[petsc-users] SLEPc - st_type cayley choice of shift and antishift

[Jose E. Roman]

I now see what is happening. In the expression of the paper the antishift has different sign compared to the expression used in SLEPc (see the users manual):

(A-sigma*B)^{-1}*(A+nu*B)x = \theta x

So nu=-sigma is a forbidden value, otherwise both factors cancel out (I will fix the interface so that this is catched).

In your case you should do -eps_target -1 -st_cayley_antishift -1

Jose


> El 27 sept 2019, a las 13:54, Michael Werner <michael.werner at dlr.de> escribió:
> 
> Yes, with sinvert its working. And using -eps_target instead of
> -st_shift didn't change anything.
> 
> I also just sent you the matrices for reproduction of the issue.
> 
> Michael
> 
> Am 27.09.19 um 13:32 schrieb Jose E. Roman:
>> Try setting -eps_target -1 instead of -st_shift -1
>> Does sinvert work with target -1?
>> Can you send me the matrices so that I can reproduce the issue?
>> 
>> Jose
>> 
>> 
>>> El 27 sept 2019, a las 13:11, Michael Werner <michael.werner at dlr.de> escribió:
>>> 
>>> Thank you for the link to the paper, it's quite interesting and pretty
>>> close to what I'm doing. I'm currently also using the "inexact" approach
>>> for my application, and in general it works, as long as the ksp
>>> tolerance is low enough. However, I was hoping to speed up convergence
>>> towards the "interesting" eigenvalues by using Cayley.
>>> 
>>> Now as a test I tried to follow the approach from your paper, choosing
>>> mu = -sigma, and mu in the order of magnitude of the imaginary part of
>>> the most amplified eigenvalue. I know the most amplified eigenvalue for
>>> my problem is -0.0398+0.724i, so I tried running SLEPc with the
>>> following settings:
>>> -st_type cayley
>>> -st_shift -1
>>> -st_cayley_antishift 1
>>> 
>>> But I never get the correct eigenvalue, instead SLEPc returns only the
>>> value of st_shift:
>>> [0]      Number of iterations of the method: 1
>>> [0]      Solution method: krylovschur
>>> [0]      Number of requested eigenvalues: 1
>>> [0]      Stopping condition: tol=1e-08, maxit=19382
>>> [0]      Number of converged eigenpairs: 16
>>> [0]     
>>> [0]              k          ||Ax-kx||/||kx||
>>> [0]      ----------------- ------------------
>>> [0]          -1.000000          0.0281754
>>> [0]          -1.000000          0.0286815
>>> [0]          -1.000000          0.0109186
>>> [0]          -1.000000           0.140883
>>> [0]          -1.000000           0.203036
>>> [0]          -1.000000         0.00801616
>>> [0]          -1.000000          0.0526871
>>> [0]          -1.000000           0.022244
>>> [0]          -1.000000          0.0182197
>>> [0]          -1.000000          0.0107924
>>> [0]          -1.000000         0.00963378
>>> [0]          -1.000000          0.0239422
>>> [0]          -1.000000         0.00472435
>>> [0]          -1.000000         0.00607732
>>> [0]          -1.000000          0.0124056
>>> [0]          -1.000000         0.00557715
>>> 
>>> Also, it doesn't matter if I'm using exact or inexact solves. Changing
>>> the values of shift and antishift also doesn't change the behaviour. Do
>>> I need to make additional adjustments to get cayley to work?
>>> 
>>> Best regards,
>>> Michael
>>> 
>>> 
>>> 
>>> Am 25.09.19 um 17:21 schrieb Jose E. Roman:
>>>>> El 25 sept 2019, a las 16:18, Michael Werner via petsc-users <petsc-users at mcs.anl.gov> escribió:
>>>>> 
>>>>> Hello,
>>>>> 
>>>>> I'm looking for advice on how to set shift and antishift for the cayley
>>>>> spectral transformation. So far I've been using sinvert to find the
>>>>> eigenvalues with the smallest real part (but possibly large imaginary
>>>>> part). For this, I use the following options:
>>>>> -st_type sinvert
>>>>> -eps_target -0.05
>>>>> -eps_target_real
>>>>> 
>>>>> With sinvert, it is easy to understand how to chose the target, but for
>>>>> Cayley I'm not sure how to set shift and antishift. What is the
>>>>> mathematical meaning of the antishift?
>>>>> 
>>>>> Best regards,
>>>>> Michael Werner
>>>> In exact arithmetic, both shift-and-invert and Cayley build the same Krylov subspace, so no difference. If the linear solves are computed "inexactly" (iterative solver) then Cayley may have some advantage, but it depends on the application. Also, iterative solvers usually are not robust enough in this context. You can see the discussion here https://doi.org/10.1108/09615530410544328
>>>> 
>>>> Jose
>>>> 
> 
> 
> 
>