of this work. Motivated by fundamental issues but also due to practical relevance for real world devices exploiting quantum effects, the entanglement dynamics of two qubits in contact with a common environment is investigated extensively. The HOPS formalism is based on the exact stochastic description of open quantum system dynamics in terms of the non-Markovian quantum state diffusion (NMQSD) theory. The distinguishing and numerically beneficial features of the HOPS approach are the stochastic nature, the implicit treatment of the environmental dynamics and, related to this, the enhanced statistical convergence (importance sampling), as well as the fact that only pure states have to be propagated. In order to claim that the HOPS approach is exact, we develop schemes to ensure that the numerical errors can be made arbitrarily small. This includes the sampling of Gaussian stochastic processes, the multi-exponential representation of the bath correlation function and the truncation of the hierarchy. Moreover, we incorporated thermal effects on the reduced dynamics by a stochastic Hermitian contribution to the system Hamiltonian. In particular, for strong system-environment couplings this is very beneficial for the HOPS. To confirm the accuracy assertion we utilize the seemingly simple, however, non-trivial spin-boson model to show agreement between the HOPS and other methods. The comparison shows the HOPS method’s versatile applicability over a broad range of model parameters including weak and strong coupling to the environment, as well as zero and high temperatures. With the gained knowledge that the HOPS method is versatile and accurately applicable, we investigate the specific case of two qubits while focusing on their entanglement dynamics. It is well known that entanglement, the relevant property when exploiting quantum effects in fields like quantum computation, communication and metrology, is fragile when exposed to environmental noise. On the other hand, a common environment can also mediate an effective interaction between the two parties featuring entanglement generation. In this work we elucidate the interplay between these competing effects, focusing on several different aspects. For the perturbative (weak coupling) regime we enlighten the difficulties inherent to the frequently used rotating wave approximation (RWA), an approximation often applied to ensure positivity of the reduced state for all times. We show that these difficulties are best overcome when simply omitting the RWA. The seemingly unphysical dynamics can still be used to approximate the exact entanglement dynamics very well. Furthermore, the influence of the renormalizing counter term is investigated. It is expected that under certain conditions (adiabatic regime) the generation of entanglement is suppressed by the presence of the counter term. It is shown, however, that for a deep sub-Ohmic environment this expectation fails. Leaving the weak coupling regime, we show that the generation of entanglement due to the influence of the common environment is a general property of the open two-spin system. Even for non-zero temperatures it is demonstrated that entanglement can still be generated and may last for arbitrary long times. Finally, we determine the maximum of the steady state entanglement as a function of the coupling strength and show how the known delocalization-to-localization phase transition is reflected in the long time entanglement dynamics. All these results require an exact treatment of the open quantum system dynamics and, thus, contribute to the fundamental understanding of the entanglement dynamics of open quantum systems. / Bei der Bestimmung der Dynamik eines Quantensystems ist die Berücksichtigung seiner Umgebung von großem Interessen, da faktisch jedes reale Quantensystem von seiner Umgebung beeinflusst wird. Es zeigt sich, dass die viel verwendeten störungstheoretischen Ansätze starken Einschränkungen unterliegen. Außerdem, da es in aktuellen Experimenten gelungen ist starke Wechselwirkung zwischen dem System und seiner Umgebung zu realisieren, gewinnt das nicht-störungstheoretischen Regime stets an Relevanz. Dementsprechend ist die Berechnung der exakten Dynamik offener Quantensysteme von grundlegender Bedeutung. Einen solchen exakten nummerischen Zugang stellt der hierarchy of pure states (HOPS) Formalismus dar. Dessen neuartige und detaillierte Herleitung, sowie diverse nummerische Aspekte werden im methodischen Teil dieser Arbeit dargelegt. In vielerlei Hinsicht relevant folgt als Anwendung eine umfangreiche Untersuchung der Verschränkungsdynamik zweier Qubits unter dem Einfluss einer gemeinsamen Umgebung. Vor allem im Hinblick auf die experimentell realisierbare starke Kopplung mit der Umgebung ist dieses Analyse von Interesse. Der HOPS Formalismus basiert auf der stochastischen Beschreibung der Dynamik offener Quantensysteme im Rahmen der non-Markovian quantum state diffusion (NMQSD) Theorie. Der stochastische Charakter der Methode, die implizite Berücksichtigung der Umgebungsdynamik, sowie das damit verbundene Importance Sampling, als auch die Tatsache dass lediglich reine Zustände propagiert werden müssen unterscheidet diese Methode maßgeblich von anderen Ansätzen und birgt numerische Vorteile. Um zu behaupten, dass die HOPS Methode exakte Ergebnisse liefert, müssen auftretenden nummerischen Fehler beliebig klein gemacht werden können. Ein grundlegender Teil der hier vorgestellten methodischen Arbeit liegt in der Entwicklung diverser Schemata, die genau das erreichen. Dazu zählen die numerische Realisierung von Gauss’schen stochastischen Prozessen, die Darstellung der Badkorrelationsfunktion als Summe von Exponentialfunktionen sowie das Abschneiden der Hierarchie. Außerdem wird gezeigt, dass sich der temperaturabhängige Einfluss der Umgebung durch einen stochastischen Hermiteschen Beitrag zum System-Hamiltonoperator berücksichtigen lässt. Vor allem bei starker Kopplung ist diese Variante besonders geeignet für den HOPS Zugang. Um die Genauigkeitsbehauptung der HOPS Methode zu überprüfen wird die Übereinstimmung mit anderen Methode gezeigt, wobei das vermeintlich einfachste, jedoch nicht triviale spin-boson-Modell als Testsystem verwendet wird. Diese Untersuchung belegt, dass die HOPS Methode für eine Vielzahl an Szenarien geeignet ist. Das beinhaltet schwache und starke Kopplung an die Umgebung, sowie Temperatur null als auch hohe Temperaturen. Mit dem gewonnenen Wissen, dass die HOPS Methode vielseitig einsetzbar ist und genaue Ergebnisse liefert wird anschließend der spezielle Fall zweier Qubits untersucht. Im Hinblick auf die Ausnutzung von Quanteneffekten in Bereichen wie Rechentechnik, Kommunikation oder Messtechnik liegt der primäre Fokus auf der Dynamik der Verschränkung zwischen den Qubits. Es ist bekannt, dass durch von außen induziertes Rauschen die Verschränkung im Laufe der Zeit abnimmt. Andererseits weiß man auch, dass eine gemeinsame Umgebung zu einer effektiven Wechselwirkung zwischen den Qubits führt, welche Verschränkung aufbauen kann. In dieser Arbeit wird das Wechselspiel zwischen diesen beiden gegensätzlichen Effekten untersucht, wobei die folgenden Aspekte beleuchtet werden. Für den Fall schwacher Kopplung, wo eine störungstheoretische Behandlung in Frage kommt, werden die Probleme der rotating wave approximation (RWA) analysiert. Diese Näherung wird häufig verwendet um die Positivität des reduzierten Zustands zu allen Zeiten zu gewährleisten. Es wird gezeigt, dass sich diese Probleme am besten vermeiden lassen, wenn die RWA einfach weggelassen wird. Die auf den ersten Blick nicht-physikalische Dynamik ist sehr gut geeignet um die exakte Verschränkungsdynamik näherungsweise wiederzugeben. Des Weiteren wird der Einfluss der Renormalisierung des sogenannten counter terms untersucht. Unter bestimmten Voraussetzungen (adiabatisches Regime) ist zu erwarten, dass der Verschränkungsaufbau durch den counter term verhindert wird. Es zeigt sich, dass für eine sehr sub-Ohm’sche Umgebung (deep sub-Ohmic regime) diese Erwartung nicht zutrifft. Weiterhin wird der Fall starker Kopplung zwischen dem zwei-Qubit-System und der Umgebung betrachtet. Die Berechnungen zeigen das generelle Bild, dass sich zwei nicht wechselwirkende Qubits durch den Einfluss einer gemeinsamen Umgebung verschränken. Selbst bei Temperaturen größer als null kann Verschränkung aufgebaut werden und auch für beliebig lange Zeiten erhalten bleiben. In einem letzten Punkt wird das Maximum der stationären Verschränkung (Langzeit-Limes) in Abhängigkeit von der Kopplungsstärke bestimmt. Dabei wird gezeigt, dass sich der bekannte Phasenübergang von Delokalisierzung zu Lokalisierung auch in der Langzeitdynamik der Verschränkung widerspiegelt. All diese Erkenntnisse erfordern eine exakte Behandlung der offenen Systemdynamik und erweitern somit das fundamentalen Verständnis der Verschränkungsdynamik offener Quantensysteme.

<p>Numerical trajectory optimization is helpful not only for mission planning but also design</p> <p>space exploration and quantifying vehicle performance. Direct methods for solving the opti-</p> <p>mal control problems, which first discretize the problem before applying necessary conditions</p> <p>of optimality, dominate the field of trajectory optimization because they are easier for the</p> <p>user to set up and are less reliant on a forming a good initial guess. On the other hand,</p> <p>many consider indirect methods, which apply the necessary conditions of optimality prior to</p> <p>discretization, too difficult to use for practical applications. Indirect methods though provide</p> <p>very high quality solutions, easily accessible sensitivity information, and faster convergence</p> <p>given a sufficiently good guess. Those strengths make indirect methods especially well-suited</p> <p>for generating large data sets for system analysis and worth revisiting.</p> <p>Recent advancements in the application of indirect methods have already mitigated many</p> <p>of the often cited issues. Automatic derivation of the necessary conditions with computer</p> <p>algebra systems have eliminated the manual step which was time-intensive and error-prone.</p> <p>Furthermore, regularization techniques have reduced problems which traditionally needed</p> <p>many phases and complex staging, like those with inequality path constraints, to a signifi-</p> <p>cantly easier to handle single arc. Finally, continuation methods can circumvent the small</p> <p>radius of convergence of indirect methods by gradually changing the problem and use previ-</p> <p>ously found solutions for guesses.</p> <p>The new optimal control problem solver Giuseppe incorporates and builds upon these</p> <p>advancements to make indirect methods more accessible and easily used. It seeks to enable</p> <p>greater research and creative approaches to problem solving by being more flexible and</p> <p>extensible than previous solvers. The solver accomplishes this by implementing a modular</p> <p>design with well-defined internal interfaces. Moreover, it allows the user easy access to and</p> <p>manipulation of component objects and functions to be use in the way best suited to solve</p> <p>a problem.</p> <p>A new technique simplifies and automates what was the predominate roadblock to using</p> <p>continuation, the generation of an initial guess for the seed solution. Reliable generation of</p> <p>a guess sufficient for convergence still usually required advanced knowledge optimal contrtheory or sometimes incorporation of an entirely separate optimization method. With the</p> <p>new method, a user only needs to supply initial states, a control profile, and a time-span</p> <p>over which to integrate. The guess generator then produces a guess for the “primal” problem</p> <p>through propagation of the initial value problem. It then estimates the “dual” (adjoint)</p> <p>variables by the Gauss-Newton method for solving the nonlinear least-squares problem. The</p> <p>decoupled approach prevents poorly guessed dual variables from altering the relatively easily</p> <p>guess primal variables. As a result, this method is simpler to use, faster to iterate, and much</p> <p>more reliable than previous guess generation techniques.</p> <p>Leveraging the continuation process also allows for greater insight into the solution space</p> <p>as there is only a small marginal cost to producing an additional nearby solutions. As a</p> <p>result, a user can quickly generate large families of solutions by sweeping parameters and</p> <p>modifying constraints. These families provide much greater insight in the general problem</p> <p>and underlying system than is obtainable with singular point solutions. One can extend</p> <p>these analyses to high-dimensional spaces through construction of compound continuation</p> <p>strategies expressible by directed trees.</p> <p>Lastly, a study into common convergence explicates their causes and recommends mitiga-</p> <p>tion strategies. In this area, the continuation process also serves an important role. Adaptive</p> <p>step-size routines usually suffice to handle common sensitivity issues and scaling constraints</p> <p>is simpler and out-performs scaling parameters directly. Issues arise when a cost functional</p> <p>becomes insensitive to the control, which a small control cost mitigates. The best perfor-</p> <p>mance of the solver requires proper sizing of the smoothing parameters used in regularization</p> <p>methods. An asymptotic increase in the magnitude of adjoint variables indicate approaching</p> <p>a feasibility boundary of the solution space.</p> <p>These techniques for indirect methods greatly facilitate their use and enable the gen-</p> <p>eration of large libraries of high-quality optimal trajectories for complex problems. In the</p> <p>future, these libraries can give a detailed account of vehicle performance throughout its flight</p> <p>envelope, feed higher-level system analyses, or inform real-time control applications.</p>

Le calcul rigoureux vise à produire des représentations certifiées pour les solutions de nombreux problèmes, notamment en analyse fonctionnelle, comme des équations différentielles ou des problèmes de contrôle optimal. En effet, certains domaines particuliers comme l’ingénierie des systèmes critiques ou les preuves mathématiques assistées par ordinateur ont des exigences de fiabilité supérieures à ce qui peut résulter de l’utilisation d’algorithmes relevant de l’analyse numérique classique.Notre objectif consiste à développer des algorithmes à la fois efficaces et validés / certifiés, dans le sens où toutes les erreurs numériques (d’arrondi ou de méthode) sont prises en compte. En particulier, nous recourons aux approximations polynomiales rigoureuses combinées avec des méthodes de validation a posteriori à base de points fixes. Ces techniques sont implémentées au sein d’une bibliothèque écrite en C, ainsi que dans un développement de preuve formelle en Coq, offrant ainsi le plus haut niveau de confiance, c’est-à-dire une implémentation certifiée.Après avoir présenté les opérations élémentaires sur les approximations polynomiales rigoureuses, nous détaillons un nouvel algorithme de validation pour des approximations sous forme de séries de Tchebychev tronquées de fonctions D-finies, qui sont les solutions d’équations différentielles ordinaires linéaires à coefficients polynomiaux. Nous fournissons une analyse fine de sa complexité, ainsi qu’une extension aux équations différentielles ordinaires linéaires générales et aux systèmes couplés de telles équations. Ces méthodes dites symboliques-numériques sont ensuite utilisées dans plusieurs problèmes reliés : une nouvelle borne sur le nombre de Hilbert pour les systèmes quartiques, la validation de trajectoires de satellites lors du problème du rendez-vous linéarisé, le calcul de polynômes d’approximation optimisés pour l’erreur d’évaluation, et enfin la reconstruction du support et de la densité pour certaines mesures, grâce à des techniques algébriques. / Rigorous numerics aims at providing certified representations for solutions of various problems, notably in functional analysis, e.g., differential equations or optimal control. Indeed, specific domains like safety-critical engineering or computer-assisted proofs in mathematics have stronger reliability requirements than what can be achieved by resorting to standard numerical analysis algorithms. Our goal consists in developing efficient algorithms, which are also validated / certified in the sense that all numerical errors (method or rounding) are taken into account. Specifically, a central contribution is to combine polynomial approximations with a posteriori fixed-point validation techniques. A C code library for rigorous polynomial approximations (RPAs) is provided, together with a Coq formal proof development, offering the highest confidence at the implementation level.After providing basic operations on RPAs, we focus on a new validation algorithm for Chebyshev basis solutions of D-finite functions, i.e., solutions of linear ordinary differential equations (LODEs) with polynomial coefficients. We give an in-depth complexity analysis, as well as an extension to general LODEs, and even coupled systems of them. These symbolic-numeric methods are finally used in several related problems: a new lower bound on the Hilbert number for quartic systems; a validation of trajectories arising in the linearized spacecraft rendezvous problem; the design of evaluation error efficient polynomial approximations; and the support and density reconstruction of particular measures using algebraic techniques.

This thesis focuses on the methodology of controlling dynamic systems in real time. It contents a review of the control theory basis and the elementary base of regulators construction. Then the list of matemathic formulaes follows as well as the math basis for the system simulations using a difeerential count and the problem of difeerential equations solving. Furthermore, there is a systematic approach to the design of general regulator enclosed, using modern simulation techniques. After the results confirmation in the Matlab system, the problematics of transport delay & quantization modelling follow.