> Secondly, the results seem to be quite dependent on the
> accuracy for the Inv(M). For the choice of 10e-08, we see a jump in
> timings (and iterations for Identity preconditioner) from the size 128
> to 256. For 10e-10, this jump occurs from the size 512 to 1024. In
> fact, for the approximate Schur complement preconditioner, this jump
> is rather too severe. To investigate further, for the largest problem
> size I additionally took the relative residual for the Inv(M) as
> 10e-12 when solving with approximate Schur complement preconditioner,
> and the largest problem could be solved even cheaply than 1024 size
> with 10e-10 accuracy for Inv(M).
>
> Now I'm curious whether these are expected results. If so, then where
> should one stop the accuracy for Inv(M) (seems smaller the better). Or
> my choice of tolerance(s) is improper?
You'll have to play around with tolerances. The basic problem is that if you
make the tolerances smaller you'll need more iterations (and more time) in
the inner solver. On the other hand, if you make the tolerance too large,
then the inner problem is not solved accurately enough. This means that every
multiplication with the Schur complement that the outer CG sees is slight
different because it is not
B^T M^-1 B
but in fact something of the form
B^T Q B
where Q is the result of an inaccurate inner solve with M. The problem now is
that the inner CG solver's actions depend on the right hand side vector, i.e.
Q=Q(r_k) where r_k is the vector with which you want to solve, which changes
from iteration k to k+1. In other words, CG thinks it is multiplying with the
Schur complement each time, but in reality the matrix changes slightly each
time because
B^T Q(r_k) B
changes. CG is not prepared to handle this situation and loses orthogonality
of its iteration vectors, which leads to drastically increased numbers of
iterations. The only way to avoid this is to (i) use a tight inner tolerance,
(ii) use a different outer solver that is not quite so sensitive, such as
GMRES.
Since you did all these experiments, would you be interested in offering some
text that we could add to the results section that summarizes what you found?
Best
W.
--
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Wolfgang Bangerth email: [email protected]
www: http://www.math.tamu.edu/~bangerth/
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