[petsc-users] Implementation of Anderson mixing

[Jonas Mairhofer]

Thank you for this fast response!


Am 22.08.2016 18:07, schrieb Barry Smith:
>> On Aug 22, 2016, at 7:36 AM, Jonas Mairhofer <mhofer at itt.uni-stuttgart.de> 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