[Dune] Setting and optimising AMG parameters in ISTL

[Markus Blatt]

Hey Eike,

On Tue, Jul 24, 2012 at 03:11:53PM -0700, Bath Uni wrote:
> 
> I'm trying to optimise the ISTL-AMG solver for the specific problem I'm solving. It is a 3d problem with a strong anisotropy in the vertical direction, the equation is
> 
> - omega^2 (\laplace_{2d} + \lambda^2 d^2/dz^2) u(x,y,z) + u(x,y,z) = RHS, where x,y,z \in [0,1]^3
> 
> with omega^2= 6.702E-4, lambda^2 = O(10^3-10^4).
> 
> omega^2 decreases with the square of the grid size, i.e. the relative size of the second order term decreases relative to the constant terms as I increase the horizontal resolution, and I always keep the vertical size fixed.
> 
> So far I've tried the default settings for AMG by using the
> ISTLBackend_CG_AMG_SSOR, and the time per iteration scales very well
> up to 65536 cores (in my weak scaling tests, where I increase the
> horizontal resolution, such that the local problem size stays
> constant), but the number of iterations is quite large (42 on 16
> cores and 45 on 1024 cores). 
>
> [...]
>
> I presume that I'm simply not using the optimal AMG parameters, so I
> would like to try to optimise them. 
> What are the best AMG parameters to use in this case and what is the
> easiest way of passing them to the AMG solver, if I'm using the ISTL
> backend? I noticed that there is a function
> setDefaultValuesAnisotropic in dune/istl/paamg/parameters.hh. Can I
> use this to set optimal default parameters for the anisotropic case?

The algorithm should do semi-coarsening by default. Because of the
piecewise constant interpolation one should make sure that the
aggregates are smaller in number of unknowns than for the isotropic
case. The relevant low level routine is
AggregationParameters::setMaxDistance(int) (inherited by the parameter
class), where distance is the maximum path length (in terms of edges)
of the shortest path between all pairs of vertices in the
aggregate. (To get aggregates of 2 vertices for an 2D anisotropic
problem this would be 1). 

BTW: Currently the setDefaultValeus{Ani|I}sotropic methods seem to
this value a little bit to high.

In the latest Pdelab you get access to the parameter class by calling
the parameters() method on the AMG backends.

> 
> Markus, I've seen that you have solved an anisotropic problem of the form
> 
> -\nabla (K \nabla u(x)) = 1 with K_{ij} = {10^{-6}} for i=j=0 and K_{ij} = 1 for i=j>0, K_{ij}=0 otherwise
> 
> in your PhD thesis, but could not figure out which parameters you
> used. The anisotropy is the other way round as in my case, but maybe
> the parameters you used are still relevant for what I'm doing. 

For the anisotropic tests I used the UnsymmetricCriterion (why is somehow beyond me):
Dune::Amg::CoarsenCriterion<Dune::Amg::UnSymmetricCriterion<BCRSMat,Dune::Amg::FirstDiagonal> >

which determines strong edges according to Vanek's criterion
-a_ij > \alpha \max_{k\not=j} -a_ik
 to determine strong edges. Unfortunately this might result in wrong
aggregation for jumping diffusion coefficients.

This criterion is not yet supported by PDElab.

Markus
-- 
Do you need more support with DUNE or HPC in general? 

Dr. Markus Blatt - HPC-Simulation-Software & Services http://www.dr-blatt.de
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