Re: [firedrake] periodic boundary conditions?

11 May 2016

      Thank you very much Jemma, Lawrence and Stephan,

Just to recap, the strong form of the problem that I am solving is

\nabla_H^2 psi - F \psi = q

Initially I am defining q to be a random variable so it mean is close to zero but not quite.

-> In a periodic channel geometry the method will converge whether F is non-zero or not.

-> In a doubly periodic domain it will only converge of F is non-zero.  I did try 1e-5 and that still converged.  

When I set the mean to zero (thanks again Jemma) I agree that does work, without having to do anything with the nullspace.  I guess the moral of the story is that if the domain is periodic it's best to have a PV with zero mean.  I will hopefully not forget this anytime soon.

I need to fix this up but in case you're interested, here is the starting to a move on 2D turbulence using firedrake.

https://youtu.be/jhAI7TMN7_w

Cheers,Francis

------------------
Francis Poulin
Associate Professor
Department of Applied Mathematics
University of Waterloo

email:           fpoulin@uwaterloo.ca
Web:            https://uwaterloo.ca/poulin-research-group/
Telephone:  +1 519 888 4567 x32637

________________________________________
From: firedrake-bounces@imperial.ac.uk [firedrake-bounces@imperial.ac.uk] on behalf of Stephan Kramer [s.kramer@imperial.ac.uk]
Sent: Wednesday, May 11, 2016 6:21 AM
To: firedrake@imperial.ac.uk
Subject: Re: [firedrake] periodic boundary conditions?

I think you're problem might be that your right-hand side is
inconsistent with your nullspace. The nullspace is not only a nullspace
on the right-hand of the matrix, but also on the left-hand. If you
consider the constant-1 test function, then the left-hand side vanishes
(regardless of psi), but if q1 doesn't integrate to 0 over the domain,
the right-hand side doesn't. At the level of the linear system, this is
equivalent to left-multiplying with a constant vector on both sides:
again the lhs disappears and the rhs doesn't. This means there is not
solution to your system. If you think q1 should actually integrate to
zero, but is not because of numerical error, you would have to project
the constant part out before going into the solve. In petsc you could do
that by setting a transpose nullspace, or explicitly calling
MatNullSpaceRemove on the rhs vector - I'm not sure Firedrake has nice
interface to do the same using the 'nullspace' you have

Cheers
Stephan

On 11/05/16 10:55, Lawrence Mitchell wrote:
...
...
On 11 May 2016, at 10:39, Shipton, Jemma <j.shipton@imperial.ac.uk> wrote:
PS In fact, it runs just fine if you delete the solver_parameters={blah} option.
Beware!  This is hiding a problem I think.  The default solver parameters are to use GMRES, not CG.
If you run with GMRES and 'ksp_monitor_true_residual': True, I get the following output.
0 KSP preconditioned resid norm 1.402486418742e-02 true resid norm 2.145922987616e-02 ||r(i)||/||b|| 1.000000000000e+00
     1 KSP preconditioned resid norm 6.859114126588e-03 true resid norm 8.235483003565e-03 ||r(i)||/||b|| 3.837734648956e-01
     1 KSP preconditioned resid norm 6.859114126588e-03 true resid norm 8.235483003565e-03 ||r(i)||/||b|| 3.837734648956e-01
     2 KSP preconditioned resid norm 4.372980631318e-03 true resid norm 4.523115353529e-03 ||r(i)||/||b|| 2.107771518191e-01
     2 KSP preconditioned resid norm 4.372980631318e-03 true resid norm 4.523115353529e-03 ||r(i)||/||b|| 2.107771518191e-01
     3 KSP preconditioned resid norm 3.083878237363e-03 true resid norm 3.181122331413e-03 ||r(i)||/||b|| 1.482402840070e-01
     3 KSP preconditioned resid norm 3.083878237363e-03 true resid norm 3.181122331413e-03 ||r(i)||/||b|| 1.482402840070e-01
     4 KSP preconditioned resid norm 2.328339850318e-03 true resid norm 2.482928092561e-03 ||r(i)||/||b|| 1.157044361280e-01
     4 KSP preconditioned resid norm 2.328339850318e-03 true resid norm 2.482928092561e-03 ||r(i)||/||b|| 1.157044361280e-01
     5 KSP preconditioned resid norm 1.850645396168e-03 true resid norm 2.076376448198e-03 ||r(i)||/||b|| 9.675913162685e-02
     5 KSP preconditioned resid norm 1.850645396168e-03 true resid norm 2.076376448198e-03 ||r(i)||/||b|| 9.675913162685e-02
     6 KSP preconditioned resid norm 1.505245304176e-03 true resid norm 1.869875298565e-03 ||r(i)||/||b|| 8.713617913390e-02
     6 KSP preconditioned resid norm 1.505245304176e-03 true resid norm 1.869875298565e-03 ||r(i)||/||b|| 8.713617913390e-02
     7 KSP preconditioned resid norm 1.228163738742e-03 true resid norm 1.716825645339e-03 ||r(i)||/||b|| 8.000406609403e-02
     7 KSP preconditioned resid norm 1.228163738742e-03 true resid norm 1.716825645339e-03 ||r(i)||/||b|| 8.000406609403e-02
     8 KSP preconditioned resid norm 9.762580464613e-04 true resid norm 1.604598260859e-03 ||r(i)||/||b|| 7.477427056418e-02
     8 KSP preconditioned resid norm 9.762580464613e-04 true resid norm 1.604598260859e-03 ||r(i)||/||b|| 7.477427056418e-02
     9 KSP preconditioned resid norm 7.187761512558e-04 true resid norm 1.509128698387e-03 ||r(i)||/||b|| 7.032538945228e-02
     9 KSP preconditioned resid norm 7.187761512558e-04 true resid norm 1.509128698387e-03 ||r(i)||/||b|| 7.032538945228e-02
    10 KSP preconditioned resid norm 4.857940540733e-04 true resid norm 1.430687141460e-03 ||r(i)||/||b|| 6.667001330972e-02
    10 KSP preconditioned resid norm 4.857940540733e-04 true resid norm 1.430687141460e-03 ||r(i)||/||b|| 6.667001330972e-02
    11 KSP preconditioned resid norm 3.400753386006e-04 true resid norm 1.370869815346e-03 ||r(i)||/||b|| 6.388252622562e-02
    11 KSP preconditioned resid norm 3.400753386006e-04 true resid norm 1.370869815346e-03 ||r(i)||/||b|| 6.388252622562e-02
    12 KSP preconditioned resid norm 2.724827673558e-04 true resid norm 1.357871715518e-03 ||r(i)||/||b|| 6.327681484164e-02
    12 KSP preconditioned resid norm 2.724827673558e-04 true resid norm 1.357871715518e-03 ||r(i)||/||b|| 6.327681484164e-02
    13 KSP preconditioned resid norm 2.023603200214e-04 true resid norm 1.348266511796e-03 ||r(i)||/||b|| 6.282921239844e-02
    13 KSP preconditioned resid norm 2.023603200214e-04 true resid norm 1.348266511796e-03 ||r(i)||/||b|| 6.282921239844e-02
    14 KSP preconditioned resid norm 1.405713028843e-04 true resid norm 1.342655679836e-03 ||r(i)||/||b|| 6.256774765844e-02
    14 KSP preconditioned resid norm 1.405713028843e-04 true resid norm 1.342655679836e-03 ||r(i)||/||b|| 6.256774765844e-02
    15 KSP preconditioned resid norm 9.092448503001e-05 true resid norm 1.337633948725e-03 ||r(i)||/||b|| 6.233373501493e-02
    15 KSP preconditioned resid norm 9.092448503001e-05 true resid norm 1.337633948725e-03 ||r(i)||/||b|| 6.233373501493e-02
    16 KSP preconditioned resid norm 5.750986093948e-05 true resid norm 1.335928348383e-03 ||r(i)||/||b|| 6.225425404790e-02
    16 KSP preconditioned resid norm 5.750986093948e-05 true resid norm 1.335928348383e-03 ||r(i)||/||b|| 6.225425404790e-02
    17 KSP preconditioned resid norm 3.802056816287e-05 true resid norm 1.335095336743e-03 ||r(i)||/||b|| 6.221543570986e-02
    17 KSP preconditioned resid norm 3.802056816287e-05 true resid norm 1.335095336743e-03 ||r(i)||/||b|| 6.221543570986e-02
    18 KSP preconditioned resid norm 2.223795106365e-05 true resid norm 1.334942349117e-03 ||r(i)||/||b|| 6.220830648727e-02
    18 KSP preconditioned resid norm 2.223795106365e-05 true resid norm 1.334942349117e-03 ||r(i)||/||b|| 6.220830648727e-02
    19 KSP preconditioned resid norm 1.199219664791e-05 true resid norm 1.334849886266e-03 ||r(i)||/||b|| 6.220399771888e-02
    19 KSP preconditioned resid norm 1.199219664791e-05 true resid norm 1.334849886266e-03 ||r(i)||/||b|| 6.220399771888e-02
    20 KSP preconditioned resid norm 6.045970214995e-06 true resid norm 1.334822540473e-03 ||r(i)||/||b|| 6.220272340509e-02
    20 KSP preconditioned resid norm 6.045970214995e-06 true resid norm 1.334822540473e-03 ||r(i)||/||b|| 6.220272340509e-02
    21 KSP preconditioned resid norm 3.357136102289e-06 true resid norm 1.334814998460e-03 ||r(i)||/||b|| 6.220237194734e-02
    21 KSP preconditioned resid norm 3.357136102289e-06 true resid norm 1.334814998460e-03 ||r(i)||/||b|| 6.220237194734e-02
    22 KSP preconditioned resid norm 2.109412994583e-06 true resid norm 1.334823835708e-03 ||r(i)||/||b|| 6.220278376303e-02
    22 KSP preconditioned resid norm 2.109412994583e-06 true resid norm 1.334823835708e-03 ||r(i)||/||b|| 6.220278376303e-02
    23 KSP preconditioned resid norm 1.697698874597e-06 true resid norm 1.334823113503e-03 ||r(i)||/||b|| 6.220275010828e-02
    23 KSP preconditioned resid norm 1.697698874597e-06 true resid norm 1.334823113503e-03 ||r(i)||/||b|| 6.220275010828e-02
    24 KSP preconditioned resid norm 1.512292251273e-06 true resid norm 1.334824567918e-03 ||r(i)||/||b|| 6.220281788403e-02
    24 KSP preconditioned resid norm 1.512292251273e-06 true resid norm 1.334824567918e-03 ||r(i)||/||b|| 6.220281788403e-02
    25 KSP preconditioned resid norm 1.250309415572e-06 true resid norm 1.334825572707e-03 ||r(i)||/||b|| 6.220286470715e-02
    25 KSP preconditioned resid norm 1.250309415572e-06 true resid norm 1.334825572707e-03 ||r(i)||/||b|| 6.220286470715e-02
    26 KSP preconditioned resid norm 7.983659954253e-07 true resid norm 1.334829784399e-03 ||r(i)||/||b|| 6.220306097201e-02
    26 KSP preconditioned resid norm 7.983659954253e-07 true resid norm 1.334829784399e-03 ||r(i)||/||b|| 6.220306097201e-02
    27 KSP preconditioned resid norm 5.434753011846e-07 true resid norm 1.334830275105e-03 ||r(i)||/||b|| 6.220308383892e-02
    27 KSP preconditioned resid norm 5.434753011846e-07 true resid norm 1.334830275105e-03 ||r(i)||/||b|| 6.220308383892e-02
    28 KSP preconditioned resid norm 3.583899393830e-07 true resid norm 1.334831849286e-03 ||r(i)||/||b|| 6.220315719574e-02
    28 KSP preconditioned resid norm 3.583899393830e-07 true resid norm 1.334831849286e-03 ||r(i)||/||b|| 6.220315719574e-02
    29 KSP preconditioned resid norm 2.531881383802e-07 true resid norm 1.334830569064e-03 ||r(i)||/||b|| 6.220309753738e-02
    29 KSP preconditioned resid norm 2.531881383802e-07 true resid norm 1.334830569064e-03 ||r(i)||/||b|| 6.220309753738e-02
    30 KSP preconditioned resid norm 1.983419151306e-07 true resid norm 1.334831092617e-03 ||r(i)||/||b|| 6.220312193496e-02
    30 KSP preconditioned resid norm 1.983419151306e-07 true resid norm 1.334831092617e-03 ||r(i)||/||b|| 6.220312193496e-02
    31 KSP preconditioned resid norm 1.685855333188e-07 true resid norm 1.334830361112e-03 ||r(i)||/||b|| 6.220308784684e-02
    31 KSP preconditioned resid norm 1.685855333188e-07 true resid norm 1.334830361112e-03 ||r(i)||/||b|| 6.220308784684e-02
    32 KSP preconditioned resid norm 1.289738938190e-07 true resid norm 1.334830636894e-03 ||r(i)||/||b|| 6.220310069824e-02
    32 KSP preconditioned resid norm 1.289738938190e-07 true resid norm 1.334830636894e-03 ||r(i)||/||b|| 6.220310069824e-02
    33 KSP preconditioned resid norm 9.235117500873e-08 true resid norm 1.334830368242e-03 ||r(i)||/||b|| 6.220308817906e-02
    33 KSP preconditioned resid norm 9.235117500873e-08 true resid norm 1.334830368242e-03 ||r(i)||/||b|| 6.220308817906e-02
    34 KSP preconditioned resid norm 6.503935947986e-08 true resid norm 1.334830235572e-03 ||r(i)||/||b|| 6.220308199666e-02
    34 KSP preconditioned resid norm 6.503935947986e-08 true resid norm 1.334830235572e-03 ||r(i)||/||b|| 6.220308199666e-02
    35 KSP preconditioned resid norm 4.003439088177e-08 true resid norm 1.334830289372e-03 ||r(i)||/||b|| 6.220308450375e-02
    35 KSP preconditioned resid norm 4.003439088177e-08 true resid norm 1.334830289372e-03 ||r(i)||/||b|| 6.220308450375e-02
    36 KSP preconditioned resid norm 2.933016192914e-08 true resid norm 1.334830350834e-03 ||r(i)||/||b|| 6.220308736789e-02
    36 KSP preconditioned resid norm 2.933016192914e-08 true resid norm 1.334830350834e-03 ||r(i)||/||b|| 6.220308736789e-02
    37 KSP preconditioned resid norm 2.457785557100e-08 true resid norm 1.334830341474e-03 ||r(i)||/||b|| 6.220308693167e-02
    37 KSP preconditioned resid norm 2.457785557100e-08 true resid norm 1.334830341474e-03 ||r(i)||/||b|| 6.220308693167e-02
    38 KSP preconditioned resid norm 2.003045563251e-08 true resid norm 1.334830452035e-03 ||r(i)||/||b|| 6.220309208385e-02
    38 KSP preconditioned resid norm 2.003045563251e-08 true resid norm 1.334830452035e-03 ||r(i)||/||b|| 6.220309208385e-02
    39 KSP preconditioned resid norm 1.592346775337e-08 true resid norm 1.334830443617e-03 ||r(i)||/||b|| 6.220309169157e-02
    39 KSP preconditioned resid norm 1.592346775337e-08 true resid norm 1.334830443617e-03 ||r(i)||/||b|| 6.220309169157e-02
    40 KSP preconditioned resid norm 1.285777369303e-08 true resid norm 1.334830523051e-03 ||r(i)||/||b|| 6.220309539319e-02
    40 KSP preconditioned resid norm 1.285777369303e-08 true resid norm 1.334830523051e-03 ||r(i)||/||b|| 6.220309539319e-02
    41 KSP preconditioned resid norm 9.550269214853e-09 true resid norm 1.334830544472e-03 ||r(i)||/||b|| 6.220309639139e-02
    41 KSP preconditioned resid norm 9.550269214853e-09 true resid norm 1.334830544472e-03 ||r(i)||/||b|| 6.220309639139e-02
    42 KSP preconditioned resid norm 6.311572416881e-09 true resid norm 1.334830559098e-03 ||r(i)||/||b|| 6.220309707295e-02
    42 KSP preconditioned resid norm 6.311572416881e-09 true resid norm 1.334830559098e-03 ||r(i)||/||b|| 6.220309707295e-02
    43 KSP preconditioned resid norm 3.652454149992e-09 true resid norm 1.334830540583e-03 ||r(i)||/||b|| 6.220309621017e-02
    43 KSP preconditioned resid norm 3.652454149992e-09 true resid norm 1.334830540583e-03 ||r(i)||/||b|| 6.220309621017e-02
    44 KSP preconditioned resid norm 2.037777932328e-09 true resid norm 1.334830540790e-03 ||r(i)||/||b|| 6.220309621981e-02
    44 KSP preconditioned resid norm 2.037777932328e-09 true resid norm 1.334830540790e-03 ||r(i)||/||b|| 6.220309621981e-02
    45 KSP preconditioned resid norm 1.128203860755e-09 true resid norm 1.334830532154e-03 ||r(i)||/||b|| 6.220309581738e-02
    45 KSP preconditioned resid norm 1.128203860755e-09 true resid norm 1.334830532154e-03 ||r(i)||/||b|| 6.220309581738e-02
So you can see that although the preconditioned norm is converging, the unpreconditioned norm is stagnating.  If you provide a nullspace to the Krylov method and use GMRES, then the solution will be the minimum residual in the least squares sense in the space orthogonal to the nullspace.  However, your solution may still have a large component in the nullspace (which is what I think is going on here).
I'm not sure what the correct fix is though, unfortunately.
Lawrence
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