Iterative Refinement of A Modified Lavrentiev Regularization Method for De-convolution of the Discrete Helmholtz Type Differential Filter

TL;DR

This paper introduces an iterative refinement of a modified Lavrentiev regularization method tailored for deconvolving the Helmholtz-type differential filter, improving accuracy and establishing optimal stopping criteria.

Contribution

The paper develops a novel iterative regularization technique exploiting Helmholtz filter properties, reducing error bounds and enhancing deconvolution performance over existing Tikhonov methods.

Findings

Error bound between original and approximate solutions is reduced.

Optimal stopping condition for iterations is derived.

Numerical examples show improved results over Tikhonov regularization.

Abstract

We propose and analyze an iterative refinement of a modified Lavrentiev regularization method for deconvolution of the discrete Helmholtz-type differential filter. The modification for the Lavrentiev regularization method exploits the properties of the Helmholtz filter, and we prove that the modification reduces the error bound between the original solution and the approximated solution. Furthermore, we derive an optimal stopping condition on the number of iterations necessary for the regularization. We provide numerical examples demonstrating the benefits of this iterative modified Lavrentiev regularization over a family of Tikhonov regularization methods.