[petsc-dev] Deflated Krylov solvers for PETSc

[Jed Brown]

On Thu, Feb 28, 2013 at 10:57 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
>    Jie,
>
>    Thanks.  How does one add a "regular" preconditioner to the deflation
> approach?
>

I think the most common deflation approach is to use estimates of the most
global eigenvectors. Those are dense, so we can construct the coarse
operator as

K_c = Y^T * (P^{-1} A) * Y

without further ado. If subdomains aggregates are used for the deflation
vectors Y, then we'd really like to exploit their sparsity. We would seem
to want preconditioner application to a special sort of sparse matrix.


>
> In the theory of domain decomposition methods one constructs
> preconditioners by composing (either additively or multiplicatively)
> solvers on subspaces of the entire solution space. For simplicity consider
> symmetric positive definite matrices.
>
> Now consider a bunch of subspaces defined by non-overlapping subdomains of
> the domain, from these one produces the block Jacobi or block Gauss-Seidel
> preconditioner.
>
> Now say we want to augment block Jacobi or block G.S. by using the solver
> on the subspace W = Z.
>
> Now for the question. How would that composed preconditioner relate to
> doing deflation with W and then block Jacobi preconditioning? Would it be
> better, worse, depends, identical? What are the formulas in the various
> cases?
>
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