[petsc-users] ARKIMEX produces incorrect values

Mon Aug 31 19:32:22 CDT 2020

Emil --

When I use PETSc on various tasks, so far, I have separated (1) how
I describe the problem structure for use by the PETSc component, and (2)
the choice of solver.  For TS I am confused about what is expected by the
design you describe.  I would like to describe my ODE system as clearly as
possible and *then* go out and try/choose solver types.

My understanding is that if I have a problem which can be written in the
form F(t,u,u') = G(t,u), and if I do not want to pre-emptively restrict to
*not* allowing IMEX, then I should put F into an IFunction and G into a
RHSFunction.  This is a *good* split, performance-wise, if in fact F
contains the stiff part, but whether good or bad I have described the ODE
system.

Any fully-implicit method should now be able to handle this form F(t,u,u')
= G(t,u), because for implicit methods there is no real distinction between
F(t,u,u')=0  and F(t,u,u')=G(t,u).  If an IMEX method is completely
flexible, and so far ROSW seems to be flexible in this way (?), then I
think it should also work with either form.

If an IMEX method is restricted further by form, e.g. requiring dF/d(u') to
be the identity, then wouldn't it make sense to have the user
programmatically indicate that structural property?  Such an indication is
not about the desired solver but about the ODE.  If the structural property
held then we could proceed with the restricted-application method, e.g.
ARKIMEX/EIMEX.

Perhaps one could have  this (proposed) functionality:

TSSetLHSHasIdentity(TS,PETSC_TRUE)

Or one might instead set an enumerate for whether dF/d(u') is I or M
(invertible) or M (noninvertible for DAE) or a general nonlinear function:

TSSetLHSStructureType(TS,TS_LHS_STRUCTURE_IDENTITY)
TSSetLHSStructureType(TS,TS_LHS_STRUCTURE_INVERTIBLE)
TSSetLHSStructureType(TS,TS_LHS_STRUCTURE_NONINVERTIBLE)
TSSetLHSStructureType(TS,TS_LHS_STRUCTURE_NONLINEAR)

Obviously the enumerate would only need to include structure which some
method could exploit.

What do you think?

In any case, I am currently having trouble with the preferred way to
describe e.g. diffusion-reaction PDEs.  It seems to me I would want to
supply all four of these
IFunction
IJacobian
RHSFunction
RHSJacobian
so as to allow full-performance for both fully-implicit methods and IMEX
methods.  (And for typical examples I certainly *can* form RHSJacobian, for
example.)  But none of the src/ts/tutorials/ examples seem to
unconditionally supply all four, and I can't tell (e.g. from -ts_view)
which parts are seen and called by the various methods.

Ed

On Mon, Aug 31, 2020 at 2:27 PM Constantinescu, Emil M. <emconsta at anl.gov>
wrote:

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> On 8/31/20 12:17 PM, Ed Bueler wrote:
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> Emil --
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> Thanks for looking at this.
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> > Hi Ed, can you please add the following
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> > TSSetEquationType(ts,TS_EQ_IMPLICIT);
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> > before calling TSSolve and try again? This is described in Table 12 in
> the pdf doc.
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> Yep, that fixes it.  After setting the TS_EQ_IMPLICIT flag
> programmatically I get:
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> It is only programmatic because it has to do with the form of RHS and
> IFunctions.
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> $ ./ex54 -ts_type arkimex -ts_arkimex_fully_implicit
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> error norm at tf = 1.000000 from 12 steps:  |u-u_exact| =  1.34500e-02
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> Without -ts_arkimex_fully_implicit we still get the wrong answer, but, as
> I understand it, we expect the wrong answer because dF/d(dudt) != I,
> correct?
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> Yes. I keep mixing F and G, but if you want to solve Mu'=H(u), then define
> the IFunction := M u_dot - H(u) then it should work with all time steppers.
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> If you want to set the RHS of your ODE in the RHS function (so that you
> can use explicit integrators, too) you have to provide:
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> IFunction := u_dot and RHSFunction := M^{-1}*H(u) [or solve Mx=H(u) in the
> RHS function].
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> Note that M u_dot - H(u) can only be solved by implicit solvers directly
> so IFunction := M u_dot and RHSFunction := H(u). Table 12 in the PDF doc
> explains these cases, but that can be improved as well.
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> So -ts_arkimex_fully_implicit does not set this flag?
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> No, its use is for when you have both IFunction (for stiff) and
> RHSfunction (for nonstiff) defined to solve Mu'=H(u) + W(u) and:
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> 1- mass is identity: IFunction:= u_dot-H(u); RHSFunction:= W(u), or
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> 2- mass is full rank, but not identity: IFunction:= M u_dot-H(u);
> RHSFunction:= M^{-1} * W(u)
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> and you have a choice of using either an IMEX scheme
> [-ts_arkimex_fully_implicit false] or just the implicit part
> [-ts_arkimex_fully_implicit true].
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> Thank you for your feedback on our short survey - it is very valuable in
> helping us crafting a less painful path to using all these options.
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> Emil
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> > So that we improve our user experience, can you tell us what are your
> usual sources/starting points
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> > when implementing a new problem:
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> > 1- PDF doc
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> Yes.  Looked briefly at the PDF manual.  E.g. I saw the tables for IMEX
> methods but my eyes glazed over.
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> > 2- tutorials (if you find a good match)
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> Yes.  Looked at various html pages including the one for TSARKIMEX.  But I
> missed the sentence "Methods with an explicit stage can only be used with
> ODE in which the stiff part G(t,X,Xdot) has the form Xdot + Ghat(t,X)."  I
> did not expect that ARKIMEX
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> had this restriction, and did not pick it up.
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> > 3- own PETSc implementations
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> Yes.  I have my own diffusion-reaction system (
> https://github.com/bueler/p4pdes/blob/master/c/ch5/pattern.c) in which
> ARKIMEX works well.  (Or at least as far
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> as I can tell.  I don't have a manufactured solution for it, for
> example.)  I am in the midst of tracking down a different kind of error,
> probably from DMDA callbacks, when I got distracted by the current issue.
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> > 4- online function doc
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> Yes.  See above comment on TSARKIMEX page.   By my memory I also looked at
> the TSSet{I,RHS}Jacobian() pages, for example, and probably others.
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> > 5- other
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> Not sure.
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> Thanks,
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> Ed
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> On Mon, Aug 31, 2020 at 6:09 AM Constantinescu, Emil M. <emconsta at anl.gov>
> wrote:
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>> On 8/30/20 6:04 PM, Ed Bueler wrote:
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>> Actually, ARKIMEX is not off the hook.  It still gets the wrong answer if
>> told the whole thing is implicit:
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>> $ ./ex54 -ts_type arkimex -ts_arkimex_fully_implicit    # WRONG  (AND
>> REALLY SLOW)
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>> error norm at tf = 1.000000 from 224 steps:  |u-u_exact| =  2.76636e+00
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>> Hi Ed, can you please add the following
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>>  TSSetEquationType <https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSSetEquationType.html#TSSetEquationType>(ts,TS_EQ_IMPLICIT <https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/TS/TSEquationType.html#TSEquationType>);
>>
>>
>>
>> before calling TSSolve and try again? This is described in Table 12 in
>> the pdf doc.
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> So that we improve our user experience, can you tell us what are your
>> usual sources/starting points when implementing a new problem:
>>
>>
>> 1- PDF doc
>>
>>
>> 2- tutorials (if you find a good match)
>>
>>
>>
>>
>> 3- own PETSc implementations
>>
>>
>> 4- online function doc
>>
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>> 5- other
>>
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>> Thanks,
>>
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>> Emil
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>> versus
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>> $ ./ex54 -ts_type arkimex      # WRONG BUT IFunction IS OF FLAGGED FORM
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>> error norm at tf = 1.000000 from 16 steps:  |u-u_exact| =  1.93229e+01
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>> $ ./ex54 -ts_type bdf   # RIGHT
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>> error norm at tf = 1.000000 from 33 steps:  |u-u_exact| =  9.29170e-02
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>> So I am not sure what "Methods with an explicit stage can only be used
>> with ODE in which the stiff part G(t,X,Xdot) has the form Xdot +
>> Ghat(t,X)." means.
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>> Ed
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>> On Sun, Aug 30, 2020 at 2:57 PM Ed Bueler <elbueler at alaska.edu> wrote:
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>>> Darn, sorry.
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>>> I realize the ARKIMEX page does say "Methods with an explicit stage can
>>> only be used with ODE in which the stiff part G(t,X,Xdot) has the form Xdot
>>> + Ghat(t,X)."  So my example does not do that.  Is there a way for
>>> ARKIMEX to detect that dG/d(Xdot) = I?
>>>
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>>> Ed
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>>> On Sun, Aug 30, 2020 at 2:44 PM Ed Bueler <elbueler at alaska.edu> wrote:
>>>
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>>>
>>>
>>>>
>>>>
>>>>
>>>> Dear PETSc --
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> I tried twice to make this an issue at the gitlab.com host site, but
>>>> both times got "something went wrong (500)".  So this is a bug report by
>>>> old-fashioned means.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> I created a TS example,
>>>>
>>>> https://github.com/bueler/p4pdes-next/blob/master/c/fix-arkimex/ex54.c
>>>> at my github, also attached.  It solves a 2D linear ODE
>>>>
>>>>
>>>> ```
>>>>
>>>>
>>>>    x' + y' = 6 y
>>>>
>>>>
>>>>         y' = x
>>>>
>>>>
>>>> ```
>>>>
>>>>
>>>> Pretty basic; the known exact solution is just exponentials.  The code
>>>> writes it as F(t,u,u')=G(t,u) and supplies all the pieces, namely
>>>> IFunction,IJacobian,RHSFunction,RHSJacobian.  Note both F and G must be
>>>> seen by TS to get the correct solution.  In summary,
>>>>
>>>> a boring (and valgrind-clean ;-)) example.
>>>>
>>>>
>>>>
>>>>
>>>> For current master branch it runs fine for the fully-implicit methods
>>>> (e.g. BDF, CN, ROSW) which can use the IFunction F, including with
>>>> finite-differenced Jacobians.  With BDF2, BDF2+-snes_fd, BDF6+tight tol.,
>>>> CN, BEULER, ROSW:
>>>>
>>>>
>>>> $ ./ex54
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 33 steps:  |u-u_exact| =  9.29170e-02
>>>>
>>>>
>>>> $ ./ex54 -snes_fd
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 33 steps:  |u-u_exact| =  9.29170e-02
>>>>
>>>>
>>>> $ ./ex54 -ts_rtol 1.0e-14 -ts_atol 1.0e-14 -ts_bdf_order 6
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 388 steps:  |u-u_exact| =  4.23624e-11
>>>>
>>>>
>>>> $ ./ex54 -ts_type beuler
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 100 steps:  |u-u_exact| =  6.71676e-01
>>>>
>>>>
>>>> $ ./ex54 -ts_type cn
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 100 steps:  |u-u_exact| =  2.22839e-03
>>>>
>>>>
>>>> $ ./ex54 -ts_type rosw
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 21 steps:  |u-u_exact| =  5.64012e-03
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> But it produces wrong values with ARKIMEX:
>>>>
>>>>
>>>> $ ./ex54 -ts_type arkimex
>>>>
>>>>
>>>> error norm at tf = 1.000000 from 16 steps:  |u-u_exact| =  1.93229e+01
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Neither tightening tolerance nor changing type (`-ts_arkimex_type`)
>>>> helps ARKIMEX.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Thanks!
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Ed
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> PS  My book is at a late proofs stage, and out of my hands.  It should
>>>> appear SIAM Press in a couple of months.  In all the examples in my book,
>>>> only my diffusion-reaction system example using F(t,u,u') = G(t,u) is
>>>> broken.  Thus the motivation for a trivial
>>>>
>>>> ODE example as above.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
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>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Ed Bueler
>>>>
>>>>
>>>> Dept of Mathematics and Statistics
>>>>
>>>>
>>>> University of Alaska Fairbanks
>>>>
>>>>
>>>> Fairbanks, AK 99775-6660
>>>>
>>>>
>>>> 306C Chapman
>>>>
>>>>
>>>>
>>>>
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>>>
>>> --
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> Ed Bueler
>>>
>>>
>>> Dept of Mathematics and Statistics
>>>
>>>
>>> University of Alaska Fairbanks
>>>
>>>
>>> Fairbanks, AK 99775-6660
>>>
>>>
>>> 306C Chapman
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
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>>>
>>>
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>> --
>>
>>
>>
>>
>>
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>>
>>
>>
>>
>>
>> Ed Bueler
>>
>>
>> Dept of Mathematics and Statistics
>>
>>
>> University of Alaska Fairbanks
>>
>>
>> Fairbanks, AK 99775-6660
>>
>>
>> 306C Chapman
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> --
>>
>> Emil M. Constantinescu, Ph.D.
>>
>> Computational Mathematician
>>
>> Argonne National Laboratory
>>
>> Mathematics and Computer Science Division
>>
>>
>>
>> Ph: 630-252-0926
>>
>> http://www.mcs.anl.gov/~emconsta
>>
>>
>>
>>
>>
>>
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> --
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>
> Ed Bueler
>
>
> Dept of Mathematics and Statistics
>
>
> University of Alaska Fairbanks
>
>
> Fairbanks, AK 99775-6660
>
>
> 306C Chapman
>
>
>
>
>
>
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>
>
>
>
>
>
>
>
>
> --
>
> Emil M. Constantinescu, Ph.D.
>
> Computational Mathematician
>
> Argonne National Laboratory
>
> Mathematics and Computer Science Division
>
>
>
> Ph: 630-252-0926
>
> http://www.mcs.anl.gov/~emconsta
>
>
>
>
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>
>

-- 
Ed Bueler
Dept of Mathematics and Statistics
University of Alaska Fairbanks
Fairbanks, AK 99775-6660
306C Chapman
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