On Modified l_1-Minimization Problems in Compressed Sensing

Sparse signal modeling has received much attention recently because of its application in medical imaging, group testing and radar technology, among others. Compressed sensing, a recently coined term, has showed us, both in theory and practice, that various signals of interest which are sparse or approximately sparse can be efficiently recovered by using far fewer samples than suggested by Shannon sampling theorem. Sparsity is the only prior information about an unknown signal assumed in traditional compressed sensing techniques. But in many applications, other kinds of prior information are also available, such as partial knowledge of the support, tree structure of signal and clustering of large coefficients around a small set of coefficients. In this thesis, we consider compressed sensing problems with prior information on the support of the signal, together with sparsity. We modify regular l_1 -minimization problems considered in compressed sensing, using this extra information. We call these modified l_1 -minimization problems. We show that partial knowledge of the support helps us to weaken sufficient conditions for the recovery of sparse signals using modified ` 1 minimization problems. In case of deterministic compressed sensing, we show that a sharp condition for sparse recovery can be improved using modified ` 1 minimization problems. We also derive algebraic necessary and sufficient condition for modified basis pursuit problem and use an open source algorithm known as l_1 -homotopy algorithm to perform some numerical experiments and compare the performance of modified Basis Pursuit Denoising with the regular Basis Pursuit Denoising

A Hierarchical Solver for Time-Harmonic Maxwell\u27s Equations

Die Kombination von Transformationsoptik und Sweeping Preconditionern sowie ein hierarchischer Ansatz ermöglichen einen Vorkonditionierer für lineare Gleichungssysteme, die bei der Diskretisierung von Maxwell\u27s Gleichungen mit der Finite Elemente Methode entstehen. Das Verfahren wird beschrieben, numerische Beispiele präsentiert und unterschiedliche Anwendungen diskutiert. Das Dokument enthält auch den dokumentierten Code als Anhang

Calculation of compressible flow about three-dimensional inlets with auxiliary inlets, slats and vanes by means of a panel method

An efficient and user oriented method was constructed for calculating flow in and about complex inlet configurations. Efficiency is attained by: (1) the use of a panel method; (2) a technique of superposition for obtaining solutions at any inlet operating condition; and (3) employment of an advanced matrix iteration technique for solving large full systems of equations, including the nonlinear equations for the Kutta condition. User concerns are addressed by the provision of several novel graphical output options that yield a more complete comprehension of the flowfield than was possible previously

Singular systems with time-varying delays

Preliminaries on time-delay singular systems -- Stability of time-delay singular systems -- State feedback controller for time-delay singular systems -- Static output feedback controller for time-delay singular systems with saturating actuators

Nonparametric least squares estimation in integer-valued GARCH models

In this thesis we consider Poisson regression models for count data. Suppose we observe a time series of count variables. Given the information about the past, each count variable has a Poisson distribution with a random intensity. The time series of intensities is unobservable, but we impose a functional relationship between the current intensity and the preceding pair of intensity and count observation. In the literature some consideration has been given to parametric models of the linear INGARCH(1,1) type or more involved ones like the log linear model. In these cases √n-consistency of the partial maximum likelihood estimator has been proven. Suppose that the relationship between a count variable and the respectively preceding pair of count and intensity variables is given by a link function that cannot be characterized by a finite-dimensional parameter. We call this model a nonparametric integer valued GARCH model. In order to obtain a suitable estimation equation in this nonparametric model, a contractive condition has to be imposed on the true link function. We analyze the rate of convergence of a least squares estimator that is inspired by the work of Meister and Kreiß (2016). We prove uniform mixing of the univariate count process and use the derived properties to apply some classical tools from empirical process theory. The size of the class of admissible functions determines the rate of convergence, which is a common property of nonparametric models. Since this estimator is computationally rather impractical, we also analyze the behavior of an approximate least squares estimator. In contrast to the analysis of the first estimator, the examination of the estimators asymptotic quality is based on the exploitation of martingale properties instead of mixing. The approximate least squares estimator is indeed computable, and we take the opportunity to conduct experiments to illustrate the proposed statistical procedure. An exposition of the experimental results will conclude this thesis