Optimized five-by-five block preconditioning for efficient GMRES convergence in curvature-based image deblurring

We introduce an enhanced preconditioning technique designed to expedite the convergence of Krylov subspace methods when dealing with non-linear systems of equations featuring a block five-by-five format. This scenario often arises in the context of cell centered finite difference discretizations applied to the mean curvature based image deblurring problem. A thorough spectral analysis of the preconditioned matrices reveals a favorable eigenvalue distribution, leading to accelerated convergence of preconditioned Generalized Minimal Residual (GMRES) methods. Furthermore, we present numerical experiments to showcase the effectiveness of this preconditioner when combined with the flexible GMRES solver for addressing non-linear systems of equations originating from a image deblurring problems. Codes can be obtained at https://github.com/shahbaz1982/Preconditioning.