Separable Inverse Problems, Blind Deconvolution, and Stray Light Correction for Extreme Ultraviolet Solar Images.

The determination of the inputs to a system given noisy output data is known as an inverse problem. When the system is a linear transformation involving unknown side parameters, the problem is called separable. A quintessential separable inverse problem is blind deconvolution: given a blurry image one must determine the sharp image and point spread function (PSF) that were convolved together to form it. This thesis describes a novel optimization approach for general separable inverse problems, a new blind deconvolution method for images corrupted by camera shake, and the first stray light correction for extreme ultraviolet (EUV) solar images from the EUVI/STEREO instruments. We present a generalization of variable elimination methods for separable inverse problems beyond least squares. Existing variable elimination methods require an explicit formula for the optimal value of the linear variables, so they cannot be used in problems with Poisson likelihoods, bound constraints, or other important departures from least squares. To address this limitation, we propose a generalization of variable elimination in which standard optimization methods are modified to behave as though a variable has been eliminated. Computational experiments indicate that this approach can have significant speed and robustness advantages. A new incremental sparse approximation method is proposed for blind deconvolution of images corrupted by camera shake. Unlike current state-of-the-art variational Bayes methods, it is based on simple alternating projected gradient optimization. In experiments on a standard test set, our method is faster than the state-of-the-art and competitive in deblurring performance. Stray light PSFs are determined for the two EUVI instruments, EUVI-A and B, aboard the STEREO mission. The PSFs are modeled using semi-empirical parametric formulas, and their parameters are determined by semiblind deconvolution of EUVI images. The EUVI-B PSFs were determined from lunar transit data, exploiting the fact that the Moon is not a significant EUV source. The EUVI-A PSFs were determined by analysis of simultaneous A/B observations from December 2006, when the instruments had nearly identical lines of sight to the Sun. We provide the first estimates of systematic error in EUV deconvolved images.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99797/1/shearerp_1.pd

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal