Lavrentiev Regularization Method Research Articles

Multilevel augmentation methods for nonlinear ill-posed problems

We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.

Optimal distributed control of the wave equation subject to state constraints

AbstractThe Lavrentiev regularization method is a tool to improve the regularity of the Lagrange multipliers in pde constrained optimal control problems with state constraints. It has already been used for problems with parabolic and elliptic systems. In this paper we consider Lavrentiev regularization for problems with a hyperbolic system, namely the scalar wave equation. We show that also in this case the regularization yields multipliers in the Hilbert space L2. We present numerical examples, where we compare the Lavrentiev regularization, Lavrentiev Prox regularization, a fixed point iteration to improve feasibility, and a penalty method.

The Lavrentiev regularization of the data completion problem

The Lavrentiev regularization method is naturally fitted to the data completion problem, put under the variational form proposed by Ben Belgacem and El Fekih (2005 Inverse Problems 21 1915). We address the important issue of selecting the regularizing parameter. We study an a priori choice and the a posteriori choice by means of the discrepancy principle written on the Kohn–Vogelius function. In both cases, we prove error estimates similar to those expected for Tikhonov's method.

A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton–Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).

Inverse electroencephalography for cortical sources

In this work the problem of recovering some characteristic of the bioelectric activity in the brain cortex is considered. Measurements of the electric potential on the scalp (EEG) generated during this activity are used. This is called an inverse electroencephalographic problem (IEP). Its solution has two great difficulties. First, its possible ill-posedness. Second, the different existing interpretations about what a characteristic of the cortical bioelectric activity could be, namely: the potential mapping, the charge density, the dipole density, or the distribution density of primary currents generated by the cortical cells activity. It happens that depending on the mathematical model employed this identification problem could either be or not be an ill-posed problem. The present work studies a simplified model whose foundation lays in the biological approach to the problem. This model, which describes the brain as a conductor medium, allows relationships between the characteristics of the bioelectrical cortical activity and the EEG to be established. Using this model, the correctness of the IEP is proved when the EEG is measured over the whole skull and the recovering of the potential mapping or the current density primary on the cortical brain is searched for. This is a remarkable result, because under actual conditions the only cause of present instability in the recovering of the cortical source is the finite number of EEG measurements. However, the identification of the cortical charge density or dipole density from EEG measurements could be an ill-posed problem, which is shown through a synthetic example. The effect due to the discretization and errors in the measurements are studied in the same synthetic example. To recover the charge densities in an stable way the spline approximation method along with the Lavrentiev regularization method are used.

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.

Lavrentiev regularization of a singularly perturbed elliptic PDE

This paper considers a reaction–diffusion equation with a small parameter multiplied with the derivatives. Considering this as an ill-posed problem, Lavrentiev regularization method is used to formulate the related well-posed problem. Pointwise a priori estimates of the regularized solutions and of the regularization error are also derived. The regularized equations are solved numerically via the central difference scheme. It is suggested that any other suitable discretization might also be used to solve the regularized problems. Also it is found that with a comparatively small number of grid points, the numerical solution of the regularized equation comes up as a good approximation to the original solution.