Conjugate Gradient Method for Indefinite Matrices

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Conjugate Gradient Method for Indefinite Matrices. Conjugate Gradient . 1) CG is a numerical method to solve a linear system of equations . 2) CG is used when A is Symmetric and Positive definite matrix (SPD). - PowerPoint PPT Presentation

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Slide 1

Conjugate Gradient Method for Indefinite Matrices1

Conjugate Gradient 1) CG is a numerical method to solve a linear system of equations

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 2

Conjugate Gradient

Standard inner product defined by:

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Preconditioning

PreconditionerNon-singular

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Non-standard Inner ProductStandard inner product defined by:

Defined by:For any real symmetric

DefinitionIs an inner productThe symmetric bilinear form

Pos. def.5

Self-AdjointSelf-adjoint in

Definition

Self-adjoint in

H-symmetric

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Bramble-Pasciak CGCG for Indefinite

Computational Fluid DynamicsOptimizationsSaddle Point ProblemPreconditioner

SymmetricIndefiniteNon-symmetricPositive definite

Is H-symmetric and positive definite

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Bramble-Pasciak CGCG for Indefinite

PreconditionerInner ProductUSE

SPD in < , >H

HHHH

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Iterative Krylov Subspace Methods

SPD

CGSymm

MINRESNon-Sym

GMRES9

Bramble-Pasciak CGCG for Indefinite

Preconditioner

Inner ProductUSE

SPD in < , >H

HHHH10

Bramble-Pasciak CG

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Bramble-Pasciak CG

200812

Bramble-Pasciak CG

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