2018 2nd International Conference on Applied Mathematics, Modeling and Simulation (AMMS 2018) ISBN: 978-1-60595-580-3
On the Shift-HSS Splitting Method for Nonsingular Saddle Point Problem
Zhuo-hong HUANG
*, Hong SU and Jing YANG
School of Science, Chongqing University of Technology Chongqing, 400054, Chongqing, P. R. China
*Corresponding author
Keywords: Shift-HSS splitting, Nnonsingular, Nonsymmetric positive definite, Saddle point problems, Krylov subspace methods.
Abstract. In this paper, the shift-HSS splitting (denoted by SFHSS) iteration method is used to solve nonsingular saddle point system with nonsymmetric positive definite (1, 1)-block. Theoretical analysis illustrates that the SFHSS iteration method converges to the unique solution of the saddle point system as the parameter satisfies certain condition.
Introduction
Consider the following saddle-point problem:
Q = = ,
0 T
A B u f
x
B v g
(1)
where QRn n is nonsymmetric positive definite i.e. the Hermitian part 1(Q Q ) 2
H
is symmetric
positive definite, BRn m (nm) has full column rank, and f Rn and gRm are given vectors. Here and in the sequel, we use ( ) T to denote the transpose, ( ) the range space and ( ) the null space of the corresponding matrix.
In recent year, a large amount of contributions have been put into developing efficient iteration algorithms and preconditioning techniques for solving the saddle point system (1) with different structures, such as HSS-type methods [1, 2, 3], block preconditioning methods [4], Uzawa-type methods [5], SOR-like method [6], constraint preconditioning methods [7], block and approximate Schur complement preconditioners [8] and (generalized) shift-splitting iteration methods [9-12], and so on.
On the Shift-HSS Splitting Iteration Method
In this section, we present the SFHSS preconditioner PSFHSS
1
( )( )
1 =
2
n n
SFHSS
T
m
I H I S B
P
B I
(2)
with H=1 A+AT 2( ) and
T
1 S= A-A
2( ) , and the SFHSS iteration matrix ( ) can be constructed as follows:
1
1 1
( )( ) ( )( )
( ) = n n n n .
T T
I H I S B I H I S B
B I B I
Let be an arbitrary eigenvalue of the iteration matrix ( ) defined as in (3) and x= ( ,u v* * *) be the corresponding eigenvector with uRn and vRm. We consider the following generalized eigenvalue problem:
1 1
( )( ) ( )( )
= .
n n n n
T T
m m
I H I S B I H I S B
x x
B I B I
(4)
After some algebra, we have
1 1
( )( ) = ( )( ) ,
= .
n n n n
T T
I H I S u Bv I H I S u Bv
B u v B u v
(5)
Lemma 3.1 ([13]). If S is a skew-Hermitian matrix, then iS is a Hermitian matrix and u Su* is a purely imaginary number or zero for all u Cn.
Lemma 3.2 ([14]). Both roots of the complex quadratic equation 2 = 0 have modulus less than one if and only if 2 <1， where denotes the conjugate complex of .
Lemma 3.3. Let ( ) be defined as in (3) with > 0 and be an any eigenvalue of ( ). Assume that B has full row rank, then 1.
Proof. Following the spirit of the proof of [12]. If = 1, then, it is straightforward to show that the equations (4) yield the following result
0
= .
0 0
T
A B u
B v
Since A is nonsymmetric positive definite and B has full row rank, then we can easily know that = 0
u and v 0. This is a contradiction as * * *
( ,u v ) is an eigenvector. Thus, we complete the proof.
Lemma 3.4. Let the conditions of Lemma 3.2 be satisfied. Assume that is an eigenvalue of ( )
as defined in (3) with > 0 and ( ,u v* * *) is the corresponding eigenvector with uRn and ,
m
v R if 0 u (BT), then ||< 1.
Proof. We demonstrate the verification of u 0. Unless, if u= 0, then it follows from the second of the equations (5) that ( 1) = 0.v According to Lemma 3.3, since 1, then
= 0.
Bv As B has full row rank, then, we further conclude v = 0. This is a contradiction since * * *
( ,u v ) is an eigenvector, so u 0.
We now turn to verify ||< 1. Assume 0 u (BT) and || || = 1u 2 , following the second of the equations (5), it is easy to see that v = 0. Following [15, Theorem 2.2] and multiplying the first of the equations (5) from the left-hand side by u*, it is obvious that
* 1 1
1 1
2
1 1
2 2
1 2
| | =| ( 2 ) ( 2 ) ( 2 )( 2 ) |
|| ( 2 ) ( 2 ) ( 2 )( 2 ) ||
|| ( 2 ) ( 2 ) || || ( 2 ) ( 2 ) ||
|| ( 2 )( 2 ) ||
2 ( )
= max
2 ( )
< 1,
n n n n
n n n n
n n n n
n n
i
i i
u I S I H I H I S u
I S I H I H I S
I H I H I S I S
I H I H
H H
Where (i H) denotes the ith eigenvalue of H. Therefore, the proof is completed.
Theorem 3.1. Let the conditions of Lemma 3.2 be satisfied. Assume that is an eigenvalue of the iteration matrix ( ) as defined in (3) and denote
* * *
* = , * = * = ,
T
u Au u HSu u BB u
a bi c di and e
u u u u u u (6) where , > 0a e and
* *
* *
1 ( ) 1 ( )
= = .
2 2
u HS SH u u HS SH u
c and d
u u i u u
If the positive iteration parameter satisfies the following condition
2 2
( ) ( ) 4
> ,
2
ac bd ac bd ed
a
(7)
then the SFHSS iteration method converges to the unique solution of the nonsymmetric saddle point problem (1), i.e.,
||< 1.
Proof. Combine Lemma 3.3 and Lemma 3.3, in order to complete this proof, we only need to verify the case B uT 0. Suppose u (BT), then the second of the equations (5) yields the following result
1
= .
( 1) T
v B u
(8) By substituting this relationship (8) into the first of the equations (5), then
2 ( 1)
( )( ) = ( )( ) .
1 T
n n n n
I H I S u I H I S u BB u
(9) Multiplying the equation (9) from the left-hand side by u*, after straightforward calculations, then the equation (8) yields the following form
* * *
2 2 2 2 2
* * *
( 1) ( 1) ( 1) ( 1) = 0.
T
u Au u HSu u BB u
u u u u u u
(10)
Following (6), a quadratic equation of is derived from the equation (10) , after some algebra, it means that
2 2 2 2
[ a c e (bd i) ] 2(e c di) a c e (b d i ) = 0. (11)
If 2a c e (b d i ) = 0, then, it easy to see that 2a c e = 0 and b d = 0.
Therefore, the equation (11) yields the following results: 2
2
( )
=
2( )
= .
2
a c e b d i
e c di
a bi e a bi
Note that , ,a e 0, hence, it is easy to see that
2 2
2 2
( ) ( )
= < 1.
(2 ) ( )
a b
e a b
In what follows, we consider this case 2 a c e (b d i) 0. From lemma 3.2, we
know that ||< 1 if and only if + 2 1. For convenience, we denote and by
2 2
2 2
2( ) ( )
= = .
( ) ( )
e c di a c e b d i
and
a c e b d i a c e b d i
After straightforward computation, we have
2 2 2 2
2
2 2 2
4 (2 ) ( ) ( )
| | | | =
( ) ( )
ed a c e b d
a c e b d
where =2(ae2aac bd ) .2 Let us assume that
2 2 2
4ae ( aac bd ) > (2ed) . (12) By straightforward computation, we have following inequality
2
2 2 2 2 2
2 2 2
2 2 2 2
2 2 2
2 2 2 2
2 2 2
| | | |
4 4 ( ) ( ) ( )
<
( ) ( )
4 ( ) ( ) ( )
=
( ) ( )
[4 ( ) ( ) ] [4 ( ) ]
=
( ) ( )
=1.
ae a ac bd a c e b d
a c e b d
ae a ac bd a c e b d
a c e b d
a e c a c e bd b d
a c e b d
This implies (7).
Combine the above analysis, we complete the proof.
Conclusions
The novelty of this present paper is the research of the shift-HSS iteration method for nonsingular saddle point systems with nonsymmetric positive definite (1,1)-block. We study the convergence property of the SFHSS iteration method and further present the accuracy and feasibility of the shift-HSS preconditioner. Future work should focus on developing the modified forms of the GSS iteration method and study the effects of iteration parameters on eigenvalue-clustering of the corresponding preconditioned matrices.
Acknowledgement
This research is supported by Chongqing Research Program of Basic Research and Frontier Technology (cstc2015jcyjA00049, cstc2018jcyjAX0685).
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