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

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our dropping rule is to drop the i-th element of k j , i.e., k j,i when the following two
inequalities holds.
|k j,i |‖a i ‖ 2 < τ d and |k j,i | < τ d ′ max
l=1,...,i−1 |k l,i|. (8.1)
Here we have another tolerance τ d ′. In the following examples, we simply fix it to
10.0 ∗ τ d . We use
‖u i ‖ 2 ≤ τ s‖A i−1 ‖ F ‖a i ‖ 2 , (8.2)
as the criterion to judge if column a i is linearly dependent on the previous columns or
not, where τ s is our switching tolerance. The stopping rule for BA-GMRES is
‖A T (b − Ax)‖ 2 ≤ 10 −8 · ‖A T b‖ 2 . (8.3)
In the following tables, we use our preconditioner to precondition GMRES and compare
with GMRES preconditioned by RIF, which is denoted by “RIF-GMRES”, CGLS
[4] preconditioned by Incomplete Modified Gram-Schmidt orthogonalization preconditioner
(IMGS) [15], which is denoted by “IMGS-CGLS”, CGLS preconditioned by using
the diagonal scaling preconditioner, which is denoted by “D-CGLS” and CGLS preconditioned
by the Incomplete Cholesky Factorization preconditioner [4], which is denoted
by “IC-CGLS”, to solve the original least squares problem min
x∈R n‖b−Ax‖ 2. The columns
τ d and τ s list the dropping tolerance and the switching tolerance we used in our preconditioning
algorithm. The column “ITS” gives the number of iterations. The columns
“Pre. T” and “Tol. T” give the preconditioning time and the total time in seconds,
respectively. For “RIF-GMRES”, “IMGS-CGLS”, “D-CGLS” and “D-CGSL”, in each
cell we give the numbers of iterations and the total CPU time. For “RIF-GMRES” and
“IMGS-CGLS”, we give the parameters we used in brackets. In the “ nnzP
nnzA ” column,
we give the ratio of the number of nonzero elements in our preconditioner and A, where
nnzP is the number of nonzero elements in K plus the number of the nonzero elements
in diagonal matrix F . The ∗ indicates the fastest method for each problem.
Table 2 Numerical Results for Full Rank Matrices
matrix τ d
nnzP
nnzA ITS Pre. T Tot. T RIF-GMRES IMGS-CGLS D-CGLS IC-CGLS
vtp base 1.e-6 18.0 4 0.03 *0.03 5, 0.04 (1.e-9) 9, 0.10 (1.e-5) 3667, 0.29 1348, 0.12
capri 1.e-3 10.0 17 0.03 0.04 9, 0.07 (1.e-4) 371, 0.04 (1.0) 496, 0.06 195, *0.03
scfxm1 1.e-3 13.3 17 0.05 0.06 7, *0.05 (1.e-5) 116, 0.08 (1.e-1) 761, 0.11 318, 0.06
scfxm2 1.e-3 16.6 27 0.12 *0.18 12, 0.26 (1.e-5) 734, 0.28 (1.0) 1332, 0.42 680, 0.23
perold 1.e-2 12.4 17 0.17 0.21 45, 0.36 (1.e-4) 134, 0.26 (1.e-1) 1058, 0.36 394, *0.14
scfxm3 1.e-4 27.0 7 0.30 *0.34 12, 0.46 (1.e-5) 878, 0.53 (1.0) 1583, 0.72 847, 0.41
maros 1.e-6 28.8 3 1.02 *1.05 10, 1.13 (1.e-7) 383, 1.47 (1.e-1) 13714, 6.71 3602, 1.8
pilot we 1.e-1 4.5 41 0.10 0.20 12, 0.4 (1.e-5) 536, 0.37 (1.0) 736, 0.38 287, *0.14
stocfor2 1.e-5 152.2 5 10.55 10.68 16, 11.5 (1.e-7) 138, 3.14 (1.e-1) 2147, 1.59 1362, *1.12
bnl2 1.e-1 3.0 114 0.4 1.22 127, 2.79 (1.e-2) 598, 1.24 (1.0) 657, 0.64 240, *0.36
pilot 1.e-2 2.9 18 1.54 1.72 359, 4.57 (1.e-1) 741, 1.49 (1.0) 822, 1.24 269, *0.75
d2q06c 1.e-3 18.7 32 4.27 5.15 19, 10.68 (1.e-5) 1209, 3.6 (1.0) 2743, 4.05 1077, *1.63
80bau3b 5.e-1 0.5 28 0.46 0.52 26, 0.54 (5.e-1) 47,1.34 (1.0) 56, *0.09 21, 0.21
For full rank matrices, we did not list τ s because we simply set τ s to be a very small
number and do not try to detect any linearly dependent columns in the coefficient
matrices. From the numerical results in Table 2, we conclude that our preconditioner
performs competitively when the matrices are ill-conditioned. We can also see that our