A convergence analysis of the Landweber iteration for nonlinear ill-posed problems

Summary. In this paper we prove that the Landweber iteration is a stable method for solving nonlinear ill-posed problems. For perturbed data with noise level we propose a stopping rule that yields the convergence rate ) under appropriate conditions. We illustrate these conditions for a few examples. Received February 15, 1993 / Revised version received August 2, 1994     

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones. Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999     

A New Third-Order Iterative Processfor Solving Nonlinear Equations

 In this paper, we build up a modification of the Midpoint method, reducing its operational cost without losing its cubical convergence. Then we obtain a semilocal convergence result for this new iterative process and by means of several examples we compare it with other iterative processes. (Received 11 April 2000; in final form 27 March 2001)     

A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems

This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations, that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results. Received November 29, 1996 / Revised version received April 25, 1997     

An iterative multi level algorithm for solving nonlinear ill–posed problems

Summary. The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations, when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization. An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The numerical performance of this multi level algorithm is compared with Landweber's iteration. Received October 21, 1996 / Revised version received July 28, 1997     

A unified approach for constructing fast two-step Newton-like methods

We introduce some new very general ways of constructing fast two-step Newton-like methods to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. We provide existence-uniqueness theorems as well as an error analysis for the iterations involved using Newton-Kantorovich-type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function defined on a closed ball centered at a certain point and of a fixed radius and with values into the positive real axis. Special choices of this function lead to favorable comparisons with results already in the literature. Some applications to the solution of nonlinear integral equations appearing in radiative transfer as well as to the solution of integral equations of Uryson-type are also provided.     

Recurrence Relations for Chebyshev-Type Methods

The convergence of new second-order iterative methods are analyzed in Banach spaces by introducing a system of recurrence relations. A system of a priori error bounds for that method is also provided. The methods are defined by using a constant bilinear operator A , instead of the second Fréchet derivative appearing in the defining formula of the Chebyshev method. Numerical evidence that the methods introduced here accelerate the classical Newton iteration for a suitable A is provided. Accepted 23 October 1998     

A Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998     

An 12 constrained best approximation problem

A problem of minimizing a quadratic function over the unit ball of l₂is considered,the motivation being a minimum norm problem for the heat equation controlled by the constrained initial condition. A constructive method for finding an ?-solution is developed and its convergence rate is estimated. The dependence of the solution, on the data is studied.     

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero. Such problems typically arise in connection with discretization of Fredholm integral equations of the first kind, and similar ill-posed problems. Some form of regularization is always required in order to compute a stabilized solution to discrete ill-posed problems. The purpose of REGULARIZATION TOOLS is to provide the user with easy-to-use routines, based on numerical robust and efficient algorithms, for doing experiments with regularization of discrete ill-posed problems. By means of this package, the user can experiment with different regularization strategies, compare them, and draw conclusions from these experiments that would otherwise require a major programming effert. For discrete ill-posed problems, which are indeed difficult to treat numerically, such an approach is certainly superior to a single black-box routine. This paper describes the underlying theory gives an overview of the package; a complete manual is also available.This work was supported by grants from Augustinus Fonden, Knud Højgaards Fond, and Civ. Ing. Frants Allings Legat.     

On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

A predictor-corrector technique for constrained least-squares regularization

Summary The paper describes a numerical strategy for the approximate solution of nonlinear, discretized, inverse problems by regularization. It is assumed that the solution of the associated direct problems and the computation of Fréchet derivatives are expensive. In order to minimize the amount of work, a predictor-corrector type algorithm is proposed. From a series of solutions to problems with a coarse discretization one obtains a starting approximation for a problem with a fine discretization.     

A Modified Alternating Direction Method for Variational Inequality Problems

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method. Accepted 4 May 2001. Online publication 19 October, 2001.     

A convergence analysis and applications for the Newton-Kantorovich method in K-normed spaces

We study the local and semilocal convergence of the Newton-Kantorovich method to a solution of a nonlinear operator equation on aK-normed space setting. Using more precise majorizing sequences than before we show that in the semilocal case finer error bounds can be determined on the distances involved and an at least as precise information on the location of the solution as in earlier results. In the local case we show that a larger radius of convergence can be obtained.     

Identification for a Nonlinear Periodic Wave Equation

This work is concerned with an approximation process for the identification of nonlinearities in the nonlinear periodic wave equation. It is based on the least-squares approach and on a splitting method. A numerical algorithm of gradient type and the numerical implementation are given. Accepted 14 January 2001. Online publication 20 June 2001.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics

We study the Lagrange Problem of Optimal Control with a functional and control-affine dynamics = f(t,x) + g(t,x)u and (a priori) unconstrained control u∈ \bf R ^m . We obtain conditions under which the minimizing controls of the problem are bounded—a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives. Accepted 15 March 1999     

On Fictitious Domain Formulations for Maxwell's Equations

We consider fictitious domain-Lagrange multiplier formulations for variational problems in the space H(curl: Ω{bf)} derived from Maxwell's equations. Boundary conditions and the divergence constraint are imposed weakly by using Lagrange multipliers. Both the time dependent and time harmonic formulations of the Maxwell's equations are considered, and we derive well-posed formulations for both cases. The variational problem that arises can be discretized by functions that do not satisfy an a-priori divergence constraint.