Lectures on Numerical Analysis by Prof. Per Christian Hansen from DTU Compute, Technical University of Denmark:
Talk 1 "Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems"
Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Professor Per Christian Hansen
DTU Compute, Technical University of Denmark
14:00-15:00 / February 25th (Tuesday), 2020
National Center of Sciences Building, 2F, Medium Conference Room 3
Solution of large-scale inverse problems call for the use of iterative methods.
The regularizing property of the Golub-Kahan bidiagonalization algorithm is powerful when the associated Krylov subspace captures the dominating components of the solution.
In some applications, we can further improve the regularized solution by means of enrichment, that is, by augmenting the Krylov subspace with a low-dimensional subspace that represents specific prior information.
Building on earlier work on Augmented GMRES and Enriched CGNR - by Calvetti & Reichel and ourselves - we demonstrate how to carry these ideas over to the bidiagonalization algorithm, and we describe how to incorporate Tikhonov regularization. This leads to a hybrid iterative method where the choice of the regularization parameter in each iteration also provides a stopping rule for the iterations.
In this talk, I will briefly survey the regularizing properties of the Krylov subspace methods, and then motivate and present our new algorithm. I will discuss some implementation details, and illustrate the algorithm with numerical examples.
This is joint work with Kuniyoshi Abe and Yiqiu Dong.
This lecture is supported by Invitational Fellowships for Research in Japan (Short-term S).
Scientific organizer: Kuniyoshi Abe (abe (at) gifu.shotoku.ac.jp)
Local organizer: Ken Hayami (hayami (at) nii.ac.jp)
Local organizer: Kensuke Aihara (aiharak (at) tcu.ac.jp)