Thesis (Ph. D.)--Technische Universität Berlin, 1998. / Includes bibliographical references (p. 52-54).

The paper presents a simple model for recovering affine shape and correspondence from two orthographic views of a 3D object. It is shown that four corresponding points along two orthographic views, taken under similar illumination conditions, determine affine shape and correspondence for all other points. The scheme is useful for purposes of visual recognition by generating novel views of an object given two model views. It is also shown that the scheme can handle objects with smooth boundaries, to a good approximation, without introducing any modifications or additional model views.

In finance, the implied volatility surface is plotted against strike price and time to maturity. The shape of this volatility surface can be identified by fitting the model to what is actually observed in the market. The metric that is used to measure the discrepancy between the model and the market is usually defined by a mean squares of error of the model prices to the market prices. A regularization term can be added to this error metric to make the solution possess some desired properties. The discrepancy that we want to minimize is usually a highly nonlinear function of a set of model parameters with the regularization term. Typically monotonic decreasing algorithm is adopted to solve this minimization problem. Steepest descent or Newton type algorithms are two iterative methods but they are local, i.e., they use derivative information around the current iterate to find the next iterate. In order to ensure convergence, line search and trust region methods are two widely used globalization techniques. Motivated by the simplicity of Barzilai-Borwein method and the convergence properties brought by globalization techniques, we propose a new Scaled Gradient (SG) method for minimizing a differentiable function plus an L1-norm. This non-monotone iterative method only requires gradient information and safeguarded Barzilai-Borwein steplength is used in each iteration. An adaptive line search with the Armijo-type condition check is performed in each iteration to ensure convergence. Coleman, Li and Wang proposed another trust region approach in solving the same problem. We give a theoretical proof of the convergence of their algorithm. The objective of this thesis is to numerically investigate the performance of the SG method and establish global and local convergence properties of Coleman, Li and Wang’s trust region method proposed in [26]. Some future research directions are also given at the end of this thesis.

In finance, the implied volatility surface is plotted against strike price and time to maturity. The shape of this volatility surface can be identified by fitting the model to what is actually observed in the market. The metric that is used to measure the discrepancy between the model and the market is usually defined by a mean squares of error of the model prices to the market prices. A regularization term can be added to this error metric to make the solution possess some desired properties. The discrepancy that we want to minimize is usually a highly nonlinear function of a set of model parameters with the regularization term. Typically monotonic decreasing algorithm is adopted to solve this minimization problem. Steepest descent or Newton type algorithms are two iterative methods but they are local, i.e., they use derivative information around the current iterate to find the next iterate. In order to ensure convergence, line search and trust region methods are two widely used globalization techniques. Motivated by the simplicity of Barzilai-Borwein method and the convergence properties brought by globalization techniques, we propose a new Scaled Gradient (SG) method for minimizing a differentiable function plus an L1-norm. This non-monotone iterative method only requires gradient information and safeguarded Barzilai-Borwein steplength is used in each iteration. An adaptive line search with the Armijo-type condition check is performed in each iteration to ensure convergence. Coleman, Li and Wang proposed another trust region approach in solving the same problem. We give a theoretical proof of the convergence of their algorithm. The objective of this thesis is to numerically investigate the performance of the SG method and establish global and local convergence properties of Coleman, Li and Wang’s trust region method proposed in [26]. Some future research directions are also given at the end of this thesis.

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For an affine Weyl group W, we explicitly determine the elements for which the Möbius function of the subposet of affine Grassmannians under the Bruhat order is non-zero by utilizing the quantum Bruhat graph of the classical Weyl group associated to W . Then we examine embedding stable and consistent statistics on the affine Weyl group of type A which permit the definition of an affine infinite symmetric group. / Doctor of Philosophy / Similar to the integers, there are groups that have both an infinite number of elements and also a way to partially order those elements. With a partial ordering, we can consider the interval between two elements. When we make a function that sums over an interval of elements, then we can invert the function by using something called the Mӧbius function. For many groups, the Mӧbius function is extremely unpredictable and calculating the inverse may require us to consider an infinite number of elements. In this paper, we focus on groups called affine Weyl groups, which are very useful in algebraic geometry. It turns out that most elements in these groups have a very predictable pattern in their Mӧbius functions which only considers a finite number of elements. The first part of this paper gives very simple rules for calculating it. The second part of this paper focuses on a special type of affine Weyl group: the affine symmetric groups. We provide an attempt at defining a large parent group, which we call the affine infinite symmetric group, that contains all the other affine symmetric groups.

La thèse consiste de deux parties. Dans la première partie, on montre la pureté des fibres de Springer affines pour $\gl_{4}$ dans le cas non-ramifié. Plus précisément, on construit une famille de pavages non standard en espaces affines de la grassmannienne affine, qui induisent des pavages en espaces affines de la fibre de Springer affine. Dans la deuxième partie, on introduit une notion de $\xi$-stabilité sur la grassmannienne affine $\xx$ pour le groupe $\gl_{d}$, et on calcule le polynôme de Poincaré du quotient $\xx^{\xi}/T$ de la partie $\xi$-stable $\xxs$ par le tore maximal $T$ par une processus analogue de la réduction de Harder-Narasimhan. / This thesis consists of two parts. In the first part, we prove the purity of affine Springer fibers for $\gl_{4}$ in the unramified case. More precisely, we have constructed a family of non standard affine pavings for the affine grassmannian, which induce an affine paving for the affine Springer fiber. In the second part, we introduce a notion of $\xi$-stability on the affine grassmannian $\xx$ for the group $G=\gl_{d}$, and we calculate the Poincaré polynomial of the quotient $\xx^{\xi}/T$ of the stable part $\xxs$ by the maximal torus $T$ by a process analogue to the Harder-Narasimhan reduction.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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