Thesis (M.A.)--Boston University / In this paper, several numerical methods, instructive for the calculation of the approximate solutions of differential equations, are exhibited to be convergent. In the first two methods (Picard's Method and the Cauchy-Euler method), the theoretical importance of numerical solutions is demonstrated by establishing existence and uniqueness theorems for the linear differential equation of the first order dy/dx = f(x,y) subject to the following conditions: The equation is considered in some region of xy space containing a point (xo,yo) and in addition to being continuous, f(x,y) is assumed to satisfy a Lipschitz condition with respect to y, i.e. |f(x,y1)-f(x,y2)| < k|y1 - y2| where k is called the Lipschitz constant [TRUNCATED]

In this digital age, it is more important than ever to have good methods for processing images. We focus on the removal of blur from a captured image, which is called the image deblurring problem. In particular, we make no assumptions about the blur itself, which is called a blind deconvolution. We approach the problem by miniming an energy functional that utilizes total variation norm and a fidelity constraint. In particular, we extend the work of Chan and Wong to use a reference image in the computation. Using the shock filter as a reference image, we produce a superior result compared to existing methods. We are able to produce good results on non-black background images and images where the blurring function is not centro-symmetric. We consider using a general Lp norm for the fidelity term and compare different values for p. Using an analysis similar to Strong and Chan, we derive an adaptive scale method for the recovery of the blurring function. We also consider two numerical methods in this disseration. The first method is an extension of Picards method for PDEs in the discrete case. We compare the results to the analytical Picard method, showing the only difference is the use of the approximation versus exact derivatives. We relate the method to existing finite difference schemes, including the Lax-Wendroff method. We derive the stability constraints for several linear problems and illustrate the stability region is increasing. We conclude by showing several examples of the method and how the computational savings is substantial. The second method we consider is a black-box implementation of a method for solving the generalized eigenvalue problem. By utilizing the work of Golub and Ye, we implement a routine which is robust against existing methods. We compare this routine against JDQZ and LOBPCG and show this method performs well in numerical testing.

The aim of the thesis is to compare the iterative methods which program RFEM 5 uses the non-linear calculations of structures, namely the analysis of large deformations and post critical analysis. Comparison should serve as a basis for which calculation method is the most accurate, fastest and most reliable in terms of getting results. Time-consuming will be judged according to the calculation of the solution and the time needed to compute one iterativ. Robustness we will compare the reliability of methods in in normal use. Accuracy of the calculation will be determined by comparing the maximum deformation structures. Comparison will be made with examples from practice.