Circulant preconditioners for mean curvature-based image deblurring problem

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler–Lagrange equations produces a nonlinear ill-conditioned system which affects the convergence of the numerical algorithms such as Krylov subspace methods (generalized minimal residual etc.) To overcome this difficulty, in this paper, we present three new circulant preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem, which combines a fixed point iteration with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new circulant preconditioners.

On single-step HSS iterative method with circulant preconditioner for fractional diffusion equations

By exploiting Toeplitz-like structure and non-Hermitian dense property of the discrete coefficient matrix, a new double-layer iterative method called SHSS-PCG method is employed to solve the linear systems originating from the implicit finite difference discretization of fractional diffusion equations (FDEs). The method is a combination of the single-step Hermitian and skew-Hermitian splitting (SHSS) method with the preconditioned conjugate gradient (PCG) method. Further, the new circulant preconditioners are proposed to improve the efficiency of SHSS-PCG method, and the computation cost is further reduced via using the fast Fourier transform (FFT). Theoretical analysis shows that the SHSS-PCG iterative method with circulant preconditioners is convergent. Numerical experiments are given to show that our SHSS-PCG method with circulant preconditioners preforms very well, and the proposed circulant preconditioners are very efficient in accelerating the convergence rate.