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

Abstract

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.