[dune-fem] Strange initialisation of tolerance in CG stopping criterion

[Sacconi, Andrea]

Hi,

1) I agree with the fact that is a relative tolerance; I was just perplexed about the usage of the norm of the RHS on the denominator. I have never seen an implementation like this, that's all.
2) the statement should print the Euclidean norm, yes, but instead prints its square. That's why I said it's "misleading": I saw on my screen something like

sqrt(Residuum) = 1e-26

which triggered my perplexity. It turns out that the real residuum is 1e-13.

Cheers,
Andrea
__________________________________________________________

Andrea Sacconi
PhD student, Applied Mathematics
AMMP Section, Department of Mathematics, Imperial College London,
London SW7 2AZ, UK
a.sacconi11 at imperial.ac.uk

________________________________________
From: dune-fem-bounces+a.sacconi11=imperial.ac.uk at dune-project.org [dune-fem-bounces+a.sacconi11=imperial.ac.uk at dune-project.org] on behalf of Andreas Dedner [a.s.dedner at warwick.ac.uk]
Sent: 27 May 2014 13:25
To: dune-fem at dune-project.org
Subject: Re: [dune-fem] Strange initialisation of tolerance in CG stopping criterion

Well, generally speaking dune-fem solvers do not really include any
preconditioning anyway (with the
exception of diagonal). They are included in the package as a "getting
started" option that does not
involve other packages (petsc) or modules (dune-istl). Also they can be
used as a template to implement
something more sophisticated. From that point of view the stopping
criteria is also not very
sophisticated.... As Claus says it is a relative error that is checked
eps > | b^n-b | / |b| with b^n=Au^n.

I don't quite see why the sqrt worries you - the statement prints the
Euclidean norm of the
residual so has to include the square root?

Andreas

On 27/05/14 13:16, Claus-Justus Heine wrote:
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> Am 27.05.14 13:26, schrieb Sacconi, Andrea:
>> Hi Dune-Fem users,
>>
>> I would like to ask you a question about the implementation of the
>> Conjugate Gradient solver. In the file cginverseoperator.hh, line
>> 399, the tolerance for the stopping criterion based on the
>> residuum is set as:
>>
>> const RangeFieldType tolerance = (epsilon_ * epsilon_) *
>> b.scalarProductDofs( b );
> Well, this means epsilon is a relative tolerance. Why not. One might
> be tempted to say, however, that then the preconditioned scalar
> product should used, not the Euclidean one. Of course, after convergence
>
> b= Au, if u is the solution.
>
> But this means that  <b,b>=<Au, Au> is a strange quantity. Applying
> the approximate inverse of A (the preconditioner) would lead to
> something which whould be closer to the energy norm of the solution,
> i.e. maybe
>
> <b, precon(b)>\approx <Au,u>
>
> should be used instead. Or am I wrong here?
>
> Cheers,
>
> Claus
>
>> Why is the square of the norm of the RHS introduced here?
>> Shouldn't the tolerance be independent on the magnitude of the RHS?
>> In addition, the std::cerr is a bit misleading, since
>>
>> std::cerr << "CG-Iteration: " << realCount_ << ", sqr(Residuum): "
>> << residuum << std::endl;
>>
>> the sqaure of the residuum, and not the residuum itself, is
>> printed.
>>
>> Cheers, Andrea
>> __________________________________________________________
>>
>> Andrea Sacconi PhD student, Applied Mathematics AMMP Section,
>> Department of Mathematics, Imperial College London, London SW7
>> 2AZ, UK a.sacconi11 at imperial.ac.uk
>>
>> _______________________________________________ dune-fem mailing
>> list dune-fem at dune-project.org
>> http://lists.dune-project.org/mailman/listinfo/dune-fem
>>
>
>
> - --
> Dr. Claus-Justus Heine
> Institut für Angewandte Analysis und Simulation
> * Numerische Mathematik für Höchstleistungsrechner
> Universität Stuttgart
>
> Fon: +49 (0) 711 - 685 65558
> Fax: +49 (0) 711 - 685 65507
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