[petsc-dev] -pc_type gamg for vector PDE

[Mark F. Adams]

Let me just add to Jeds comments that equivalent real form does increase the spectra by reflecting it about an axis -- which axis depends on which form you use.  So you want to pick the right form to minimize this growth.  I talk about it in "Algebraic multigrid techniques for strongly indefinite linear systems from direct frequency response analysis in solid mechanics." Computional Mechanics (2007).

Mark

On Jan 20, 2012, at 6:37 AM, Alexander Grayver wrote:

> Mark, Jed,
> 
> Thanks a lot for your response.
> 
>> It looks like you have a largish \omega\mu\sigma because, as Jed mentioned, one would not expect GAMG to work well for the curl-curl part.
> 
> Well in my case it is the other way around. I solve this equation for low \omega and possibly very low \sigma. The curl-curl term dominates always.
> 
>> 
>> A few things to try:
>> 
>> 1) '-pc_gamg_type sa' will provide what should be a better solver for SPD problems.
> 
> Unfortunately, the convergence with this option is worse for several operators I've tried. 
> Would it help to specify coordinates vertices?
> 
>> 
>> 2) Try:
>> 
>> -pc_type hypre    ! instead of -pc_type gamg
>> -pc_hypre_type boomeramg 
>> -pc_hypre_boomeramg_strong_threshold 0.6
> 
> When configuring with hypre I get "Cannot use hypre with complex numbers it is not coded for this capability".
> I can reformulate my problem in terms of real matrix:
> 
> If C = A + iB 
> Cx=b  ==  [A -B; B A] [xr; xi] = [br; bi]
> 
> But I'm not sure that making spectra two times larger would not influence my condition number?
> 
>> 
>> 3) You say 'staggard' but I just see E here.  Do you have E on faces?  I forget how staggering works here.  If E is cell centered then you have a system of 3x3 blocks (with the right ordering) and GAMG might benefit from setting the block size to tell it this:
>> 
>> MatSetBlockSize(mat,3); 
> 
> I have fields on the edges, but I can formulate another staggering scheme where fields are on the faces. Would it help?
> My matrix is 3x3 block. For instance, if I have 100^3 grid, then matrix is 3*10^6 and each block is of 10^6 size. 
> Which block size should I pass to MatSetBlockSize? 10^6 or 3? Because from its description it is not obvious.
> 
>> And Jed's answer addresses your 2nd question about null-space.  These solvers will degrade as \omega\mu\sigma gets smaller.
> 
> I do observe this for low \omega and low \sigma (e.g. in the air). I was thinking how could one project out this null-space. 
> Hitpmair, as Jed pointed, gives some clues. However it requires some efforts to implement and wanted first to try petsc built in stuff.
> 
>> 
>> Mark
>> 
>> On Jan 19, 2012, at 5:37 PM, Jed Brown wrote:
>> 
>>> On Wed, Jan 18, 2012 at 08:22, Alexander Grayver <agrayver at gfz-potsdam.de> wrote:
>>> Hello petsc team,
>>> 
>>> I solve 3D vector Helmholtz equation like following:
>>> 
>>> \nabla \times \nabla \times E + i\omega\mu\sigma E  = -J
>>> 
>>> Multigrid methods for curl-curl problems are pretty specialized. ML and Hypre have support for specific discretizations, I don't know if they support an imaginary shift. The PETSc interface to these packages does not currently support their special Maxwell interfaces.
>>> 
>>> The methods are generally based on edge relaxation or auxiliary space preconditioning, see Hiptmair or Arnold, Falk, and Winther for the mathematical background.
>>>  
>>> 
>>> I use structured staggered grid and FD. The solution is a vector that consists of three parts E = {Ex Ey Ez}. The operator is symmetric matrix with complex numbers on diagonal.
>>> I'm interested in solving this system with iterative techniques. I applied newly presented gamg and it gives promising results, but all I did is just:
>>> -ksp_type tfqmr -pc_type gamg
>>> 
>>> I played with different ksp_type and gamg options which are listed on PCGAMG doc page, but nothing improved convergence.
>>> Could you please guide me a bit through usage of this technique?
>>> The precise questions are:
>>> 1. Do I have to do something to say petsc that my equation is a vector equation? Is it important for gamg pc?
>>> 2. Should I take into account null-space using KSPSetNullSpace? Since it is well known that as \omega or \sigma get small, null-space of geometric term (curl curl operator) starts to dominate and system gets more ill-conditioned.
>>> 3. Which options for gamg may improve convergence in my case?
>>> 
>>> 
>>> Thanks a lot in advance.
>>> 
>>> -- 
>>> Regards,
>>> Alexander
>>> 
>>> 
>> 
> 
> 
> -- 
> Regards,
> Alexander

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