[petsc-users] Matrix-free vs finite differenced Jacobian approximation

[Alexander Lindsay]

Ok, I'm going to go back on my original statement...the physics being run
here is a sub-set of a much larger set of physics; for the current set the
hand-coded Jacobian actually appears to be quite good.

With hand-coded Jacobian, -pc_type lu, the convergence is perfect:

 0 Nonlinear |R| = 2.259203e-02
      0 Linear |R| = 2.259203e-02
      1 Linear |R| = 1.129089e-10
 1 Nonlinear |R| = 6.295583e-11

So yea I guess at this point I'm just curious about the different behavior
between `-snes_fd` and `-snes_fd -snes_mf_operator`. Does the hand-coded
result change your opinion Matt that the rules for FormFunction/Jacobian
might be being violated?

I understand that a finite difference approximation of the true Jacobian *is
an approximation. *However, in the absence of possible complications like
Matt suggested where an on-the-fly calculation might stand a better chance
of capturing the behavior, I would expect both `-snes_mf_operator -snes_fd`
and `-snes_fd` to suffer from the same approximations, right?

On Tue, Dec 12, 2017 at 9:43 AM, Matthew Knepley <knepley at gmail.com> wrote:

> On Tue, Dec 12, 2017 at 11:30 AM, Alexander Lindsay <
> alexlindsay239 at gmail.com> wrote:
>
>> I'm not using any hand-coded Jacobians.
>>
>
> This looks to me like the rules for FormFunction/Jacobian() are being
> broken. If the residual function
> depends on some third variable, and it changes between calls independent
> of the solution U, then
> the stored Jacobian could look wrong, but one done every time on the fly
> might converge.
>
>    Matt
>
>
>> Case 1 options: -snes_fd -pc_type lu
>>
>> 0 Nonlinear |R| = 2.259203e-02
>>       0 Linear |R| = 2.259203e-02
>>       1 Linear |R| = 7.821248e-11
>>  1 Nonlinear |R| = 2.258733e-02
>>       0 Linear |R| = 2.258733e-02
>>       1 Linear |R| = 5.277296e-11
>>  2 Nonlinear |R| = 2.258733e-02
>>       0 Linear |R| = 2.258733e-02
>>       1 Linear |R| = 5.993971e-11
>> Nonlinear solve did not converge due to DIVERGED_LINE_SEARCH iterations 2
>>
>> Case 2 options: -snes_fd -snes_mf_operator -pc_type lu
>>
>>  0 Nonlinear |R| = 2.259203e-02
>>       0 Linear |R| = 2.259203e-02
>>       1 Linear |R| = 2.258733e-02
>>       2 Linear |R| = 3.103342e-06
>>       3 Linear |R| = 6.779865e-12
>>  1 Nonlinear |R| = 7.497740e-06
>>       0 Linear |R| = 7.497740e-06
>>       1 Linear |R| = 8.265413e-12
>>  2 Nonlinear |R| = 7.993729e-12
>> Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 2
>>
>> On Tue, Dec 12, 2017 at 9:12 AM, zakaryah . <zakaryah at gmail.com> wrote:
>>
>>> When you say "Jacobians are bad" and "debugging the Jacobians", do you
>>> mean that the hand-coded Jacobian is wrong?  In that case, why would you be
>>> surprised that the finite difference Jacobians, which are "correct" to
>>> approximation error, perform better?  Otherwise, what does "Jacobians are
>>> bad" mean - ill-conditioned?  Singular?  Not symmetric?  Not positive
>>> definite?
>>>
>>
>>
>
>
> --
> 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
>
> https://www.cse.buffalo.edu/~knepley/ <http://www.caam.rice.edu/~mk51/>
>
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