On 01/31/2017 11:24 AM, Ingo Kligge wrote:
during my work with the Krylov subspace method QMRS in deal.ii I met
some problems: the implementation of the method is, as stated in the
class description, adapted from Algorithm 5.1 in /Freund/Nachtigal:
Software for simplified Lanczos and QMR algorithms, Appl. Num. Math. 19
(1995), pp. 319-341/ (e.g http://dl.acm.org/citation.cfm?id=223396)
specialized for the solution of right-preconditioned symmetric linear
systems. The deal.ii version however calculates (at least up to version
8.4.1) the *preconditioned* initial residual - I think, referring to the
cited paper and any other right-preconditioned Krylov subspace method,
it should be the *unpreconditioned* one. I've noticed very bad
convergence behaviour and breakdowns as well. What do you say? Do you
agree with my conjecture?
I implemented this ~17 years and have no recollection. I would be
surprised if anyone else has looked at it in the meantime. If you say
that there is a bug, I have no reason to doubt that :-)
Beside of that: I would suggest a different implementation of the
symmetric QMR-method. In an earlier paper
(https://www.researchgate.net/publication/234171461_A_new_Krylov-subspace_method_for_symmetric_indefinite_linear_system
, 1994) Freund and Nachtigal propose the essentially same algorithm
especially for solving symmetric indefinite linear systems with a
symmetric precondition matrix, that is a) flexible for right, left and
split preconditioning and b) compared to the existing deal.ii code
requires only ONE application of the precondition matrix per iteration
in the case of right preconditioning (instead of two). The best, I
think, is a hybrid of both, avoiding squaring the scalars for updating
the iterates because of possible rounding errors.
Maybe I will rewrite the method on my own. if I find the time.
Since you seem to know far more about this than I suspect any of the
other developers, I think it would be great if you could "fix" the
implementation we have, or write a better one. Any help would definitely
be much appreciated!
Best
Wolfgang
--
Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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