1 / 13

Conjugate Gradient Method for Indefinite Matrices

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).

selma
Download Presentation

Conjugate Gradient Method for Indefinite Matrices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Conjugate Gradient Method for Indefinite Matrices

  2. 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

  3. Conjugate Gradient Standard inner product defined by:

  4. Preconditioning Non-singular Preconditioner

  5. Non-standard Inner Product Standard inner product defined by: Definition For any real symmetric Defined by: The symmetric bilinear form Is an inner product Pos. def.

  6. Self-Adjoint Definition Self-adjoint in Self-adjoint in H-symmetric

  7. Bramble-Pasciak CG CG for Indefinite Computational Fluid Dynamics • Symmetric • Indefinite Optimizations Saddle Point Problem • Non-symmetric • Positive definite Preconditioner Is H-symmetric and positive definite

  8. Bramble-Pasciak CG CG for Indefinite USE Preconditioner Inner Product H H SPD in < , >H H H

  9. Iterative Krylov Subspace Methods SPD Symm Non-Sym CG MINRES GMRES

  10. Bramble-Pasciak CG CG for Indefinite USE Preconditioner Inner Product H H SPD in < , >H H H

  11. Bramble-Pasciak CG

  12. Bramble-Pasciak CG 2008

  13. Bramble-Pasciak CG

More Related