LAVRENTIEV'S REGULARIZATION METHOD FOR NONLINEAR ILL-POSED EQUATIONS IN BANACH SPACES

In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the imple- mentation of Lavrentiev regularization method. Using general HSlder type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock （2005） for choosing the regularization parameter.

Iterative Lavrentiev regularization for symmetric kernel-driven operator equations: with application to digital image restoration problems

The symmetric kernel-driven operator equations play an important role in mathematical physics, engineering, atmospheric image processing and remote sensing sciences. Such problems are usually ill-posed in the sense that even if a unique solution exists, the solution need not depend continuously on the input data. One common technique to overcome the difficulty is applying the Tikhonov regularization to the symmetric kernel operator equations, which is more generally called the Lavrentiev regularization. It has been shown that the iterative implementation of the Tikhonov regularization can improve the rate of convergence. Therefore in this paper, we study the iterative Lavrentiev regularization method in a similar way when applying it to symmetric kernel problems which appears frequently in applications, say digital image restoration problems. We first prove the convergence property, and then under the widely used Morozov discrepancy principle(MDP), we prove the regularity of the method. Numerical performance for digital image restoration is included to confirm the theory. It seems that the iterated Lavrentiev regularization with the MDP strategy is appropriate for solving symmetric kernel problems.

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent,robust,and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented.We first consider a Lavrentiev regularization of the state-constrained optimization problem.Then,a multigrid scheme is designed for the numerical solution of the regularized optimality system.Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration.Results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

Multigrid Methods for Elliptic Optimal Control Problems with Pointwise State Constraints

An elliptic optimal control problem with constraints on the state variable is considered.The Lavrentiev-type regularization is used to treat the constraints on the state variable.To solve the problem numerically,the multigrid for optimization(MGOPT)technique and the collective smoothing multigrid(CSMG)are implemented.Numerical results are reported to illustrate and compare the efficiency of both multigrid strategies.