Numerical methods for Data Assimilation

• Left preconditioning : solve .

• GMRES, inexact left preconditioning : .

• Inexact Arnoldi relation reads , with

• Set , then .

in GMRES}

• Application of theorem for relaxed GMRES

• Let , such that and consider the use of the strategy that generates perturbations in the preconditioning step by introducing residuals which norm are monitored by

  

  The corresponding relaxed left preconditioned GMRES algorithm is such that at the breakdown of the method, the norm of the preconditioned backward error is less than .

• Right preconditioning : solve , then .

• GMRES, inexact right preconditioning : .

• Inexact Arnoldi relation reads , with

• Set , then .

in GMRES

Let , such that and consider the use of the strategy that generates perturbations in the preconditioning step by introducing residual which norm is monitored by

.

The corresponding relaxed right preconditioned GMRES algorithm is such that at the breakdown of the method, the norm of the backward error of computed at the preconditioned variable is less than .

To get the solution to the original system an additional preconditioning operation has to be performed, and let . Suppose that the relaxed right-preconditioned GMRES is run on using the strategy of the previous theorem and that is the corresponding estimate of . Suppose in addition that , with .

The backward error of considered as a solution of satisfies

,

where .

GMRES with relaxed right preconditioner , Matrix , .

Outer scheme : solve with the Conjugate Gradient (CG).

each matrix-vector product involves local linear systems

solved in turn by CG:

Up to the first order:

• Anis - subdomains

• Each subdomain : grid points

  :

• Inner preconditioner : Incomplete Cholesky (IC)

• Targeted accuracy:

Comparison with a