Multilevel Methods for Ill-Posed Problems

Lothar Reichel (Kent University)

Multilevel methods are popular for the solution of well-posed problems, such
as certain boundary value problems for partial differential equations and
Fredholm integral equations of the second kind. However, little is known
about the behavior of multilevel methods when applied to the solution of
linear ill-posed problems, such as Fredholm integral equations of the first
kind, with a right-hand side that is contaminated by error. We discuss
properties of cascadic multilevel methods with a conjugate gradient-type
method as basic iterative scheme. In particular, we consider applications
to image deblurring, denoising, and segmentation. The multilevel methods
blend
linear algebra and partial differential equation techniques. Deblurring and
denoising are achieved by using edge-preserving prolongation operators
based on partial differential equations defined by Perona-Malik models.
The talk presents joint work with Serena Morigi, Fiorella Sgallari, and
Andriy Shyshkov.