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

[Claus-Justus Heine]

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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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