Preconditioning Technique for the Total Fractional-Order Variation for an Image Deblurring Problem

This proposal is for 11-months project submitted under the "junior faculty Grant". It is an internal grant intended to meet the needs of faculty who have recently joined KFUPM or recently received their Ph.D.s. The proposed research aims to provide an efficient iterative solver for a large linear system of equations. This system is resulted from discretizing the Euler Lagrange equations associated with the image deblurring problem when using the total fractional-order variation model. The coefficient matrix of this system is a huge, dense and it has a large condition number. Hence, solving this system is very difficult. In this project, we propose block tridiagonal preconditioners for this system. The importance of these preconditioners is to accelerate the convergence of the iterative methods such as Krylov subspace methods. Moreover, using exact preconditioners lead to preconditioned matrices with exactly two distinct eigenvalues. This means that we need at most two MINRES iterations to converge to the exact solution. We study the bounds of the eigenvalues of the preconditioned matrix. The efficiency of our preconditioners will be investigated using some digital images