Inverse Toeplitz Preconditioners for Ill-Posed Problems
M. Hanke and J. Nagy
It has been shown
recently that iterative regularization using
conjugate gradient type methods for image restoration problems
can be effectively preconditioned with circulant approximations.
Here it is shown that the theoretical properties of this
approach are not restricted to circulant matrices.
a Toeplitz approximate inverse preconditioning scheme for
discrete ill-posed problems is considered.
It is proved that the preconditioned system approximates the
prolate matrix, and that this property implies
that fast convergence of conjugate gradient type methods
can be expected. In addition, it is shown that
these results can be generalized to two-dimensional problems.
An image restoration application is used to demonstrate the
properties of the preconditioner.