Preconditioning for saddle point problems

Jed Brown jed at
Wed Apr 30 01:58:54 CDT 2008

On Wed 2008-04-30 09:11, Dave May wrote:
> Hey Jed,
>     I took a look through the link you provided and wanted to ask a couple
> of questions to clarity a couple of things.
> 1) How well does the the preconditioner you describe work with the power
> law rheology implemented? What is the maximum viscosity contrast between
> elements and across that box you've tried?

The preconditioner is weakest on the first Newton step after bumping the
continuation.  For small regularization parameters, it is better to make it
stronger by doing a few more iterations on the Schur complement.  The method
converges with effective viscosity contrast of 10^7 although it takes more
continuation steps and the number of required iterations per solve goes up
(slowly) with the strength of the nonlinearity.  I'm experimenting with
preconditioners for S to improve this behavior.

Note that this code has only one element---it is just spectral collocation.  It
is also not parallel since we must DCT the entire domain to apply the spectral

> 2) When you say you use AMG for the action of y = A^{-1} x, it appears from
> your README that this is defined using hypre. Is this correct? Have you found
> boomeramg to be superior to ML for the A block? In the past I tried both
> preconditioners on the A block for variable viscosity Stokes flow, but I never
> had much success. They either required massive amounts of setup time or the
> solution time has not obtained in O(N). Was there something fancy in the
> options configuration required to make hypre work for you?

I haven't really played with AMG tuning.  I found BoomerAMG to be stronger than
ML with default options.  A strong preconditioner is important since it is
applied at the innermost iteration, therefore we can afford somewhat higher
setup costs.  Note that applying the spectral operator requires O(n log n).

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