Es wird ein Ansatz und ein Algorithmus zur Lösung von stochastischen Stoppproblemen vorgestellt, der auf einer dualen Formulierung zum klassischen Lösungsansatz für Stoppprobleme mittels Variationsungleichungen basiert. Unter bestimmten Voraussetzungen kann man für diese duale Formulierung ein äquivalentes unendlichdimensionales lineares Programm aufstellen, das die Momente des Aufenthaltsmaßes des stochastischen Prozesses bis zum Stoppzeitpunkt und die Momente der Verteilung des Prozesses zum Zeitpunkt des Stoppens als Variablen enthält. Für dieses unendlichdimensionale Problem werden endlichdimensionale Approximationen formuliert und gelöst, wobei die Momente nur bis zu einer endlichen Ordnung berücksichtigt werden. Die Güte der numerischen Resultate hängt davon ab, wie genau der Träger des Maßes zum Stoppzeitpunkt identifiziert werden kann. Aus diesem Grund wird ein Verfeinerungsalgorithmus entwickelt, mit dem diese Identifizierung in einer Reihe von Fällen gelingt und sich sehr genaue Ergebnisse erzielen lassen. Der für Stoppprobleme entwickelte Algorithmus kann auch bei der Ermittlung von optimalen Steuerungen für stetige stochastische Prozesse angewandt werden. Für einzelne Beispiele wird gezeigt, welche Resultate dabei erzielt werden können. / We present an approach to, and an algorithm for solving optimal stopping problems. The approach is based on a dual formulation of the classical method for solving stopping problems using variational inequalities. Under suitable conditions it is possible to express the dual formulation as an infinite-dimensional linear program. This linear program uses the moments of the occupation measure and the moments of the stopping measure as variables. We formulate and solve finite-dimensional approximations to this infinite-dimensional program by restricting the number of moments. The accuracy of the numerical results depend on how well the support of the stopping measure can be identified. To this end we develop an iterative procedure which works very well in many cases. In the second part of the dissertation we show how the algorithm, developed for stopping problems, can be used for solving stochastic control problems.

Diese Dissertation behandelt sowohl die Theorie, als auch beobachtetes Verhalten in Stoppproblemen. In einem Stoppproblem, beobachtet ein Agent die Entwicklung eines stationären, stochastischen Prozesses über die Zeit. Zu jedem Zeitpunkt genießt der Agent das Recht den Prozess zu stoppen, um eine Auszahlung einzustreichen die Funktion des gegenwärtigen und der vergangenen Realisationen des Prozesses sind. Das Ziel des Agenten ist es den Stoppzeitpunkt so zu wählen, dass die erwartete Auszahlung oder der erwartete Verlust durch Stoppen maximiert bzw. minimiert wird. Stoppprobleme dieser Art konstituieren können als die einfachsten, jedoch wirklich dynamischen Entscheidungsprobleme in der ökonomischen Theorie angesehen werden Das erste Kapitel legt neue theoretische Resultate hinsichtlich der optimalen Stoppstrategien unter Erwartungsnutzentheorie, sog. gain-loss utilities und Bedauerungspräferenzen vor. Das zweite Kapitel behandelt sodann die Ergebnisse eines Laborexperiments in dem die theoretischen Vorhersagen getestet werden. Kapitel drei beschäftigt sich mit der Situation in der die Agenten nicht vollständig über Wahrscheinlichkeiten für künftige Ereignisse informiert sind, d.h. es herrscht Ambiguität. / This dissertation is concerned with theory and behavior in stopping problems. In a stopping problem an agent or individual observes the realization of some exogenous and stationary stochastic process over time. At every point in time, she has the right or the once-only option to stop the process in order to earn a function of the past and current values of the process. The agent''s objective then is to choose the point in time to exercise the option in order to maximize an expected reward or to minimize an expected loss. Such problems constitute the most rudimentary, yet truly dynamic class of choice problems that is studied in economics. The first chapter provides new theoretical results about optimal stopping both under expected utility, as well as gain-loss utility and regret preferences. The second chapter presents a laboratory experiment that tests several of the theoretical predictions about behavior made in the first chapter. The third chapter is concerned with stopping behavior in a setting, where the probability law that drives the observed process is not perfectly known to the decision maker, i.e. there is ambiguity.

Gegenstand der vorliegenden Dissertation sind stochastische Kontrollprobleme, denen sich Agenten im Zusammenhang mit Entscheidungen auf Finanzmärkten gegenübersehen. Der erste Teil der Arbeit behandelt die Maximierung des erwarteten Nutzens des Endvermögens eines Finanzmarktinvestors. Für den Investor ist eine Beschreibung der optimalen Handelsstrategie, die zur numerischen Approximation geeignet ist sowie eine Stabilitätsanalyse der optimalen Handelsstrategie bzgl. kleinerer Fehlspezifikationen in Nutzenfunktion und Anfangsvermögen, von höchstem Interesse. In stetigen Marktmodellen beweisen wir Stabilitätsresultate für die optimale Handeslsstrategie in geeigneten Topologien. Für hinreichend differenzierbare Nutzenfunktionen und zeitstetige Marktmodelle erhalten wir eine Beschreibung der optimalen Handelsstrategie durch die Lösung eines Systems von stochastischen Vorwärts-Rückwärts-Differentialgleichungen (FBSDEs). Der zweite Teil der Arbeit beschäftigt sich mit optimalen Stopproblemen für einen Agenten, dessen Ertragsprozess von einem Ausfallsereignis abhängt. Unser Hauptinteresse gilt der Beschreibung der Lösungen vor und nach dem Ausfallsereignis und damit dem besseren Verständnis des Verhaltens des Agenten bei Auftreten eines Ausfallsereignisses. Wir zeigen wie sich das optimale Stopproblem in zwei einzelne Teilprobleme zerlegen lässt: eines, für das der zugrunde liegende Informationsfluss das Ausfallereignis nicht beinhaltet, und eines, in welchem der Informationsfluss das Ausfallereignis berücksichtigt. Aufbauend auf der Zerlegung des Stopproblems und der Verbindung zwischen der Optimalen Stoptheorie und der Theorie von reflektierenden stochastischen Rückwärts-Differentialgleichungen (RBSDEs), leiten wir einen entsprechenden Zerlegungsansatz her, um RBSDEs mit genau einem Sprung zu lösen. Wir beweisen neue Existenz- und Eindeutigkeitsresultate von RBSDEs mit quadratischem Wachstum. / This thesis studies stochastic control problems faced by agents in financial markets when making decisions. The first part focuses on the maximization of expected utility from terminal wealth for an investor trading in a financial market. Of utmost concern to the investor is a description of optimal trading strategy that is amenable to numerical approximation, and the stability analysis of the optimal trading strategy w.r.t. "small" misspecification in his utility function and initial capital. In the setting of a continuous market model, we prove stability results for the optimal wealth process in the Emery topology and the uniform topology on semimartingales, and stability results for the optimal trading strategy in suitable topologies. For sufficiently differentiable utility functions, we obtain a description of the optimal trading strategy in terms of the solution of a system of forward-backward stochastic differential equations (FBSDEs). The second part of the thesis deals with the optimal stopping problem for an agent with a reward process exposed to a default event. Our main concern is to give a description of the solutions before and after the default event and thereby better understand the behavior of the agent in the presence of default. We show how the stopping problem can be decomposed into two individual stopping problems: one with information flow for which the default event is not visible, and another one with information flow which captures the default event. We build on the decomposition of the optimal stopping problem, and the link between the theories of optimal stopping and reflected backward stochastic differential equations (RBSDEs) to derive a corresponding decomposition approach to solve RBSDEs with a single jump. This decomposition allows us to establish existence and uniqueness results for RBSDEs with drivers of quadratic growth.

The central theme of this thesis is the design of optimal balanced black-box stopping criteria in iterative solvers of symmetric positive-definite, symmetric indefinite, and nonsymmetric linear systems arising from finite element approximation of stochastic (parametric) partial differential equations. For a given stochastic and spatial approximation, it is known that iteratively solving the corresponding linear(ized) system(s) of equations to too tight algebraic error tolerance results in a wastage of computational resources without decreasing the usually unknown approximation error. In order to stop optimally-by avoiding unnecessary computations and premature stopping-algebraic error and a posteriori approximation error estimate must be balanced at the optimal stopping iteration. Efficient and reliable a posteriori error estimators do exist for close estimation of the approximation error in a finite element setting. But the algebraic error is generally unknown since the exact algebraic solution is not usually available. Obtaining tractable upper and lower bounds on the algebraic error in terms of a readily computable and monotonically decreasing quantity (if any) of the chosen iterative solver is the distinctive feature of the designed optimal balanced stopping strategy. Moreover, this work states the exact constants, that is, there are no user-defined parameters in the optimal balanced stopping tests. Hence, an iterative solver incorporating the optimal balanced stopping methodology that is presented here will be a black-box iterative solver. Typically, employing such a stopping methodology would lead to huge computational savings and in any case would definitely rule out premature stopping. The constants in the devised optimal balanced black-box stopping tests in MINRES solver for solving symmetric positive-definite and symmetric indefinite linear systems can be estimated cheaply on-the- fly. The contribution of this thesis goes one step further for the nonsymmetric case in the sense that it not only provides an optimal balanced black-box stopping test in a memory-expensive Krylov solver like GMRES but it also presents an optimal balanced black-box stopping test in memory-inexpensive Krylov solvers such as BICGSTAB(L), TFQMR etc. Currently, little convergence theory exists for the memory-inexpensive Krylov solvers and hence devising stopping criteria for them is an active field of research. Also, an optimal balanced black-box stopping criterion is proposed for nonlinear (Picard or Newton) iterative method that is used for solving the finite dimensional Navier-Stokes equations. The optimal balanced black-box stopping methodology presented in this thesis can be generalized for any iterative solver of a linear(ized) system arising from numerical approximation of a partial differential equation. The only prerequisites for this purpose are the existence of a cheap and tight a posteriori error estimator for the approximation error along with cheap and tractable bounds on the algebraic error.

Dual-energy computed tomography (DECT) has shown great potential in the reduction of uncertainties of proton ranges and low energy photon cross section estimation used in radiation therapy planning. The work presented herein investigated three contributions for advancing DECT applications. 1) A linear and separable two-parameter DECT, the basis vector model (BVM) was used to estimate proton stopping power. Compared to other nonlinear two-parameter models in the literature, the BVM model shows a comparable accuracy achieved for typical human tissues. This model outperforms other nonlinear models in estimations of linear attenuation coefficients. This is the first study to clearly illustrate the advantages of linear model not only in accurately mapping radiological quantities for radiation therapy, but also in providing a unique model for accurate linear forward projection modelling, which is needed by the statistical iterative reconstruction (SIR) and other advanced DECT reconstruction algorithms. 2) Accurate DECT requires knowledge of x-ray beam properties. Using the Birch-Marshall1 model and beam hardening correction coefficients encoded in a CT scanner’s sinogram header files, an efficient and accurate way to estimate the x-ray spectrum is proposed. The merits of the proposed technique lie in requiring no physical transmission measurement after a one-time calibration against an independently measured spectrum. This technique can also be used in monitoring the aging of x-ray CT tubes. 3) An iterative filtered back projection with anatomical constraint (iFBP-AC) algorithm was also implemented on a digital phantom to evaluate its ability in mitigating beam hardening effects and supporting accurate material decomposition for in vivo imaging of photon cross section and proton stopping power. Compared to iFBP without constraints, both algorithms demonstrate high efficiency of convergence. For an idealized digital phantom, similar accuracy was observed under a noiseless situation. With clinically achievable noise level added to the sinograms, iFBP-AC greatly outperforms iFBP in prediction of photon linear attenuation at low energy, i.e., 28 keV. The estimated mean errors of iFBP and iFBP-AC for cortical bone are 1% and 0.7%, respectively; the standard deviations are 0.6% and 5%, respectively. The achieved accuracy of iFBP-AC shows robustness versus contrast level. Similar mean errors are maintained for muscle tissue. The standard deviation achieved by iFBP-AC is 1.2%. In contrast, the standard deviation yielded by iFBP is about 20.2%. The algorithm of iFBP-AC shows potential application of quantitative measurement of DECT. The contributions in this thesis aim to improve the clinical performance of DECT.

The Web is a vast repository of data, and information on almost any subject can be found with the aid of search engines. Although the Web is international, the majority of research on finding of information has a focus on languages such as English and Chinese. In this thesis, we investigate information retrieval techniques for Indonesian. Although Indonesia is the fourth most populous country in the world, little attention has been given to search of Indonesian documents. Stemming is the process of reducing morphological variants of a word to a common stem form. Previous research has shown that stemming is language-dependent. Although several stemming algorithms have been proposed for Indonesian, there is no consensus on which gives better performance. We empirically explore these algorithms, showing that even the best algorithm still has scope for improvement. We propose novel extensions to this algorithm and develop a new Indonesian stemmer, and show that these can improve stemming correctness by up to three percentage points; our approach makes less than one error in thirty-eight words. We propose a range of techniques to enhance the performance of Indonesian information retrieval. These techniques include: stopping; sub-word tokenisation; and identification of proper nouns; and modifications to existing similarity functions. Our experiments show that many of these techniques can increase retrieval performance, with the highest increase achieved when we use grams of size five to tokenise words. We also present an effective method for identifying the language of a document; this allows various information retrieval techniques to be applied selectively depending on the language of target documents. We also address the problem of automatic creation of parallel corpora --- collections of documents that are the direct translations of each other --- which are essential for cross-lingual information retrieval tasks. Well-curated parallel corpora are rare, and for many languages, such as Indonesian, do not exist at all. We describe algorithms that we have developed to automatically identify parallel documents for Indonesian and English. Unlike most current approaches, which consider only the context and structure of the documents, our approach is based on the document content itself. Our algorithms do not make any prior assumptions about the documents, and are based on the Needleman-Wunsch algorithm for global alignment of protein sequences. Our approach works well in identifying Indonesian-English parallel documents, especially when no translation is performed. It can increase the separation value, a measure to discriminate good matches of parallel documents from bad matches, by approximately ten percentage points. We also investigate the applicability of our identification algorithms for other languages that use the Latin alphabet. Our experiments show that, with minor modifications, our alignment methods are effective for English-French, English-German, and French-German corpora, especially when the documents are not translated. Our technique can increase the separation value for the European corpus by up to twenty-eight percentage points. Together, these results provide a substantial advance in understanding techniques that can be applied for effective Indonesian text retrieval.

Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system occurs. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case. When applying an iterative solver to an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence performance of Landweber-type iteration, and propose new ways to specify the relaxation parameters. These are computed so as to control the propagated error. We also describe a class of stopping rules for Landweber-type iteration for solving linear inverse problems. The class includes the well known discrepancy principle, and the monotone error rule. We unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections. Kaczmarz's method, also called ART (Algebraic Reconstruction Technique) is often used for solving the linear system which appears in image reconstruction. This is a fully sequential method. We examine and compare ART and its symmetric version. It is shown that the cycles of symmetric ART, unlike ART, converge to a weighted least squares solution if and only if the relaxation parameter lies between zero and two. Further we show that ART has faster asymptotic rate of convergence than symmetric ART. Also a stopping criterion is proposed and evaluated for symmetric ART. We further investigate a class of block-iterative methods used in image reconstruction. The cycles of the iterative sequences are characterized in terms of the original linear system. We define symmetric block-iteration and compare the behavior of symmetric and non-symmetric block-iteration. The results are illustrated using some well-known methods. A stopping criterion is offered and assessed for symmetric block-iteration.

Nous avons montré que l'irradiation aux ions d'argon d'énergie comprise entre 1 et 15 MeV cause des dommages en surface du titane et du zirconium, qui prennent la forme d'une oxydation accélérée et/ou d'une cratérisation dont les effets évoluent en fonction de l'énergie du projectile et de l'atmosphère de recuit (température et pression), simulant les conditions environnementales représentatives de l'interface gaine/combustible d'un réacteur REP. Par AFM, nous avons montré que la surface du titane et du zirconium était attaquée par bombardement aux ions d'argon, à haute température (jusqu'à 500°C) en milieu faiblement oxydant (sous pression d'air sec raréfié comprise entre 5,7 10-5 Pa et 5 10-3 Pa) et à une dose moyenne fixée à environ 5 1014 ions.cm-2. On observe ainsi la formation de cratères nanométriques sur toute la surface du titane irradié entre 2 et 9 MeV et celle du zirconium irradié à 4 MeV, dont les caractéristiques varient en fonction de la température et de la pression. Dans le cas du couple Ar/Ti, l'efficacité d'endommagement superficiel augmente lorsque l'énergie du projectile diminue de 9 à 2 MeV. Par ailleurs, alors que la surface du titane apparaît transparente au faisceau d'ions à 15 MeV, celle du zirconium révèle de nombreux cratères micrométriques entourés d'un large halo sombre. Les caractéristiques des cratères (taille et densité superficielle) diffèrent de façon significative de celles observées à la fois dans le domaine des basses énergies (keV) où le dépôt d'énergie est contrôlé par les collisions balistiques (Sn) et dans le domaine des hautes énergies (MeV - GeV) où le dépôt d'énergie est contrôlé par les excitations électroniques (Se), ce qui n'était pas complètement inattendu dans le domaine d'énergie intermédiaire étudié, dans lequel des effets de pouvoir d'arrêt combiné Sn - Se sont envisageables. Par XPS couplé au décapage ionique, nous avons montré qu'il existe un effet d'irradiation sur l'oxydation thermique du titane, exacerbée sous faisceau d'ions d'argon entre 2 et 9 MeV, et qu'il existe aussi un effet d'énergie sur l'épaisseur et la stœchiométrie de l'oxyde. L'étude menée par Ellipsométrie Spectroscopique sur les films d'oxyde formés entre 1 et 9 MeV a corroboré ces résultats et montré précisément qu'il existe un pic d'oxydation en fonction de l'énergie d'ions d'argon, qui présente un maximum à 3 MeV, dans les conditions de l'expérience. Les mesures de gain d'oxygène effectuées par NBS confirment l'existence de ce pic d'oxydation. Les résultats acquis à ce jour par NBS concernant l'oxydation thermique du zirconium sous irradiation à l'argon de 4 et 9 MeV confirment les travaux antérieurs du groupe Aval du Cycle Electronucléaire de l'Institut de Physique Nucléaire de Lyon, et suggèrent fortement l'existence d'un pic d'oxydation dans la même plage d'énergie de projectile que pour le titane.

Two standard graph representations for linear codes are the Tanner graph and the tailbiting trellis. Such graph representations allow the decoding problem for a code to be phrased as a computational problem on the corresponding graph and yield graph theoretic criteria for good codes. When a Tanner graph for a code is used for communication across a binary erasure channel (BEC) and decoding is performed using the standard iterative decoding algorithm, the maximum number of correctable erasures is determined by the stopping distance of the Tanner graph. Hence the computational problem of determining the stopping distance of a Tanner graph is of interest. In this thesis it is shown that computing stopping distance of a Tanner graph is NP hard. It is also shown that there can be no (1 + є ) approximation algorithm for the problem for any є > 0 unless P = NP and that approximation ratio of 2(log n)1- є for any є > 0 is impossible unless NPCDTIME(npoly(log n)). One way to construct Tanner graphs of large stopping distance is to ensure that the graph has large girth. It is known that stopping distance increases exponentially with the girth of the Tanner graph. A new elementary combinatorial construction algorithm for an almost regular LDPC code family with provable Ώ(log n) girth and O(n2) construction complexity is presented. The bound on the girth is close within a factor of two to the best known upper bound on girth. The problem of linear time exact maximum likelihood decoding of tailbiting trellis has remained open for several years. An O(n) complexity approximate maximum likelihood decoding algorithm for tail-biting trellises is presented and analyzed. Experiments indicate that the algorithm performs close to the ideal maximum likelihood decoder.

Annotation. This thesis presents some analysis of car braking process: a number of analythical formulas and graphic illustrations, discussion on different modes of braking, wider analysis of the main factors, having influence on the braking process. The main parameters and characteristic features, such as steady deceleration, driver’s reaction time, time of deceleration increase, restopping time and others, are also examined, values of most of them were specified during experimental research. A number of dependency graphs and numerical data, recommended for use in calculations performed during the examination of road accidents (and so improving the accuracy of the methods) was also presented.