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

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