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

[Olivier Jamond]

Dear Jed,

The code I am working on is quite generic and at the solve step, the 
matrix C can be 'whatever' (but is supposed to be full rank). But in 
practice, in 99% of the cases, C contain MPCs that refers to boundary 
conditions applied to subsets of the mesh boundary. These MPCs can 
couple several dofs, and a given dofs can be involved in several MPCs. 
For example, one could impose that the average of the solution in the 
x-direction is null on a part of the boundary, and that this part of the 
boundary is in contact with another part of the boundary.

So yes, CCt is block diagonal, where each block is a set of MPCs that 
share dofs, and CtC is also block diagonal, where each block is a set of 
dofs that share MPCs. For the vast majority of cases, these blocks 
involve dofs/MPCs attached to a subset of the boundary, so they are 
small with respect to the total number of dofs (and their size grows 
slower than the total number of dofs when the mesh is refined).

I am not sure to understand what you mean by compute the MPCs 
explicitly: do you mean eliminating them? For very simple dirichlet 
conditions I see how to do that, but in a more generic case I don't see 
(but there may be some techniques I don't know about!).

I don't understand also what you mean by cleaning them in a small number 
of krylov iterations?

Many thanks,
Olivier

On 01/10/2020 20:47, Jed Brown wrote:
> 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.