Lecture on Numerical Analysis by Prof. Zhong-Zhi Bai from Chinese Academy of Sciences:
"Recent Advances on the Randomized Kaczmarz Method"
Recent Advances on the Randomized Kaczmarz Method
Professor Zhong-Zhi Bai
State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing, P.R. China
11:00-12:00 / September 17th (Tue.), 2019
Room 1212, 12th floor, NII
For solving large scale system of linear equations by iteration methods, we introduce an effective probability criterion for selecting the working rows from the coefficient
matrix and construct a greedy randomized Kaczmarz method.
It is proved that this method converges to the unique least-norm solution of the linear system when it is consistent. Theoretical analysis demonstrates that the convergence rate of the greedy randomized Kaczmarz method is much faster than the randomized Kaczmarz method, and numerical results show that the greedy randomized Kaczmarz method is more efficient than the randomized Kaczmarz method, too.
In addition, by introducing a relaxation parameter in the involved probability criterion, we further generalize the greedy randomized Kaczmarz method, obtaining a class of relaxed greedy randomized Kaczmarz methods.
Both theoretical validation and numerical verification show that these methods can be more efficient than the greedy randomized Kaczmarz method if the relaxation parameter is chosen appropriately.
Ken Hayami ( hayami(at)nii.ac.jp )>