Matrix-free constructions of circulant and block circulant preconditioners

A framework for constructing circulant and block circulant preconditioners (C) for a symmetric linear system Ax=b arising from certain signal and image processing applications is presented in this paper. The proposed scheme does not make explicit use of matrix elements of A. It is ideal for applications in which A only exists in the form of a matrix vector multiplication routine, and in which the process of extracting matrix elements of A is costly. The proposed algorithm takes advantage of the fact that for many linear systems arising from signal or image processing applications, eigenvectors of A can be well represented by a small number of Fourier modes. Therefore, the construction of C can be carried out in the frequency domain by carefully choosing its eigenvalues so that the condition number of C{sup T} AC can be reduced significantly. We illustrate how to construct the spectrum of C in a way such that the smallest eigenvalues of C{sup T} AC overlaps with those of A extremely well while the largest eigenvalues of C{sup T} AC are smaller than those of A by several orders of magnitude. Numerical examples are provided to demonstrate the effectiveness of the preconditioner on accelerating the solution of linear systems arising from image reconstruction application