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Improved ball convergence of Newton's method under general conditions

Ioannis K. Argyros, Hongmin Ren (2012)

Applicationes Mathematicae

We present ball convergence results for Newton's method in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our hypotheses involve very general majorants on the Fréchet derivatives of the operators involved. In the special case of convex majorants our results, compared with earlier ones, have at least as large radius of convergence, no less tight error bounds on the distances involved, and no less precise information on the uniqueness of...

Improving the convergence of iterative methods

Jan Zítko (1983)

Aplikace matematiky

The author considers the operator equation x = T x + b . Methods for acceleration of convergence of the iterative process x n + 1 ) = T x n + b are investigated.

Inexact Newton method under weak and center-weak Lipschitz conditions

I. K. Argyros, S. K. Khattri (2013)

Applicationes Mathematicae

The paper develops semilocal convergence of Inexact Newton Method INM for approximating solutions of nonlinear equations in Banach space setting. We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis. The results obtained compare favorably with earlier ones in at least the case of Newton's Method (NM). Numerical examples, where our convergence criteria are satisfied but the earlier ones are not, are also explored.

Inexact Newton methods and recurrent functions

Ioannis K. Argyros, Saïd Hilout (2010)

Applicationes Mathematicae

We provide a semilocal convergence analysis for approximating a solution of an equation in a Banach space setting using an inexact Newton method. By using recurrent functions, we provide under the same or weaker hypotheses: finer error bounds on the distances involved, and an at least as precise information on the location of the solution as in earlier papers. Moreover, if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained. Furthermore, numerical...

Interior proximal method for variational inequalities on non-polyhedral sets

Alexander Kaplan, Rainer Tichatschke (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Interior proximal methods for variational inequalities are, in fact, designed to handle problems on polyhedral convex sets or balls, only. Using a slightly modified concept of Bregman functions, we suggest an interior proximal method for solving variational inequalities (with maximal monotone operators) on convex, in general non-polyhedral sets, including in particular the case in which the set is described by a system of linear as well as strictly convex constraints. The convergence analysis of...

Inverse problems in spaces of measures

Kristian Bredies, Hanna Katriina Pikkarainen (2013)

ESAIM: Control, Optimisation and Calculus of Variations

The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate 𝒪(n-1)...

Inverted finite elements : a new method for solving elliptic problems in unbounded domains

Tahar Zamène Boulmezaoud (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ℝ n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Inverted finite elements: a new method for solving elliptic problems in unbounded domains

Tahar Zamène Boulmezaoud (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ℝ n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

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