6 research outputs found
Mixed spatially varying -BV regularization of inverse ill-posed problems
Several generalizations of the traditional Tikhonov-Phillips regularization
method have been proposed during the last two decades. Many of these
generalizations are based upon inducing stability throughout the use of
different penalizers which allow the capturing of diverse properties of the
exact solution (e.g. edges, discontinuities, borders, etc.). However, in some
problems in which it is known that the regularity of the exact solution is
heterogeneous and/or anisotropic, it is reasonable to think that a much better
option could be the simultaneous use of two or more penalizers of different
nature. Such is the case, for instance, in some image restoration problems in
which preservation of edges, borders or discontinuities is an important matter.
In this work we present some results on the simultaneous use of penalizers of
and of bounded variation (BV) type. For particular cases, existence and
uniqueness results are proved. Open problems are discussed and results to
signal restoration problems are presented.Comment: 18 pages, 12 figure
On the existence of global saturation for spectral regularization methods with optimal qualification
A family of real functions {g_\alpha} defining a spectral regularization
method with optimal qualification is considered. Sufficient condition on the
family and on the optimal qualification guaranteeing the existence of
saturation are established. Appropriate characterizations of both the
saturation function and the saturation set are found and some examples are
provided.Comment: 19 page
Existence, uniqueness and stability of solutions of generalized Tikhonov-Phillips functionals
The Tikhonov-Phillips method is widely used for regularizing ill-posed
inverse problems mainly due to the simplicity of its formulation as an
optimization problem. The use of different penalizers in the functionals
associated to the corresponding optimization problems has originated a variety
other methods which can be considered as "variants" of the traditional
Tikhonov-Phillips method of order zero. Such is the case for instance of the
Tikhonov-Phillips method of order one, the total variation regularization
method, etc. In this article we find sufficient conditions on the penalizers in
generalized Tikhonov-Phillips functionals which guarantee existence and
uniqueness and stability of the minimizers. The particular cases in which the
penalizers are given by the bounded variation norm, by powers of seminorms and
by linear combinations of powers of seminorms associated to closed operators,
are studied. Several examples are presented and a few results on image
restoration are shown.Comment: 24 pages, 8 figure
Global Saturation of Regularization Methods for Inverse Ill-Posed Problems
In this article the concept of saturation of an arbitrary regularization
method is formalized based upon the original idea of saturation for spectral
regularization methods introduced by A. Neubauer in 1994. Necessary and
sufficient conditions for a regularization method to have global saturation are
provided. It is shown that for a method to have global saturation the total
error must be optimal in two senses, namely as optimal order of convergence
over a certain set which at the same time, must be optimal (in a very precise
sense) with respect to the error. Finally, two converse results are proved and
the theory is applied to find sufficient conditions which ensure the existence
of global saturation for spectral methods with classical qualification of
finite positive order and for methods with maximal qualification. Finally,
several examples of regularization methods possessing global saturation are
shown.Comment: 29 page