[petsc-users] Simultaneously compute Residual+Jacobian in SNES

[Matthew Knepley]

On Sun, Dec 11, 2016 at 2:43 PM, Derek Gaston <friedmud at gmail.com> wrote:

> Thanks Barry - I'll try it and get back to you.
>
> Matt: There are lots of cases where this could be a large savings.  Here
> are a few examples:
>
> 1.  If you have automatic differentiation.  With my newest code, just
> computing a residual computes the Jacobian as a side effect.  If you throw
> away that Jacobian that's obviously a waste.  If you compute one residual
> without computing the Jacobian (which isn't always possible, depending on
> how you have your automatic differentiation setup) then you still have to
> compute _another_ residual to compute the Jacobian... so you're directly
> doing double the residual computations that are necessary.
>

I consider this bad code management more than an analytical case for the
technique, but I can see the point.


> 2.  Anytime you have extremely expensive work to do "per element" that
> would need to be done for both the residual and Jacobian.  A few examples:
>   - Extremely complex, heavy shape function evaluation (think super high
> order with first and second derivatives needing to be computed)
>

I honestly do not understand this one. Maybe I do not understand high order
since I never use it. If I want to compute an integral, I have
the basis functions tabulated. I understand that for high order, you use a
tensor product evaluation, but you still tabulate in 1D. What is
being recomputed here?


>   - Extremely heavy material property computations that need to happen at
> each quadrature point.  Think: multiscale.  Maybe you have an expensive
> lower-length-scale solve to do at every quadrature point (yes, we've
> actually done this).
>

Yes. I have to think about this one more.


>   - MANY coupled variables (we've run thousands).  Each of those variables
> needs to have value, gradient and (possibly) second derivatives computed at
> every quadrature point.  These values are exactly the same for the residual
> and Jacobian.
>

Ah, so you are saying that the process of field evaluation at the
quadrature points is expensive because you have so many fields.
It feels very similar to the material case, but I cannot articulate why. I
guess my gut says that really expensive material properties,
much more expensive than my top level model, should be modeled by something
simpler at that level. Same feeling for using
thousands of fields.

However, science proceeds by brute force, not clever should have beens. I
can see in these cases that a combined evaluation would
save a lot of time. However, our Newton does not really know whether it
needs a Jacobian or residual at the same time. Its hard to make
it work in my head. For example,

  1) I compute a Jacobian with every residual. This sucks because line
search and lots of other things use residuals.

  2) I compute a residual with every Jacobian. This sound like it could
work because I compute both for the Newton system, but here I
      am reusing the residual I computed to check the convergence criterion.

Can you see a nice way to express Newton for this?

  Matt


> These cases could be so extreme that these heavy "element" calculations
> actually dominate your residual/jacobian assembly time.  That would mean
> that by computing the residual and Jacobian simultaneously you could
> directly cut your assembly time in _half_.  That could be significant for
> many applications.  In my current application that essentially cuts the
> whole runtime of the application in half (runtime is very much assembly
> dominated).
>
> Derek
>
> On Fri, Dec 9, 2016 at 3:11 PM Matthew Knepley <knepley at gmail.com> wrote:
>
>> On Fri, Dec 9, 2016 at 2:10 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
>>
>>
>> > On Dec 9, 2016, at 1:50 PM, Derek Gaston <friedmud at gmail.com> wrote:
>> >
>> > Oh man!  Sorry Barry!  I swear I looked around before I sent the
>> email.  I should have checked the FAQ a little more closely!
>> >
>> > I can understand the reasoning in the FAQ... but I still wonder if it
>> might not be useful to provide all three options (Function, Jacobian,
>> FunctionJacobian).  In my case I could fill in each one to do the right
>> thing.  That way PETSc could call the "FunctionJacobian" one when it knew
>> it needed both
>>
>>    Derek,
>>
>>       The code literally never knows if it will need a Jacobian following
>> the function evaluation, yes at the first function evaluation it will need
>> the Jacobian unless the function norm is sufficiently small but after that
>> it is only a question of probabilities (which it can't know) whether it
>> will need the Jacobian.
>>
>> > (by default that could just farm out to the individual calls).  But you
>> guys have definitely thought a lot more about this than I have.
>> >
>> > So, do you still recommend what's suggested in the FAQ?  Save off the
>> Jacobian computation during the residual computation and then use that when
>> SNES asks for a Jacobian?
>>
>>    Yes, try it. I think you can get away with simply putting the new
>> Jacobian matrix values into the same Jacobian matrix that is regularly used
>> so there is no need to "stash the values" somewhere else and copy them over
>> later.
>>
>>    I'd be interested in hearing how the performance works out, compute
>> always or compute only when requested.
>>
>>
>> Can anyone write down a simple model for a concrete algorithm where this
>> is more efficient? I would like to see the high level reasoning.
>>
>>   Thanks,
>>
>>     Matt
>>
>>
>>
>>   Barry
>>
>> >  In the case of automatic differentiation this could make a pretty huge
>> difference in time...
>> >
>> > Derek
>> >
>> > On Fri, Dec 9, 2016 at 1:49 PM Barry Smith <bsmith at mcs.anl.gov> wrote:
>> >
>> >   Sorry the title in the FAQ is a bit tongue-in-check.
>> >
>> >   http://www.mcs.anl.gov/petsc/documentation/faq.html#functionjacobian
>> >
>> >
>> > > On Dec 9, 2016, at 12:45 PM, Derek Gaston <friedmud at gmail.com> wrote:
>> > >
>> > > Is there a way to tell SNES to simultaneously compute both the
>> residual and the Jacobian in one callback?
>> > >
>> > > My code can compute both simultaneously and it will be more efficient
>> (think FE where you can reuse the shape-functions, variables, material
>> properties, etc. for both residual and Jacobian computation).  In addition,
>> I also have automatic differentiation as an option which _definitely_
>> computes both efficiently (and actually computes residuals, by themselves,
>> much slower).
>> > >
>> > > I was thinking that I may just save off the Jacobian whenever the
>> initial residual computation is asked for by SNES... and then just return
>> that Jacobian when SNES asks for it.  This may be a bit dicey though as
>> SNES can ask for residual computations at many different points during the
>> solve.
>> > >
>> > > Thanks for any help!
>> > >
>> > > Derek
>> >
>>
>>
>>
>>
>> --
>> 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
>>
>


-- 
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
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