We consider the use of multigrid methods for solving certain differential-convolution equations which arise in regularized image deconvolution problems. We first point out that the usual smoothing procedures (e.g. relaxation smoothers) do nto work well for these types of problems because the high frequency error components are nost smoothed out. To overcome this problem, we propose to use optimal fast-transform preconditioned conjugate gradient smoothers. The motivation is to combine the advantages of multigrid (mesh independence) and fast transform based methods (clustering of eigenvalues for the convolution operator). Numerical results for Tikhonov regularization with the identity and the Laplacian operators show that the resulting method is effective. However, preliminary results for total variation regularization show that this case is much more difficult and further analysis is required.
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