[petsc-users] A question on finite difference Jacobian

[Zou (Non-US), Ling]

well, it has the same risk that Newton direction is not good due to the
simplification. However, it worth my trying.

On Wed, Oct 7, 2015 at 8:54 AM, Zou (Non-US), Ling <ling.zou at inl.gov> wrote:

> Thank you Barry.
>
> The background I am asking this question is that I want to reduce (or you
> can say optimize) the cost of my finite difference Jacobian evaluation,
> which is used for preconditioning purpose. The concept is based on my
> understanding of the problem I am solving, but I am not sure if it will
> work, thus I want to do some test.
>
> Here is the concept, assume that my residual reads,
>
> F(\vec{U}) = F[\vec{U}, g(\vec{U})]
>
>
> in which, g(\vec{U}) is a quite complicated and thus expensive function
> evaluation. This function, however, is not very sensitive to \vec{U}, i.e.,
> \partial{g(\vec{U})}/\partial{g(\vec{U})} is not that important.
>
>
>
> Normally, a finite difference Jacobian is evaluated as (as discussed in
> PETSc manual),
>
>
> J(\vec{u}) \approx \frac{F(\vec{U}+\epsilon \vec{v}) - F(\vec{U})}
> {\epsilon}
>
>
> In my case, it reads,
>
>
> J(\vec{u}) \approx \frac{F[(\vec{U}+\epsilon \vec{v}), g(\vec{U}+\epsilon
> \vec{v})] - F[(\vec{U}), g(\vec{U})]} {\epsilon}
>
>
> Because \partial{g(\vec{U})}/\partial{g(\vec{U})} is not important, the
> simplification I want to make is, when finite difference Jacobian (as
> preconditioner) is evaluated, it can be further simplified as,
>
>
> J(\vec{u}) \approx \frac{F[(\vec{U}+\epsilon \vec{v}), g(\vec{U})] -
> F[(\vec{U}), g(\vec{U})]} {\epsilon}
>
>
> Thus, the re-evaluation on g(\vec{U}+\epsilon \vec{v}) is removed. It
> seems to me that I need some kind of signal from PETSc so I can tell the
> code not to update g(\vec{U}). However, I never tested it and I don't know
> if anybody did similar things before.
>
>
> Thanks,
>
>
> Ling
>
>
>
> On Tue, Oct 6, 2015 at 7:09 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
>
>>
>> > On Oct 6, 2015, at 4:22 PM, Zou (Non-US), Ling <ling.zou at inl.gov>
>> wrote:
>> >
>> >
>> >
>> > On Tue, Oct 6, 2015 at 2:38 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
>> >
>> > > On Oct 6, 2015, at 3:29 PM, Zou (Non-US), Ling <ling.zou at inl.gov>
>> wrote:
>> > >
>> > > Hi All,
>> > >
>> > > If the non-zero pattern of a finite difference Jacobian needs 20
>> colors to color it (20 comes from MatFDColoringView, the non-zero pattern
>> is pre-determined from mesh connectivity), is it true that PETSc needs 40
>> functions evaluation to get the full Jacobian matrix filled? This is
>> because that a perturbation per color needs two function evaluation
>> according to PETSc manual (ver 3.6, page 123, equations shown in the middle
>> of the page).
>> > > But I only see 20 function evaluations. I probably have some
>> misunderstanding somewhere. Any suggestions?
>> >
>> >    PETSc uses forward differencing to compute the derivatives, hence it
>> needs a single function evaluation at the given point (which has almost
>> always been previously computed in Newton's method)
>> > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>> > Is it a potential problem if the user chooses to use a different (e.g.
>> simplified) residual function as the function for MatFDColoringSetFunction?
>>
>>    Yes, you can do that. But this may result in a "Newton" direction that
>> is not a descent direction hence Newton stalls. If you have 20 colors I
>> doubt that it would be a good idea to use a cheaper function there. If you
>> have several hundred colors then you can use a simpler function PLUS
>> -snes_mf_operator to insure that the Newton direction is correct.
>>
>>
>>   Barry
>>
>> >
>> >
>> > and then one function evaluation for each color. This is why it reports
>> 20 function evaluations.
>> >
>> >   Barry
>> >
>> >
>> > >
>> > > Ling
>> >
>> >
>>
>>
>
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