> On Aug 22, 2016, at 7:36 AM, Jonas Mairhofer <[email protected]>
> wrote:
>
> Dear PETSc-Team,
>
> I am having a bit of trouble understanding the way the Anderson mixing method
> is implemented inPETSc.
>
> On the corresponding website
> (http://www.mcs.anl.gov/petsc/petsc-3.5/docs/manualpages/SNES/SNESAnderson.html)
> it says in the
> "Option database" section
>
> X_{update} = X + \beta F
This is wrong, it is the "mixing" parameter. I will fix it.
>
> which looks to me as if the new solution is calculated from the old solution
> plus something which depends on the old residual F.
>
> However, in the "Notes" section it says that the new solution is found by
> combining "m previous solutions" which I interpret as the last m values of X.
>
> In most other sources on Anderson mixing (e.g. the original one referd to on
> the website mentioned above) the update procedure looks something like
>
> X^{k+1} = (1-\beta) \sum_{i=0}^{m} \alpha_i^k X^{k-m+1} + \beta \sum_{i=0}^m
> \alpha_i^k G(X^{k-m+i}) where G = F + X.
>
> Therefore, the new value of X is calculated from the last m values of X and F.
>
> Unfortunately, I was not able to understand and follow the actual source code
> in src/snes/impls/ngmres/anderson.c
The computation of the \sum_{i=0}^{m} \alpha_i^k X^{k-m+1} and \sum_{i=0}^m
\alpha_i^k G(X^{k-m+i} take place in
146:
SNESNGMRESFormCombinedSolution_Private(snes,ivec,l,XM,FM,fMnorm,X,XA,FA);
in particular the
ierr = VecMAXPY(XA,l,beta,Xdot);CHKERRQ(ierr);
(note this is NOT the same \beta it is more like the \alpha above).
ierr = VecMAXPY(FA,l,beta,Fdot);CHKERRQ(ierr);
>
> Could you please tell me whether the last values of X, F or both are used in
> the update scheme of PETSc's Anderson mixing implementation?
Both are, see above.
>
> And is \beta used in a "regular line search way" where X^{new} = X^{old} +
> \beta* "undamped update"
No
>
> or as in the equation above
>
> X^{new} = (1-\beta) \sum "function of last m X" + \beta \sum "function of
> last m G"
Yes.
142: VecAXPY(XM,-ngmres->andersonBeta,FM);
>
> which would explain to me why beta is not set via -snes_linesearch_damping
> but using the extra option -snes_anderson_beta.
>
>
> Thank you very much for your help!
> Jonas
