[petsc-dev] Deflated Krylov solvers for PETSc

[Jed Brown]

Interesting. I've always thought of deflation as being about the
construction of the deflation vectors rather than application of the
projection, which I think can be implemented by composing PCComposite and
PCGalerkin (or with a degenerate Galerkin two-level multigrid cycle). Note
that the DGMRES construction has some similarities to Bootstrap AMG in that
low-energy eigenvectors are processed incrementally and used to define the
new coarse grid.

On Tue, Feb 26, 2013 at 11:58 PM, Jie Chen <jiechen at mcs.anl.gov> wrote:

> To whomever is interested in deflated Krylov solvers:
>
> Deflation is very useful technique for accelerating the convergence of
> Krylov iterations. It seems to me that deflation is not supported in PETSc,
> except for the special case DGMRES, where eigenvectors are used to deflate
> GMRES. As a matter of fact, one does not need to use eigenvectors for
> deflation, nor deflation is dependent on a specific choice of the iterative
> method.
>
> Coincidentally, Kees Vuik pointed to me an earlier paper of his:
>
> http://ta.twi.tudelft.nl/nw/users/vuik/papers/Vui01F.pdf
>
> which claims that the convergence of using block Jacobi/ILU(0) [the
> default preconditioner for PETSc] becomes in general better if deflation is
> included. In his paper, deflation means "coarse grid acceleration"; see
> page 935, the paragraph of equation (2).
>
> So I think (including my personal experience), deflation is a general
> enough technique that may benefit every user of the KSP solver.
> Implementation-wise, my personal take is that the library needs to change
> the A-multiply to (A-AZ(Z'AZ)^{-1}Z'A)-multiply, where Z is the deflation
> matrix: tall, skinny, and with a small number of columns. PETSc can ask the
> user to define Z, and it can also predefine several choices of Z, one
> example choice is from the above mentioned paper. In principle, deflation
> should be independent of the KSP types, but I don't see how exactly it can
> be implemented without touching the code of every krylov solvers. Perhaps
> adding one more layer of abstraction is needed.
>
> So for anyone who is interested in implementing deflation, I can provide a
> more detailed tutorial. The design of PETSc is becoming more and more
> complicated for me to digest, thus I think it may be unrealistic for me to
> implement deflation myself for the moment.
>
> Jie
>
>
>
> --
> Jie Chen
> Mathematics and Computer Science Division
> Argonne National Laboratory
> Address: 9700 S Cass Ave, Bldg 240, Lemont, IL 60439
> Phone: (630) 252-3313
> Email: jiechen at mcs.anl.gov
> Homepage: http://www.mcs.anl.gov/~jiechen
>
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