A short survey on Kantorovich-like theorems for Newton's method

We survey influential quantitative results on the convergence of the Newton iterator towards simple roots of continuously differentiable maps defined over Banach spaces. We present a general statement of Kantorovich's theorem, with a concise proof from scratch, dedicated to wide audience. From it, we quickly recover known results, and gather historical notes together with pointers to recent articles

Difference equations and iterative processes

Divergence equations and iterative processe

Lie series for celestial mechanics, accelerators, satellite stabilization and optimization

Lie series applications to celestial mechanics, accelerators, satellite orbits, and optimizatio

Iterative Linear Algebra for Parameter Estimation

The principal goal of this thesis is the development and analysis of effcient numerical methods for large-scale nonlinear parameter estimation problems. These problems are of high relevance in all sciences that predict the future using big data sets of the past by fitting and then extrapolating a mathematical model. This thesis is concerned with the fitting part. The challenges lie in the treatment of the nonlinearities and the sheer size of the data and the unknowns. The state-of-the-art for the numerical solution of parameter estimation problems is the Gauss-Newton method, which solves a sequence of linearized subproblems. One of the contributions of this thesis is a thorough analysis of the problem class on the basis of covariant and contravariant k-theory. Based on this analysis, it is possible to devise a new stopping criterion for the iterative solution of the inner linearized subproblems. The analysis reveals that the inner subproblems can be solved with only low accuracy without impeding the speed of convergence of the outer iteration dramatically. In addition, I prove that this new stopping criterion is a quantitative measure of how accurate the solution of the subproblems needs to be in order to produce inexact Gauss- Newton sequences that converge to a statistically stable estimate provided that at least one exists. Thus, this new local approach results to be an inexact Gauss-Newton method that requires far less inner iterations for computing the inexact Gauss-Newton step than the classical exact Gauss-Newton method based on factorization algorithm for computing the Gauss-Newton step that requires to perform 100% of the inner iterations, which is computationally prohibitively expensive when the number of parameters to be estimated is large. Furthermore, we generalize the local ideas of this local inexact Gauss-Newton approach, and introduce a damped inexact Gauss-Newton method using the Backward Step Control for global Newton-type theory of Potschka. We evaluate the efficiency of our new approach using two examples. The first one is a parameter identification of a nonlinear elliptical partial differential equation, and the second one is a real world parameter estimation on a large-scale bundle adjustment problem. Both of those examples are ill conditioned. Thus, a convenient regularization in each one is considered. Our experimental results show that this new inexact Gauss- Newton approach requires less than 3% of the inner iterations for computing the inexact Gauss-Newton step in order to converge to a statistically stable estimate

Estimation and variational methods for gradient algorithm generation.

Thesis. 1977. M.S.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.Bibliography: leaves 110-113.M.S

Poincaré and the Three Body Problem

The purpose of the thesis is to present an account of Henri Poincare's famous memoir on the three body problem, the final version of which was published in Acta Mathematica in 1890 as the prize-winning entry in King Oscar II's 60th birthday competition. The memoir is reknowned both for its role in providing the foundations for Poincare's celebrated three volume Méthodes Nouvelles de la Mécanique Céleste, and for containing the first mathematical description of chaotic behaviour in a dynamical system. A historical context is provided both through consideration of the problem itself and through a discussion of Poincaré's earlier work which relates to the mathematics developed in the memoir. The organisation of the Oscar competition, which was undertaken by Gösta Mittag-Leffler, is also described. This not only provides an insight into the late 19th century European mathematical community but also reveals that after the prize had been awarded Poincare found an important error in his work and substantially revised the memoir prior to its publication in Acta. The discovery of a printed version of the original memoir personally annotated by Poincaré has allowed for a detailed comparative study of the mathematics contained in both versions of the memoir. The error is explained and it is shown that it was only as a result of its correction that Poincaré discovered the chaotic behaviour now associated with the memoir. The contemporary reception of the memoir is discussed and Poincaré's subsequent work in celestial mechanics and related topics is examined. Through the consideration of sources up to 1920 the influence and impact of the memoir on the progress of the three body problem and on dynamics in general is assessed

On studies in the field of space flight and guidance theory progress report no. 4 <20 dec. 1962 - 18 jul. 1963<

Trajectories, orbital calculations, and adaptive guidanc