Academic literature on the topic 'Smoothing problems'

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.<br>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.

The Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Ensemble Kalman Filter (EnKF) are commonly implemented practical solutions for solving nonlinear state space estimation problems; all based on the linear state space estimator, the Kalman Filter. Often, the UKF and EnKF are cited as a superior methods to the EKF with respect to error-based performance criteria. The UKF in turn has the advantage over the EnKF of smaller computational complexity. In practice however the UKF often fails to live up to this expectation, with performance which does not surpass the EKF and estimates which are not as robust as the EnKF. This work explores the geometry of alternative sigma point sets, which form the basis of the UKF, contributing several new sets along with novel methods used to generate them. In particular, completely novel systems of sigma points that preserve higher order statistical moments are found and evaluated. Additionally a new method for scaling and problem specific tuning of sigma point sets is introduced as well as a discussion of why this is necessary, and a new way of thinking about UKF systems in relation to the other two Kalman Filter methods. An Iterated UKF method is also introduced, similar to the smoothing iterates developed previously for the EKF. The performance of all of these methods is demonstrated using problem exemplars with the improvement of the contributed methods highlighted.

Rare events are part of the real world but inevitably environmental extreme events may have a massive impact on everyday life. We are familiar, for example, with the consequences and damage caused by hurricanes and floods etc. Consequently, there is considerable attention in studying, understanding and predicting the nature of such phenomena and the problems caused by them, not least because of the possible link between extreme climate events and global warming or climate change. Thus the study of extreme events has become ever more important, both in terms of probabilistic and statistical research. This thesis aims to provide statistical modelling and methods for making inferences about extreme events for two types of process. First, non-stationary univariate processes; second, spatial stationary processes. In each case the statistical aspects focus on model fitting and parameter estimation with applications to the modelling of environmental processes including, in particular, nonstationary extreme temperature series and spatially recorded rainfall measures.

In many instances arising prominently, but not exclusively, in imaging problems, it is important to condense the salient information so as to obtain a low-dimensional approximant of the data. This thesis is concerned with two basic situations which call for such a dimension reduction. The first of these is the statistical recovery of smooth edges in regression and density surfaces. The edges are understood to be contiguous curves, although they are allowed to meander almost arbitrarily through the plane, and may even split at a finite number of points to yield an edge graph. A novel locally-parametric nonparametric method is proposed which enjoys the benefit of being relatively easy to implement via a `tracking' approach. These topics are discussed in Chapters 2 and 3, with pertaining background material being given in the Appendix. In Chapter 4 we construct concomitant confidence bands for this estimator, which have asymptotically correct coverage probability. The construction can be likened to only a few existing approaches, and may thus be considered as our main contribution. ¶ Chapter 5 discusses numerical issues pertaining to the edge and confidence band estimators of Chapters 2-4. Connections are drawn to popular topics which originated in the fields of computer vision and signal processing, and which surround edge detection. These connections are exploited so as to obtain greater robustness of the likelihood estimator, such as with the presence of sharp corners. ¶ Chapter 6 addresses a dimension reduction problem for spatial data where the ultimate objective of the analysis is the discrimination of these data into one of a few pre-specified groups. In the dimension reduction step, an instrumental role is played by the recently developed methodology of functional data analysis. Relatively standar non-linear image processing techniques, as well as wavelet shrinkage, are used prior to this step. A case study for remotely-sensed navigation radar data exemplifies the methodology of Chapter 6.

In this thesis, quadratic hexahedral edge elements have been applied to the three dimensional for open region electromagnetic scattering problems. For this purpose, a semi-automatic all-hexahedral mesh generation algorithm is developed and implemented. Material properties inside the elements and along the edges are also determined and prescribed during the mesh generation phase in order to be used in the solution phase. Based on the condition number quality metric, the generated mesh is optimized by means of the Particle Swarm Optimization (PSO) technique. A framework implementing hierarchical hexahedral edge elements is implemented to investigate the performance of linear and quadratic hexahedral edge elements. Perfectly Matched Layers (PMLs), which are implemented by using a complex coordinate transformation, have been used for mesh truncation in the software. Sparse storage and relevant efficient matrix ordering are used for the representation of the system of equations. Both direct and indirect sparse matrix solution methods are implemented and used. Performance of quadratic hexahedral edge elements is deeply investigated over the radar cross-sections of several curved or flat objects with or without patches. Instead of the de-facto standard of 0.1 wavelength linear element size, 0.3-0.4 wavelength quadratic element size was observed to be a new potential criterion for electromagnetic scattering and radiation problems.

Le but principal de cette thèse est d'obtenir des résultats d'existence et d'unicité pour des équations aux dérivées partielles dispersives avec conditions aux limites non homogènes. L'approche privilégiée est l'adaptation de techniques issues de la théorie classique des problèmes aux limites hyperboliques (que l'on rappelle au chapitre 1, en améliorant légèrement un résultat). On met en évidence au chapitre 3 une classe d'équations linéaires qu'on peut qualifier de dispersives satisfaisant des critères “minimaux”, et des résultats d'existence et d'unicité pour le problème aux limites associé à celles-ci sont obtenus au chapitre 4.Le fil rouge du mémoire est le modèle d'Euler-Korteweg, pour lequel on aborde l'analyse du problème aux limites sur une version linéarisée au chapitre 2. Toujours pour cette version linéarisée, on prouve un effet Kato-régularisant au chapitre 3. Enfin l'analyse numérique du modèle est abordée au chapitre 5. Pour cela, on commence par utiliser les résultats précédents pour décrire une manière simple d'obtenir les conditions aux limites dites transparentes dans le cadre des équations précédemment décrites puis on utilise ces conditions aux limites pour le modèle d'Euler-Korteweg semi-linéaire afin d'observer la stabilité/instabilité des solitons, ainsi qu'un phénomène d'explosion en temps fini<br>The main aim of this thesis is to obtain well-posedness results for boundary value problems especially with non-homogeneous boundary conditions. The approach that we chose here is to adapt technics from the classical theory of hyperbolic boundary value problems (for which we give a brief survey in the first chapter, and a slight generalization). In chapter 3 we delimitate a class of linear dispersive equations, and we obtain well-posedness results for corresponding boundary value problems in chapter 4.The leading thread of this memoir is the Euler-Korteweg model. The boundary value problem for a linearized version is investigated in chapter 2, and the Kato-smoothing effect is proved (also for the linearized model) in chapter 3. Finally, the numerical analysis of the model is made in chapter 5. To begin with, we use the previous abstract results to show a simple way of deriving the so-called transparent boundary conditions for the equations outlined in chapter 3, and those conditions are then used to numerically solve the semi-linear Euler-Korteweg model. This allow us to observe the stability and instability of solitons, as well as a finite time blow up