Most preconditioners for Toeplitz systems A(n)(f) arising in the discretization of ill-posed problems give rise to instability and noise amplification. Indeed, since these preconditioners are constructed from linear approximation processes of the generating function f, they inherit the ill-posedness of the problem.Here we first identify a novel set of approximation processes which regularizes the inversion of real functions. Then, such processes are used as a basic tool for the computation of preconditioners endowed with regularizing properties. We show that these preconditioners provide fast convergence and noise control of iterative methods for discrete ill-posed Toeplitz systems.
Regularization processes for real functions and ill-posed Toeplitz problems
Estatico, C
2005-01-01
Abstract
Most preconditioners for Toeplitz systems A(n)(f) arising in the discretization of ill-posed problems give rise to instability and noise amplification. Indeed, since these preconditioners are constructed from linear approximation processes of the generating function f, they inherit the ill-posedness of the problem.Here we first identify a novel set of approximation processes which regularizes the inversion of real functions. Then, such processes are used as a basic tool for the computation of preconditioners endowed with regularizing properties. We show that these preconditioners provide fast convergence and noise control of iterative methods for discrete ill-posed Toeplitz systems.
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