Matrix free example snes/ex20.c

[Barry Smith]

On Dec 31, 2007, at 6:06 PM, Vijay M wrote:

> Barry,
>
> Thanks for the detailed explanation. That sure is a tricky and  
> interesting
> problem. I did run the problem with the options you suggested and  
> see what
> you mean.
>
> I just have one another question though that is not quite related to  
> the
> example: Say when you do J-free Newton-Krylov iteration, then is it  
> correct
> to say that the F.D calculation of the action of Jacobian on a  
> vector is
> more accurate than using a numerical  Jacobian (not analytical)  
> found at the
> start of a Newton iteration ? Because even though in both cases, the
> Jacobian is technically found by perturbation about the last Newton
> iteration, it seems to me that there is some gain in this convergence
> respect with J-free immaterial of the problem being solved.

    I would say no; this is just a fluke thing.

   Barry


> Now is that
> confusing or am I making sense ? I'll be glad to explain more on  
> that and
> awaiting to hear your comments.
>
> Well, happy new year to you Barry and all the PETSc team !!
>
> Cheers,
> Vijay
>
> -----Original Message-----
> From: owner-petsc-users at mcs.anl.gov [mailto:owner-petsc-users at mcs.anl.gov 
> ]
> On Behalf Of Barry Smith
> Sent: Sunday, December 30, 2007 4:51 PM
> To: petsc-users at mcs.anl.gov
> Subject: Re: Matrix free example snes/ex20.c
>
>
>   Vijay,
>
>     This is a very cool problem.
>
>     Because of the exact symmetry of the domain the EXACT Jacobian
> at each step has exactly 9 different eigenvalues. This means the
> GMRES will take exactly 9 iterations (and "completely" converge in
> the ninth iteration) if the "exact" Jacobian is used.  You can run  
> with
> -pc_type none -snes_mf -ksp_monitor_singular_value -
> ksp_plot_eigenvalues -display :0.0 -draw_pause -1
> to see the 9 eigenvalues.
>
> Now run without the -snes_mf option. You will see the first Newton
> iteration's
> eigenvalues still look like 9; but starting at the second Newton
> iteration the
> "identical" eigenvalues are now not all identically placed so GMRES
> needs
> more iterations.
>
>   The question then becomes how come the matrix-free application of
> the Jacobian is more accurate than actually computing it as a sparse
> matrix then applying it? Here is my non-rigorous answer; the
> multiplication
> of the sparse matrix values (even if very accurate) against the vector
> introduces
> some rounding error that screws up the eigenvalues slightly. For some
> reason
> for this problem the matrix-free application is accurate enough not to
> perturb the eigenvalues.
>
>   Barry
>
>
>
>
>
> On Dec 30, 2007, at 2:19 PM, Vijay M wrote:
>
>> I ran both the cases with -ksp_monitor on and have attached the
>> output in
>> two different files. 1.txt is the Jfree case and 2.txt is the
>> analytical
>> case.
>>
>> Barry, the example problem is ex20 from the snes tutorial directory.
>> The
>> petsc version is 2.3.3-p7 if that helps to clear things a little.
>> Now I
>> haven't yet completely checked for a bug in the analytical Jacobian
>> but I
>> would imagine that if it were incorrect, wouldn't that affect only
>> how the
>> nonlinear iteration converges and not the linear iteration since the
>> matrix
>> sparsity structure is still the same (well assuming the condition
>> number is
>> not very different from the exact Jacobian !). Just my 2 cents.
>>
>> Anyway, I will look into the code for ex20 and then see if something
>> is
>> messed up. Let me know if you find out the problem from the output.
>>
>> Thanks,
>> Vijay
>>
>>> I understand that both the methods will not give me the same number
>>> of total
>>> linear iterations but a factor of 2 seems a little odd to me.
>>
>>   Yes, this is surprising.
>>
>>   Run with -ksp_monitor how are the linear convergence different?
>>
>>> This leads to
>>> another question whether the user can actually change the epsilon
>>> used for
>>> computing the perturbation in J-free scheme or is this fixed in
>>> PETSc ?
>>
>>   Yes, see the manual page for MatMFFDSetFromOptions() and related
>> manual
>> pages.
>>
>>>
>>>
>>> If not, then what do you think is the reason for this ?
>>
>>   Bug in your analytic Jacobian? Run with -snes_monitor and -
>> ksp_monitor and
>> send all output.
>>
>>   Barry
>>
>>> Do let me know your
>>> comments when you get some time. Thanks.
>>>
>>> Vijay
>>>
>>> -----Original Message-----
>>> From: owner-petsc-users at mcs.anl.gov [mailto:owner-petsc-users at mcs.anl.gov
>>> ]
>>> On Behalf Of Matthew Knepley
>>> Sent: Saturday, December 29, 2007 9:05 PM
>>> To: petsc-users at mcs.anl.gov
>>> Subject: Re: Matrix free example snes/ex20.c
>>>
>>> On Dec 29, 2007 8:07 PM, Vijay M <vijay.m at gmail.com> wrote:
>>>> Hi all,
>>>>
>>>> I was trying to compile and run the ex20.c example code in the
>>>> tutorial
>>>> section of SNES. Although it does not explicitly specify that -
>>>> snes_mf
>>>> option can be used, my understanding is that as long as a nonlinear
>>> residual
>>>> function is written correctly, PETSc will calculate via finite
>>>> difference
>>>> the action of the Jacobian on a given vector. Is that correct ?
>>>
>>> Yes.
>>>
>>>> Now if that is the case, then please observe the discrepancy in the
>>>> number
>>>> of linear iterations taken with an analytical Jacobian and matrix-
>>>> free
>>>> option. What puzzles me is that the SNES function norm are quite
>>>> close for
>>>> both the methods but the linear iterations differ by a factor of 3.
>>>> Why
>>>> exactly is this ?
>>>
>>> There is no PC when using -snes_mf whereas the default is ILU for  
>>> the
>>> analytic
>>> Jacobian.
>>>
>>> Matt
>>>
>>>> Here's the output to make this clearer.
>>>>
>>>> vijay :mpirun -np 1 ex20 -ksp_type gmres -snes_monitor
>>>>
>>>> 0 SNES Function norm 2.271442542876e-01
>>>>
>>>> 1 SNES Function norm 6.881516100891e-02
>>>>
>>>> 2 SNES Function norm 1.813939751552e-02
>>>>
>>>> 3 SNES Function norm 2.354176462207e-03
>>>>
>>>> 4 SNES Function norm 3.063728077362e-05
>>>>
>>>> 5 SNES Function norm 3.106106268946e-08
>>>>
>>>> 6 SNES Function norm 5.344742712545e-12
>>>>
>>>> 0 SNES Function norm 2.271442542876e-01
>>>>
>>>> 1 SNES Function norm 6.881516100891e-02
>>>>
>>>> 2 SNES Function norm 1.813939751552e-02
>>>>
>>>> 3 SNES Function norm 2.354176462207e-03
>>>>
>>>> 4 SNES Function norm 3.063728077362e-05
>>>>
>>>> 5 SNES Function norm 3.106106268946e-08
>>>>
>>>> 6 SNES Function norm 5.344742712545e-12
>>>>
>>>> Number of Newton iterations = 6
>>>>
>>>> Number of Linear iterations = 18
>>>>
>>>> Average Linear its / Newton = 3.000000e+00
>>>>
>>>>
>>>>
>>>> vijay :mpirun -np 1 ex20 -ksp_type gmres -snes_monitor -snes_mf
>>>>
>>>> 0 SNES Function norm 2.271442542876e-01
>>>>
>>>> 1 SNES Function norm 6.870629867542e-02
>>>>
>>>> 2 SNES Function norm 1.804335379848e-02
>>>>
>>>> 3 SNES Function norm 2.290074339682e-03
>>>>
>>>> 4 SNES Function norm 3.082384186373e-05
>>>>
>>>> 5 SNES Function norm 3.926396277038e-09
>>>>
>>>> 6 SNES Function norm 3.754922566585e-16
>>>>
>>>> 0 SNES Function norm 2.271442542876e-01
>>>>
>>>> 1 SNES Function norm 6.870629867542e-02
>>>>
>>>> 2 SNES Function norm 1.804335379848e-02
>>>>
>>>> 3 SNES Function norm 2.290074339682e-03
>>>>
>>>> 4 SNES Function norm 3.082384186373e-05
>>>>
>>>> 5 SNES Function norm 3.926396277038e-09
>>>>
>>>> 6 SNES Function norm 3.754922566585e-16
>>>>
>>>> Number of Newton iterations = 6
>>>>
>>>> Number of Linear iterations = 54
>>>>
>>>> Average Linear its / Newton = 9.000000e+00
>>>>
>>>>
>>>>
>>>> Thanks,
>>>>
>>>> Vijay
>>>>
>>>>
>>>
>>>
>>>
>>> -- 
>>> What most experimenters take for granted before they begin their
>>> experiments is infinitely more interesting than any results to which
>>> their experiments lead.
>>> -- Norbert Wiener
>>>
>> <1.txt><2.txt>
>