Approximate Inverse Circulant-plus-Diagonal Preconditioners for Toeplitz-plus-Diagonal Matrices
SIAM Journal on Scientific Computing
Fast algorithms for l1 norm/mixed l1 and l2 norms for image restoration
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and Its Applications - Volume Part IV
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In this paper, we study factorized banded inverse preconditioners for matrices with Toeplitz structure. We show that if a Toeplitz matrix T has certain off-diagonal decay property, then the factorized banded inverse preconditioner approximates T-1 accurately, and the spectra of these preconditioned matrices are clustered around 1. In nonlinear image restoration applications, Toeplitz-related systems of the form I + T* D T are required to solve, where D is a positive nonconstant diagonal matrix. We construct inverse preconditioners for such matrices. Numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners. A two-dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner.