HyBR

Hybrid Bidiagonalization Regularization

Background

Lanczos-hybrid regularization methods have been proposed as effective approaches for solving large-scale ill-posed inverse problems. Lanczos methods restrict the solution to lie in a Krylov subspace, but they are hindered by semi-convergence behavior, in that the quality of the solution first increases and then decreases. Hybrid methods apply a standard regularization technique, such as Tikhonov regularization, to the projected problem at each iteration. Thus, regularization in hybrid methods is achieved both by Krylov filtering and by appropriate choice of a regularization parameter at each iteration.

HyBR Implementation

Here we provide a MATLAB implementation of the Lanczos-hybrid regularization, which we refer to as HyBR.

The software for this may be obtained from:
- Compressed unix tar file.
- A zip file.
Included with the above archives are examples:
- Using HyBR with problems from Per Christian Hansen's Regularization Tools
- Using HyBR with image deblurring problems from RestoreTools
We are grateful to Professor Per Christian Hansen for allowing us to include Regularization Tools with our codes.

In our implementation, we use a "weighted GCV" scheme to choose regularization parameters. A discussion of our approach is given in:

J. Chung , J. Nagy , and D. O'Leary ,
A Weighted GCV Method for Lanczos Hybrid Regularization,
Emory Unviersity, Math/CS Technical Report TR-2007-004-A.
(pdf version of paper)

Acknowledgements

This work has been supported by the National Science Foundation under Grants DMS 05-11454 and CCF 05-14214, and by the DOE under a Computational Sciences Graduate Research Fellowship. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or the Department of Energy.