Conjugate Gradient Method for Indefinite
Matrices
Conjugate Gradient 1) CG is a numerical method to solve a linear system of equations bAx
2) CG is used when A is Symmetric and Positive definite matrix (SPD)
3) CG of Hestenes and Stiefel [1] is an effective popular method for solving large, sparse symmetric positive definite (SPD).
[1] M. R. Hestenes and E. Stiefel. Methods of conjugate gradient for solving linear systems. Journal of Research of the Natural Bureau of Standards, 49:409-435, 1952
Conjugate Gradient
, Standard inner product defined by: yxyx T ,
Preconditioning
bAx bPAxP 11 P
Preconditioner
Non-singular
bxA ~~
1)~( A
Non-standard Inner ProductStandard inner product defined by: yxyx T , ,
Defined by:For any real symmetric ,HH ,
Hyxyx TH ,
Definition
Is an inner product
The symmetric bilinear form
HH ,
Pos. def.
Self-AdjointSelf-adjoint inA ,
Defin
ition
AyxyAx ,,
symmetric is ,, AAAAyxyAxAyxyAx TTTT
Self-adjoint inA H , HH AyxyAx ,,
HAyxHyAxAyxyAx TTTHH ,, HAHAT
H-symmetricA HAHAT
Bramble-Pasciak CGCG for Indefinite
2
1
2
1
0 bb
xx
BBA T
IB
AP
00
IAA
H0
00
Computational Fluid Dynamics
Optimizations
Saddle Point Problem
Preconditioner
0~ 1
BBA
PAT
SymmetricIndefinite
Non-symmetricPositive definite
Is H-symmetric and positive definite
A~
Bramble-Pasciak CGCG for Indefinite
2
1
2
1
0 bb
xx
BBA T
IB
AP
00
IAA
H0
00
Preconditioner Inner Product
USE
A~ SPD in < , >H
H
H
H
H
Iterative Krylov Subspace Methods
bAx SPDA
CG
SymmA
MINRES
Non-SymA
GMRES
Bramble-Pasciak CGCG for Indefinite
2
1
2
1
0 bb
xx
BBA T
IB
AP
00
IAA
H0
00
Preconditioner Inner Product
USE
A~ SPD in < , >H
H
H
H
H
Bramble-Pasciak CG
IB
AP
00
IAA
H0
00
Bramble-Pasciak CG
IB
AP
00
IAA
H0
00
2008
Bramble-Pasciak CG
IB
AP
00
IAA
H0
00
11 , HP
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