decomposition with static pivoting M. Arioli, I. S. Duff, S. - PowerPoint PPT Presentation

GMRES preconditioned by a perturbed LDL T decomposition with static pivoting M. Arioli, I. S. Duff, S. Gratton, and S. Pralet http://www.numerical.rl.ac.uk/people/marioli/marioli.html Harrachov, 2007 – p.1/40

Outline Multifrontal Static pivoting GMRES and Flexible GMRES Flexible GMRES: a roundoff error analysis GMRES right preconditioned: a roundoff error analysis Test problems Numerical experiments Harrachov, 2007 – p.2/40

Linear system We wish to solve large sparse systems Ax = b R N × N is symmetric indefinite where A ∈ I Harrachov, 2007 – p.3/40

Linear system A particular and important case arises in saddle-point problems where the coefficient matrix is of the form    H A  A T 0 Since we want accurate solutions, we would prefer to use a direct method of solution and our method of choice uses a multifrontal approach. Harrachov, 2007 – p.4/40

Multifrontal method ASSEMBLY TREE Harrachov, 2007 – p.5/40

Multifrontal method AT EACH NODE ASSEMBLY TREE F F 12 11 T F F 22 12 Harrachov, 2007 – p.5/40

Multifrontal method AT EACH NODE ASSEMBLY TREE F F 12 11 T F F 22 12 12 F − 1 F 22 ← F 22 − F T 11 F 12 Harrachov, 2007 – p.5/40

Multifrontal method From children to parent Harrachov, 2007 – p.6/40

Multifrontal method From children to parent Gather/Scatter ASSEMBLY operations (indirect address- ing) Harrachov, 2007 – p.6/40

Multifrontal method From children to parent ASSEMBLY Gather/Scatter operations (indirect addressing) ELIMINATION Full Gaussian elimination, Level 3 BLAS (TRSM, GEMM) Harrachov, 2007 – p.6/40

Multifrontal method From children to parent ASSEMBLY Gather/Scatter operations (indirect addressing) ELIMINATION Full Gaussian elimination, Level 3 BLAS (TRSM, GEMM) Harrachov, 2007 – p.6/40

Multifrontal method F F 12 11 T F F 22 12 Pivot can only be chosen from F 11 block since F 22 is NOT fully summed. Harrachov, 2007 – p.7/40