[petsc-users] Null space and preconditioners

[Mark Adams]

I'll bet CG will work fine on it. CG is pretty robust and as Phil would
say this operator is morally symmetric.

On Mon, Mar 21, 2022 at 3:42 PM Barry Smith <bsmith at petsc.dev> wrote:

>
>
> On Mar 21, 2022, at 3:33 PM, Mark Adams <mfadams at lbl.gov> wrote:
>
> (I did suggest CG, he just has a pressure solve, which is a Laplacian,
> right?)
>
>
>    His finite volume scheme is finite difference-ish in that it produces a
> non-symmetric matrix. The non-symmetric part A-A' is actually very cool
> looking; one of the coolest matrices I've ever seen :-)
>
>
>
> Ugh, this is pretty bad. The logic might be a bit convoluted, but
> if SetOperator is called after SetFromOptions, as is usual I think, it
> could check if it has left a left PC, if the operator has a null space.
>
>
>
> On Mon, Mar 21, 2022 at 2:48 PM Barry Smith <bsmith at petsc.dev> wrote:
>
>>
>>   Marco,
>>
>>      I have confirmed your results.
>>
>>      Urgg, it appears we do not have something well documented. The
>> removal of the null space only works for left preconditioned solvers and
>> FGMRES only works with right preconditioning. Here is the reasoning.
>>
>>      The Krylov space for left preconditioning is built from [r, BAr,
>> (BA)^2 r, ...] and the solution space is built from this basis. If A has a
>> null space of n then the left preconditioned Krylov methods simply remove n
>> from the "full" Krylov space after applying each B preconditioner and the
>> resulting "reduced" Krylov space has no components in the n directions
>> hence the solution built by GMRES naturally has no component in the n.
>>
>>      But with right preconditioning the Krylov space is [s ABs (AB)^2 s,
>> ....] We would need to remove B^-1 n from the Krylov space so that (A B)
>> B^-1 n = 0 In general we don't have any way of applying B^-1 to a vector so
>> we cannot create the appropriate "reduced" Krylov space.
>>
>>      If I run with GMRES (which defaults to left preconditioner) and the
>> options ./testPreconditioners -pc_type gamg -ksp_type gmres
>> -ksp_monitor_true_residual -ksp_rtol 1.e-12 -ksp_view -mg_coarse_pc_type svd
>>
>>   Then  it handles the null space correctly and the solution has Solution
>> mean = 4.51028e-17
>>
>> Is there any reason to use FGMRES instead of GMRES? You just cannot use
>> GMRES as the smoother inside GAMG if you use GMRES on the outside, but for
>> pressure equations you don't want use such a strong smoother anyways.
>>
>>   Barry
>>
>>   I feel we should add some information to the documentation on the
>> removal of the null space to the user's manual when using right
>> preconditioning and maybe even have an error check in the code so that
>> people don't fall into this trap. But I am not sure exactly what to do.
>> When the A and B are both symmetric I think special stuff happens that
>> doesn't require providing a null space; but I am not sure.
>>
>>
>>
>>
>>
>> On Mar 21, 2022, at 12:41 PM, Marco Cisternino <
>> marco.cisternino at optimad.it> wrote:
>>
>> Thank you, Mark.
>> However, doing this with my toy code
>> mpirun -n 1 ./testpreconditioner -pc_type gamg
>> -pc_gamg_use_parallel_coarse_grid_solver -mg_coarse_pc_type jacobi
>> -mg_coarse_ksp_type cg
>>
>> I get 16 inf elements. Do I miss anything?
>>
>> Thanks again
>>
>> Marco Cisternino
>>
>>
>> *From:* Mark Adams <mfadams at lbl.gov>
>> *Sent:* lunedì 21 marzo 2022 17:31
>> *To:* Marco Cisternino <marco.cisternino at optimad.it>
>> *Cc:* petsc-users at mcs.anl.gov
>> *Subject:* Re: [petsc-users] Null space and preconditioners
>>
>> And for GAMG you can use:
>>
>> -pc_gamg_use_parallel_coarse_grid_solver -mg_coarse_pc_type jacobi
>> -mg_coarse_ksp_type cg
>>
>> Note if you are using more that one MPI process you can use 'lu' instead
>> of 'jacobi'
>>
>> If GAMG converges fast enough it can solve before the constant creeps in
>> and works without cleaning in the KSP method.
>>
>> On Mon, Mar 21, 2022 at 12:06 PM Mark Adams <mfadams at lbl.gov> wrote:
>>
>> The solution for Neumann problems can "float away" if the constant is not
>> controlled in some way because floating point errors can introduce it even
>> if your RHS is exactly orthogonal to it.
>>
>> You should use a special coarse grid solver for GAMG but it seems to be
>> working for you.
>>
>> I have lost track of the simply way to have the KSP solver clean the
>> constant out, which is what you want.
>>
>> can someone help Marco?
>>
>> Mark
>>
>>
>>
>>
>>
>> On Mon, Mar 21, 2022 at 8:18 AM Marco Cisternino <
>> marco.cisternino at optimad.it> wrote:
>>
>> Good morning,
>> I’m observing an unexpected (to me) behaviour of my code.
>> I tried to reduce the problem in a toy code here attached.
>> The toy code archive contains a small main, a matrix and a rhs.
>> The toy code solves the linear system and check the norms and the mean of
>> the solution.
>> The problem into the matrix and the rhs is the finite volume
>> discretization of the pressure equation of an incompressible NS solver.
>> It has been cooked as tiny as possible (16 cells!).
>> It is important to say that it is an elliptic problem with homogeneous
>> Neumann boundary conditions only, for this reason the toy code sets a null
>> space containing the constant.
>>
>> The unexpected (to me) behaviour is evident by launching the code using
>> different preconditioners, using -pc-type <pctype>
>> I tested using PCNONE (“none”), PCGAMG (“gamg”) and PCILU (“ilu”). The
>> default solver is KSPFGMRES.
>> Using the three PC, I get 3 different solutions. It seems to me that they
>> differ in the mean value, but GAMG is impressive.
>> PCNONE gives me the zero mean solution I expected. What about the others?
>>
>> Asking for residuals monitor, the ratio ||r||/||b|| shows convergence for
>> PCNONE and PCILU (~10^-16), but it stalls for PCGAMG (~10^-4).
>> I cannot see why. Am I doing anything wrong or incorrectly thinking about
>> the expected behaviour?
>>
>> Generalizing to larger mesh the behaviour is similar.
>>
>> Thank you for any help.
>>
>> Marco Cisternino
>>
>>
>>
>
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