Preconditioner

Table 1: The size of the matrices A, B, C and D.

IVS model
l	n	m	size of A	size of B	size of C	size of D
Test 2	1503	14	1503 ×1503	1503 × 14	14× 1503	14× 14
Test 3	127760	380	127760×127760	127760× 380	380× 127760	380× 380

Generic information of the test problems, including n and m, are given in Table 1.

Table 2: Numerical results of Schur complement approximation.

Sˆ
Test 1	CPUShur: 9.76e-03
Test 2	CPUShur: 1.71e+01

In Table 2, we reported the results of Schur complement approximation in terms of CPU times.

Table 3: Numerical results for the three preconditioned GMRES methods.

Test		^Pα,S^ˆ	PT	PD
Test1	Iter	8	23	10
	CPU	1.70e-03	5.65e-03	2.66e-03
	RES	3.02e-14	3.79e-13	1.72e-12
Test2	Iter CPU RES	2 1.36 7.94e-22	† † †	† † †

Table 4: Numerical results for the three preconditioned FGMRES methods.

Test		^Pα,S^ˆ	PT	PD
Test1	Iter	12	30	29
	CPU	2.86e-03	6.79e-03	4.78e-03
	RES	2.20e-24	2.00e-11	2.00e-11
Test2	Iter CPU RES	2 6.12e-01 5.37e-26	† † †	† † †

Table 5: Numerical results for the three preconditioned BICGSTAB methods.

Test		^Pα,S^ˆ	PT	PD
Test1	Iter	15	22	15
	CPU	3.68e-03	5.59e-03	3.40e-03
	RES	3.83e-15	6.75e-15	5.40e-15
Test2	Iter CPU RES	2 5.94e-01 6.27e-28	† † †	† † †

In Tables 3, 4 and 5 we report the results for the preconditioned GMRES, FGMRES and BICGSTAB iterative methods. From numerical results listed in Tables, we can

conclude that the P_α,Sˆ preconditioned GMRES, FGMRES and BICGSTAB methods

P P
require less iterations and has faster CPU times than P_T and P_D in all trials. The P_D and _T preconditioner do not converge in case of Test 2, whereas the _α,Sˆ preconditioner converge in case of both Test 1 and Test 2.

5. Conclusion

In the present work, we have developed and studied numerically parallel block pre- conditioner for a class of linear systems arising from IVS model. Parallel block precon- ditioner have been proposed based on the approximate Schur complement (Block-Schur) and on a regularization technique. Several numerical experiments have been conducted in parallel on a parallel computer architecture in order to study the performance of the iterative solvers in terms of Krylov subspace methods iterations and computational time.

P P
Numerical results worked out in Section 5 (Tables 3, 4 and 5) reveal that the regular- ized parallel block preconditioned Krylov subspace methods with suitable parameter has great superiority compared with _D and _T preconditioned Krylov subspace meth- ods in terms of the iterations and CPU times, and illustrate that the regularized parallel block preconditioned Krylov method is a very efficient method for solving (1).
However, I should mention that this new preconditioner involved the parameter α. How to choose the optimal parameters for the regularized preconditioner is a very practical and interesting problem that needs to be further in-depth studied.

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