In this work we research convergence rates of densities of independent random extreme values. Using [4] and [2] work results, we will get nonuniform estimate of convergence rates in local theorem of extreme values of independent random variables exploring different distribution functions. [4], [2] work results we will generalize exploring nonuniform normalized extreme values convergence.

The boundary layer problems in periodic homogenization arise naturally from the quantitative analysis of convergence rates. Formally they are second-order linear elliptic systems with periodically oscillating coefficient matrix, subject to periodically oscillating Dirichelt or Neumann boundary data. In this dissertation, for either Dirichlet problem or Neumann problem, we establish the homogenization results and obtain the nearly sharp convergence rates, provided the domain is strictly convex. Also, we show that the homogenized boundary data is in W1,p for any p ∈ (1,∞), which implies the Cα-Hölder continuity for any α ∈ (0,1).

Since the mid 1980's many statisticians have studied methods for combining parametric and nonparametric esimates to improve the quality of fits in a regression problem. Notably in 1987, Einsporn and Birch proposed the Model Robust Regression estimate (MRR1) in which estimates of the parametric function, ƒ, and the nonparametric function, 𝑔, were combined in a straightforward fashion via the use of a mixing parameter, λ. This technique was studied extensively at small samples and was shown to be quite effective at modeling various unusual functions. In 1995, Mays and Birch developed the MRR2 estimate as an alternative to MRR1. This model involved first forming the parametric fit to the data, and then adding in an estimate of 𝑔 according to the lack of fit demonstrated by the error terms. Using small samples, they illustrated the superiority of MRR2 to MRR1 in most situations. In this dissertation we have developed asymptotic convergence rates for both MRR1 and MRR2 in OLS and GLS (maximum likelihood) settings. In many of these settings, it is demonstrated that the user of MRR1 or MRR2 achieves the best convergence rates available regardless of whether or not the model is properly specified. This is the "Golden Result of Model Robust Regression". It turns out that the selection of the mixing parameter is paramount in determining whether or not this result is attained. / Ph. D.

No description available.

<p>The dissertation focuses on solving some important theoretical and methodological problems associated with Bayesian modeling of infinite dimensional `objects', popularly called nonparametric Bayes. The term `infinite dimensional object' can refer to a density, a conditional density, a regression surface or even a manifold. Although Bayesian density estimation as well as function estimation are well-justified in the existing literature, there has been little or no theory justifying the estimation of more complex objects (e.g. conditional density, manifold, etc.). Part of this dissertation focuses on exploring the structure of the spaces on which the priors for conditional densities and manifolds are supported while studying how the posterior concentrates as increasing amounts of data are collected.</p><p>With the advent of new acquisition devices, there has been a need to model complex objects associated with complex data-types e.g. millions of genes affecting a bio-marker, 2D pixelated images, a cloud of points in the 3D space, etc. A significant portion of this dissertation has been devoted to developing adaptive nonparametric Bayes approaches for learning low-dimensional structures underlying higher-dimensional objects e.g. a high-dimensional regression function supported on a lower dimensional space, closed curves representing the boundaries of shapes in 2D images and closed surfaces located on or near the point cloud data. Characterizing the distribution of these objects has a tremendous impact in several application areas ranging from tumor tracking for targeted radiation therapy, to classifying cells in the brain, to model based methods for 3D animation and so on. </p><p> </p><p> The first three chapters are devoted to Bayesian nonparametric theory and modeling in unconstrained Euclidean spaces e.g. mean regression and density regression, the next two focus on Bayesian modeling of manifolds e.g. closed curves and surfaces, and the final one on nonparametric Bayes spatial point pattern data modeling when the sampling locations are informative of the outcomes.</p> / Dissertation

In the last decade l1-regularization became a powerful and popular tool for the regularization of Inverse Problems. While in the early years sparse solution were in the focus of research, recently also the case that the coefficients of the exact solution decay sufficiently fast was under consideration. In this paper we seek to show that l1-regularization is applicable and leads to optimal convergence rates even when the exact solution does not belong to l1 but only to l2. This is a particular example of over-smoothing regularization, i.e., the penalty implies smoothness properties the exact solution does not fulfill. We will make some statements on convergence also in this general context.

No description available.

The subject of this thesis is a class of fast inpainting methods (image or video) based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels (or video voxels) are filled by assigning them a color equal to a weighted average of either their already filled neighbors (the ``direct'' form of the method) or those neighbors plus additional neighbors within the current shell (the ``semi-implicit'' form). In the direct form, pixels (voxels) in the current shell may be filled independently, but in the semi-implicit form they are filled simultaneously by solving a linear system. We focus in this thesis mainly on the image inpainting case, where the literature contains several methods corresponding to the {\em direct} form of the method - the semi-implicit form is introduced for the first time here. These methods effectively differ only in the order in which pixels (voxels) are filled, the weights used for averaging, and the neighborhood that is averaged over. All of them are very fast, but at the same time all of them leave undesirable artifacts such as ``kinking'' (bending) or blurring of extrapolated isophotes. This thesis has two main goals. First, we introduce new algorithms within this class, which are aimed at reducing or eliminating these artifacts, and also target a specific application - the 3D conversion of images and film. The first part of this thesis will be concerned with introducing 3D conversion as well as Guidefill, a method in the above class adapted to the inpainting problems arising in 3D conversion. However, the second and more significant goal of this thesis is to study these algorithms as a class. In particular, we develop a mathematical theory aimed at understanding the origins of artifacts mentioned. Through this, we seek is to understand which artifacts can be eliminated (and how), and which artifacts are inevitable (and why). Most of the thesis is occupied with this second goal. Our theory is based on two separate limits - the first is a {\em continuum} limit, in which the pixel width →0, and in which the algorithm converges to a partial differential equation. The second is an asymptotic limit in which h is very small but non-zero. This latter limit, which is based on a connection to random walks, relates the inpainted solution to a type of discrete convolution. The former is useful for studying kinking artifacts, while the latter is useful for studying blur. Although all the theoretical work has been done in the context of image inpainting, experimental evidence is presented suggesting a simple generalization to video. Finally, in the last part of the thesis we explore shell-based video inpainting. In particular, we introduce spacetime transport, which is a natural generalization of the ideas of Guidefill and its predecessor, coherence transport, to three dimensions (two spatial dimensions plus one time dimension). Spacetime transport is shown to have much in common with shell-based image inpainting methods. In particular, kinking and blur artifacts persist, and the former of these may be alleviated in exactly the same way as in two dimensions. At the same time, spacetime transport is shown to be related to optical flow based video inpainting. In particular, a connection is derived between spacetime transport and a generalized Lucas-Kanade optical flow that does not distinguish between time and space.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.