Greville's Method for Preconditioning Least Squares ... - Projects

22
RIF preconditioned GMRES did not converge in 2000 steps for all the dropping tolerances
τ d we tried. The column “deficiency detected” gives the linearly dependent
columns detected by our algorithm. −1260 means that the 1260th column, which is
a rank deficient column, is missed by our algorithm. For example, when τ d = 10 −6 ,
τ s = 10 −10 our preconditioning algorithm detected 12 rank deficient columns, and
missed the 1239-th, 1261-th, and 1278-th columns, and did not detect wrong linearly
dependent columns. Hence, Assumption 1 is satisfied. When τ d = 10 −6 , τ s = 10 −7
our preconditioning algorithm found exactly the 15 linearly dependent columns. When
τ d = 10 −6 , τ s = 10 −5 many more than 15 columns were detected. Thus, assumption
1 is not satisfied. Hence, the preconditioned GMRES did not converge. From this example
we can see that the numerical results coincide with our theory very well. From
Figure 1, we obtain a better insight into this situation. We observe that when the
0
Residual Curve of lp_cycle, τ d
= 10 −6
−2
log 10
||A T r|| 2
/||A T b|| 2
−4
−6
−8
−10
τ s
=10 −5
τ s
=10 −7
τ s
=10 −10
−12
−14
0 50 100 150 200
Iteration Number
Fig. 1 Convergence Curve for lp cycle
switching tolerance τ s = 10 −5 , ‖AT r‖ 2
‖A T maintains the level 10 −1 in the end, which
b‖ 2
means that the solution is not a solution to the original least squares problem. This
phenomenon illustrates that Assumption 1 is necessary.
9 Conclusion
In this paper, we proposed a new preconditioner for least squares problems. When
matrix A is full column rank, our preconditioning method is similar to the RIF preconditioner
[3]. When A is rank deficient, our preconditioner is also rank deficient.
We proved that under Assumption 1, using our preconditioners, the preconditioned