A shifted nested splitting iterative method with applications to ill-posed problems and image restoration

We present a shifted nested iterative method for solving systems of linear equations with a coefficient matrix that contains a dominant skew-Hermitian part. This new scheme is practically the inner/outer iterations, which employs the CGNR method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrix. Convergence properties of the new scheme are studied in depth and possible choices of the shift parameter are discussed. Moreover, an adapted version of the method is used for ill-posed problems and image restoration. At the last, numerical examples are used to further examine the effectiveness and robustness of the new method.