[petsc-users] Ainsworth formula to solve saddle point problems / preconditioner for shell matrices

[Jed Brown]

Olivier Jamond <olivier.jamond at cea.fr> writes:

> Dear all,
>
> I am working on a finite-elements/finite-volumes code, whose distributed 
> solver is based on petsc. For FE, it relies on Lagrange multipliers for 
> the imposition of various boundary conditions or interactions (simple 
> dirichlet, contact, ...). This results in saddle point problems:
>
> [S Ct][x]=[f]
> [C 0 ][y] [g]
>
> As discussed in this mailing list ("Saddle point problem with nested 
> matrix and a relatively small number of Lagrange multipliers"), the 
> fieldsplit/PC_COMPOSITE_SCHUR approach involves (2 + 'number of 
> iterations of the KSP for the Schur complement') KSPSolve(S, Sp). I 
> would like to try the formula given by Ainsworth in [1] to solve this 
> problem:
>
> x = (Sp)^(-1) * fp
> y = Rt * (f - S*x)
>
> where:
> Sp= Ct*C + Qt*S*Q

I just want to observe here that Ct*C lives in the big space and is low rank.  It's kinda like what you would get from an augmented Lagrangian approach.

The second term involves these commutators that destroy sparsity in general, but the context of the paper (as I interpreted it in a quick skim) is such that C*Ct consists of small decoupled blocks associated with each MPC.  The suggestion is that these can either be computed explicitly (possibly at the element level) or cleaned up in a small number of Krylov iterations.

> Q = I - P
> P = R * C
> R = Ct * (C*Ct)^(-1)
>
> My input matrices (S and C) are MPIAIJ matrices. I create a shell matrix 
> for Sp (because it involves (C*Ct)^(-1) so I think it may be a bad idea 
> to compute it explicitly...) with the MatMult operator to use it in a 
> KSPSolve. The C matrix and g vector are scaled so that the condition 
> number of Sp is similar to the one of S.
>
> It works, but my main problem is that because Sp is a shell matrix, as 
> far as I understand, I deprive myself of all the petsc 
> preconditioners... I tried to use S as a preconditioning matrix, but 
> it's not good: With a GAMG preconditioner, my iteration number is about 
> 4 times higher than in a "debug" version where I compute Sp explicitly 
> as a MPIAIJ matrix and use it as preconditioning matrix.

Are your coupling constraints nonlocal, such that C*Ct is not block diagonal?

> Is there a way to use the petsc preconditioners for shell matrices or at 
> least to define a shell preconditioner that internally calls the petsc 
> preconditioners?
>
> In the end I would like to have something like GAMG(Ct*C + Qt*S*Q) as a 
> preconditioner (here Q is a shell matrix), or something 
> like Qt*GAMG(S)*Q (which from matlab experimentation could be a 
> good preconditioner).
>
> Many thanks,
> Olivier
>
> [1]: Ainsworth, M. (2001). Essential boundary conditions and multi-point 
> constraints in finite element analysis. Computer Methods in Applied 
> Mechanics and Engineering, 190(48), 6323-6339.