Dissertation--Uppsala, 2009. / Bibliogr. p. 293-298.

Thesis--Lund. / Extra t.p. with thesis statement, inserted. Summary in English. Bibliography: p. 303-307.

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Several projects are currently underway to obtain large galaxy redshift surveys over the course of the next decade. The aim of this thesis is to study how well the resultant three-dimensional maps of the galaxy distribution will be able to constrain the various parameters of the standard Big Bang cosmology. The work is driven by the need to deal with data of far better quality than has previously been available. Systematic biases in the treatment of existing datasets have been dwarfed by random errors due to the small size of the sample, but this will not be the case with the wealth of data that will shortly become available. We employ a set of high-resolution /V-body simulations spanning a range of cosmologies and galaxy biasing schemes. We use the power spectrum of the galaxy density field, measured using the fast Fourier transform process, to develop models and statistics for extracting cosmological information. In particular, we examine the distortion of the power spectrum by galaxy peculiar velocities when measurements are made in redshift space. Mock galaxy catalogues are drawn from these simulations, mimicking the geometries and selection functions of the large surveys we wish to model. Applying the same models to the mock catalogues is not a trivial task, as geometrical effects distort the power spectrum, and measurement errors are determined by the survey volume. We develop methods for assessing these effects and present an in-depth analysis of the likely confidence intervals we will obtain from the surveys on the parameters that determine the power spectrum. Real galaxy catalogues are prone to additional biases that must be assessed and removed. One of these is the effect of extinction by dust in the Milky Way, which imprints its own angular clustering signal on the measured power spectrum. We investigate the strength of this effect for the SDSS survey.

On considère le problème de contrôle optimal de temps minimal pour des systèmes affine et mono-entrée en dimension finie, avec conditions initiales et finales fixées, où le contrôle scalaire prend ses valeurs dans un intervalle fermé. Lors de l'application d'une méthode de tir pour résoudre ce problème, on peut rencontrer des obstacles numériques car la fonction de tir n'est pas lisse lorsque le contrôle est bang-bang. Pour ces systèmes, dans le cas bang-bang, un concept théorique de temps conjugué a été défini, toutefois les algorithmes de calcul direct sont difficiles à appliquer. En outre, les questions théoriques et pratiques de la théorie du temps conjugué sont bien connues dans le cas lisse, et des outils efficaces de mise en oeuvre sont disponibles. On propose une procédure de régularisation pour laquelle les solutions du problème de temps minimal dépendent d'un paramètre réel positif suffisamment petit et sont définis par des fonctions lisses en temps, ce qui facilite l'application de la méthode de tir simple. Sous des hypothèses convenables, nous prouvons un résultat de convergence forte des solutions du problème régularisé vers la solution du problème initial, lorsque le paramètre réel tend vers zéro. Le calcul des temps conjugués pour les trajectoires localement optimales du problème régularisé est standard. Nous prouvons, sous des hypothèses appropriées, la convergence du premier temps conjugué du problème régularisé vers le premier temps conjugué du problème de contrôle bang-bang initial, quand le paramètre réel tend vers zéro. Ainsi, on obtient une procédure algorithmique efficace pour calculer les temps conjugués dans le cas bang-bang. / In this thesis we consider a minimal time control problem for single-input control-affine systems in finite dimension with fixed initial and final conditions, where the scalar control take values on a closed interva1. When applying a shooting method for solving this problem, one may encounter numerical obstacles due to the fact that the shooting function is non smooth whenever the control is bang-bang. For these systems a theoretical concept of conjugate time has been defined in the bang-bang case, however direct algorithms of computation are difficult to apply. Besides, theoretical and practical issues for conjugate time theory are well known in the smooth case, and efficient implementation tools are available. We propose a regularization procedure for which the solutions of the minimal time problem depend on a small enough real positive parameter and are defined by smooth functions with respect to the time variable, facilitating the application of a single shooting method. Under appropriate assumptions, we prove a strong convergence result of the solutions of the regularized problem towards the solution of the initial problem, when the real parameter tends to zero. The conjugate times computation of the locally optimal trajectories for the regularized problem falls into the standard theory. We prove, under appropriate assumptions, the convergence of the first conjugate time of the regularized problem towards the first conjugate time of the initial bang-bang control problem, when the real parameter tends to zero. As a byproduct, we obtain an efficient algorithmic way to compute conjugate times in the bang-bang case.

Dissertation--Oslo, 1995. / Bibliogr. p. 213-223.

Diss. Uppsala universitet.

Motiviert durch das Problem der optimalen Strategie beim Handel einer großen Aktienposition, behandelt diese Arbeit ein stochastisches Kontrollproblem mit zwei besonderen Eigenschaften. Zum einen wird davon ausgegangen, dass das Kontrollproblem eine exponentielle Verzögerung in der Kontrollvariablen beinhaltet, zum anderen nehmen wir an, dass die Koeffizienten des Kontrollproblems linear in der Kontrollvariablen sind. Wir erhalten ein degeneriertes stochastisches Kontrollproblem, dessen Lösung - sofern sie existiert - Bang-Bang-Charakter hat. Die resultierende Unstetigkeit der optimalen Kontrolle führt dazu, dass die Existenz einer optimalen Lösung nicht selbstverständlich ist und bewiesen werden muss. Es wird eine Folge von stochastischen Kontrollproblemen mit Zustandsprozessen konstruiert, deren jeweilige Diffusionsmatrix invertierbar ist und die ursprüngliche degenerierte Diffusionsmatrix approximiert. Außerdem stellen die Kostenfunktionale der Folge eine konvexe Approximation des ursprünglichen linearen Kostenfunktionals dar. Um die Konvergenz der Lösungen dieser Folge zu zeigen, stellen wir die Kontrollprobleme in Form von stochastischen Vorwärts-Rückwärts-Differential-gleichungen (FBSDEs) dar. Wir zeigen, dass die zu der konstruierten Folge von Kontrollproblemen gehörigen Lösungen der Vorwärts-Rückwärtsgleichungen – zumindest für eine Teilfolge - in Verteilung konvergieren. Mit Hilfe einer Konvexitätsannahme der Koeffizienten ist es möglich, einen Kontroll-prozess auf einem passenden Wahrscheinlichkeitsraum zu konstruieren, der optimal für das ursprüngliche stochastische Kontrollproblem ist. Neben der damit bewiesenen Existenz einer optimalen (Bang-Bang-) Lösung, wird damit auch eine glatte Approximation der unstetigen Bang-Bang-Lösung erreicht, welche man für die numerische Approximation des Problems verwenden kann. Die Ergebnisse werden schließlich dann in Form von numerischen Simulationen auf das Problem der optimalen Handels¬ausführung angewendet. / Motivated by the problem of how to optimally execute a large stock position, this thesis considers a stochastic control problem with two special properties. First, the control problem has an exponential delay in the control variable, and so the present value of the state process depends on the moving average of past control decisions. Second, the coefficients are assumed to be linear in the control variable. It is shown that a control problem with these properties generates a mathematically challenging problem. Specifically, it becomes a stochastic control problem whose solution (if one exists) has a bang-bang nature. The resulting discontinuity of the optimal solution creates difficulties in proving the existence of an optimal solution and in solving the problem with numerical methods. A sequence of stochastic control problems with state processes is constructed, whose diffusion matrices are invertible and approximate the original degenerate diffusion matrix. The cost functionals of the sequence of control problems are convex approximations of the original linear cost functional. To prove the convergence of the solutions, the control problems are written in the form of forward-backward stochastic differential equations (FBSDEs). It is then shown that the solutions of the FBSDEs corresponding to the constructed sequence of control problems converge in law, at least along a subsequence. By assuming convexity of the coefficients, it is then possible to construct from this limit an admissible control process which, for an appropriate reference stochastic system, is optimal for our original stochastic control problem. In addition to proving the existence of an optimal (bang-bang) solution, we obtain a smooth approximation of the discontinuous optimal bang-bang solution, which can be used for the numerical solution of the problem. These results are then applied to the optimal execution problem in form of numerical simulations.

<p> A new radar tracking detector using fixed delay lines in place of continuously variable delay lines is described. The fixed delays are switched in and out depending on the output of a correlator. Results of a working system are shown using bang-bang feedback and analog-to-digital feedback, for tracking a time-varying delay. </p> / Thesis / Master of Engineering (MEngr)

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