Generalization of Strang's preconditioner with applications to iterative deconvolution

Abstract

In this paper, we proposed a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The [n/2]th column of our circulant preconditioner Sn is equal to the [n/2]th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as (S*nSn) 1/2 . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. Comparisons of our preconditioner Sn with other circulant-based preconditioners are carried out for some 1D Toeplitz least squares problems: min||b - Ax||2. Preliminary numerical results show that S n performs quite well. Test results are also reported for a 2D deconvolution problem arising in ground-based atmospheric imaging.