Simplified Levenberg–Marquardt method in Banach spaces for nonlinear ill-posed operator equations

Modified Noor iterations for nonlinear equations in Banach spaces

Recurrence relations for a Newton-like method in Banach spaces

Recently, Mahale and Nair considered a simplified generalized Gauss-Newton iterative method for getting an approximate solution for the nonlinear ill-posed operator equation under the modified general source condition. The advantage of this method and the source condition over the classical Gauss-Newton iterative method is that the iterations and source condition involve calculation of the Fréchet derivative only at the point x 0, i.e., at the initial approximation for the exact solution x † of the nonlinear ill-posed operator equation F(x) = y. Motivated by the work of Qinian Jin and Tautenhan, error analysis of the simplified Gauss-Newton iterative method is done in this article under a Morozove-type stopping rule, which is much simpler than the stopping rule considered in the article of Mahale and Nair. An order optimal error estimate is obtained under a modified general source condition which also involves calculation of the Fréchet derivative at the point x 0.

We study an iterative method with order 1 + $\sqrt 2$ for solving nonlinear operator equations in Banach spaces. We present the received new results of local convergence, when the first‐order divided differences of a nonlinear operator are Hölder continuous. Method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two theorems on an iteration method of order three -- usually referred to as Chebyshev's method -- for the approximate solution of nonlinear operator equations in Banach spaces are given. Very simple and weak assumptions are obtained as well as error estimates which are much better compared with known results. Some worked examples of operator equations of different types are given.

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

This paper concerns the dissipativity of nonlinear Volterra functional differential equations (VFDEs) in Banach space and their numerical discretization. We derive sufficient conditions for the dissipativity of nonlinear VFDEs. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type appearing in practice. Then the dissipativity property of the backward Euler method for VFDEs is investigated. It is shown that the method can inherit the dissipativity of the underlying system. The close relationship between the absorbing set of the numerically discrete system generated by the backward Euler method and that of the underlying system is revealed.

Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces

Preface. 1. Introduction and Preliminaries. 2. Existence Theory for Nonlinear Fredholm and Volterra Integrodifferential Equations. 3. Solution Sets of Abstract Volterra Equations. 4. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Compact Intervals. 5. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Half-Open Intervals. 6. Existence Theory for Nonlinear Nonresonant Operator and Integral Equations. 7. Existence Theory for Nonlinear Resonant Operator and Integral Equations. 8. Integral Inclusions. 9. Approximation of Solutions of Operator Equations on the Half Line. 10. Operator Equations in Banach Spaces Relative to the Weak Topology. 11. Stochastic Integral Equations. 12. Periodic Solutions for Operator Equations. Index.

A projective Newton method is presented for the numerical solution of semilinear operator equations in Banach spaces and its convergence properties are studied. Applications to boundary value problems for nonlinear ordinary and partial differential equations are discussed

General local convergence theory for a class of iterative processes and its applications to Newton’s method

: The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

On the local convergence of a deformed Newton's method under Argyros-type condition

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.