|
|
|
Professor:
Prof. Martin H. Gutknecht
Date: 20.10.2007
Project Title:
An Efficient and Reliable Block GMRES Algorithm
Summary:
In the thesis of Vital (1990), the generalized minimal residual method
(GMRES) of Saad and Schultz (1986) was extended to a block version
for linear systems with multiple right-hand sides.
But while the formal extension of CG or GMRES to block CG (BlCG)
and block GMRES (BlGMRES) is quite easy, safeguards are required for
a robust implementation.
The problem is that the vectors in the blocks associated to the
multiple right-hand sides may become linear dependent, in which case
one should reduce the block size.
This is often referred to as (implicit) deflation.
While the reduction of block size (deflation) in the block Lanczos
and block conjugate gradient algorithms has been discussed in detail
in the literature, this is only partially true for BlGMRES, despite
the fact that deflation may strongly reduce the number of matrix-vector
products needed. Moreover, the necessity to restart gives us the
option to choose a more generous deflation tolerance than in other
block methods or to restrict deflation to restarts.
But there is also a risk that goes with such strategies.
The aim of the project is to find the right compromise.