TY - JOUR
T1 - An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems
AU - Marco Donatelli
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 43
EP - 61
PY - 2012
DA - 2012/05
SN - 5
DO - http://doi.org/10.4208/nmtma.2011.m12si03
UR - https://global-sci.org/intro/article_detail/nmtma/5927.html
KW - Multigrid methods, Toeplitz matrices, discrete ill-posed problems.
AB - <p style="text-align: justify;">Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichlet
boundary conditions are imposed the deblurring matrix has a Toeplitz
structure and it is potentially full. A crucial task of a multilevel
strategy is to preserve the Toeplitz structure at the coarse levels
which can be exploited to obtain fast computations. The smoother has
to be an iterative regularization method. The grid transfer operator
should preserve the regularization property of the smoother.
This paper improves the iterative multigrid method proposed in
[11] introducing a wavelet soft-thresholding denoising
post-smoother. Such post-smoother avoids the noise amplification
that is the cause of the semi-convergence of iterative
regularization methods and reduces ringing effects. The resulting
iterative multigrid regularization method stabilizes the iterations
so that the imprecise (over) estimate of the stopping iteration does
not have a deleterious effect on the computed solution. Numerical
examples of signal and image deblurring problems confirm the
effectiveness of the proposed method.</p>