[petsc-users] Ghost particles for DMSWARM (or similar)

Tue Oct 1 12:22:02 CDT 2024

Hi Miguel,

On Tue, 1 Oct 2024 at 09:05, MIGUEL MOLINOS PEREZ <mmolinos at us.es> wrote:

> Sorry, I forgot to add:
>
> You think VecScatterCreate is the way to go?
>

No.

The semantics you seek are very similar to what VecGhost provides.
I was only using VecGhostUpdateBegin/End by way of example.

Cheers,
Dave

> Thanks,
> Miguel
>
> On Oct 1, 2024, at 5:56 PM, MIGUEL MOLINOS PEREZ <mmolinos at us.es> wrote:
>
> Hi Dave,
>
> Would something like that work?
>
> Yes, this should work! Any idea on where to look so I can try to implement
> it myself?
>
> Best,
> Miguel
>
> On Oct 1, 2024, at 5:22 PM, Dave May <dave.mayhem23 at gmail.com> wrote:
>
> Hi Miguel,
>
> On Tue 1. Oct 2024 at 07:56, MIGUEL MOLINOS PEREZ <mmolinos at us.es> wrote:
>
>> Thank you Matt, it works!
>>
>> The implementation is straightforward:
>> - 1º Define the paddle regions using DMGetLocalBoundingBox with the
>> background DMDA mesh as an auxiliary mesh for the domain-partitioning.
>> - 2º Create an integer to count the local number of particles to be used
>> as ghost particle for other processors (N_ghost). One particle can be
>> counted more than one time. At the same time, fill two arrays:
>> - one with the index of the "main particle” (local particle),
>> - and the other with the target rank of the "main particle”.
>> - 3º Create the new particles using DMSwarmAddNPoints
>> - 4º Fill the new particles with the information of the “main particle”
>> but set the internal variable DMSwarmField_rank with the target rank.
>> - 5º Call DMSwarmMigrate(*,PETSC_TRUE). Therefore, we send the ghost
>> particles to the corresponding processors and we delete them from the
>> original   processor.
>> - 6º Do stuff…
>> - 7º Delete ghost particles. This is very easy, we just have to call
>> DMSwarmRemovePoint N_ghost times.
>>
>> I think this can be easily implemented as closed routine for the DMSwarm
>> class.
>>
>> The remaining question is: how to do the communication between the
>> “original" particle and the ghost particles? For instance, if we update
>> some particle variable (locally) inside of a SNES context, this same
>> variable should be updated in the ghost particles at the other processors.
>>
>
> I think what you are asking about is an operation similar to
> VecGhostUpdate{Begin,End}(). In the case of a DMSwarm I’m not sure how to
> define the InsertMode = ADD_VALUES? Some swarm fields do not make sense to
> be added. INSERT_VALUES is fine.
>
> One solution might be to have something like this
>
> DMSwarmCollectViewUpdateGhostOwners(DM dm, InsertMode mode, PetscInt
> nfields, const char *fieldNames[]);
>
> where one can specify the insert mode and the fields on which the insert
> mode will apply.
>
> Would something like that work?
>
> Cheers,
> Dave
>
>
>
>
>
>> PS: Hope this helps someone in the future :-)
>>
>>
>> On Sep 27, 2024, at 10:50 AM, MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>> wrote:
>>
>> Thank you Matt, let me give it try.
>>
>> Miguel
>>
>> On Sep 27, 2024, at 3:44 AM, Matthew Knepley <knepley at gmail.com> wrote:
>>
>> On Thu, Sep 26, 2024 at 7:18 PM MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>> wrote:
>>
>>> I see, you mean:
>>>
>>> Create the ghost particles at the local cell with the same properties as
>>> particle 1 (duplicate the original particle) but different value
>>> DMSwarmField_rank. Then, call DMSwarmMigrate(*,PETSC_FALSE) so we do
>>> the migration and delete the local copies of the particle 1.  Right?
>>>
>>
>> Yep. I think it will work, from what I know about BASIC.
>>
>>   Thanks,
>>
>>      Matt
>>
>>
>>> Thanks,
>>> Miguel
>>>
>>> On Sep 26, 2024, at 11:09 PM, Matthew Knepley <knepley at gmail.com> wrote:
>>>
>>> On Thu, Sep 26, 2024 at 11:20 AM MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>>> wrote:
>>>
>>>> Thank you Matt.
>>>>
>>>> Okey, let me have a careful look to the DMSwarmMigrate_Push_Basic implementation
>>>> to see if there is some workaround.
>>>>
>>>> The idea of adding new particles is interesting. However, in that case,
>>>> we need to initialize the new (ghost) particles using the fields of
>>>> the “real” particle, right? This can be done using something like:
>>>>
>>>> VecGhostUpdateBegin <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/VecGhostUpdateBegin/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63job1UA$ >(Vec <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/Vec/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk6wexE7L8$ > globalout,InsertMode <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Sys/InsertMode/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63Qjcg-m$ > ADD_VALUES <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Sys/ADD_VALUES/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk6364wZBs$ >, ScatterMode <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/ScatterMode/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk670gaETR$ > SCATTER_REVERSE <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/SCATTER_REVERSE/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk69jjn4Yl$ >);VecGhostUpdateEnd <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/VecGhostUpdateEnd/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63EFcTKh$ >(Vec <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/Vec/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk6wexE7L8$ > globalout,InsertMode <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Sys/InsertMode/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63Qjcg-m$ > ADD_VALUES <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Sys/ADD_VALUES/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk6364wZBs$ >, ScatterMode <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/ScatterMode/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk670gaETR$ > SCATTER_REVERSE <https://urldefense.us/v3/__https://petsc.org/release/manualpages/Vec/SCATTER_REVERSE/__;!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk69jjn4Yl$ >);
>>>>
>>>> for the particle fields (?).
>>>>
>>>
>>> I think we can just copy from the local particle. For example, suppose I
>>> decide that particle 1 should go to rank 5, 12, and 27. Then
>>> I first set p1.rank = 5, then I add two new particles with the same
>>> values as particle 1, but with rank = 12 and 27. Then when I call migrate,
>>> it will move these three particles to the correct processes, and delete the
>>> original particles and the copies from the local set.
>>>
>>>   Thanks,
>>>
>>>      Matt
>>>
>>>
>>>> Thanks,
>>>> Miguel
>>>>
>>>>
>>>> On Sep 26, 2024, at 3:53 PM, Matthew Knepley <knepley at gmail.com> wrote:
>>>>
>>>> On Thu, Sep 26, 2024 at 6:31 AM MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>>>> wrote:
>>>>
>>>>> Hi Matt et al,
>>>>>
>>>>> I’ve been working on the scheme that you proposed to create ghost
>>>>> particles (atoms in my case), and it works! With a couple of caveats:
>>>>> -1º In general the overlap particles will be migrate from their own
>>>>> rank to more than one neighbor rank, this is specially relevant for those
>>>>> located close to the corners. Therefore, you'll need to call DMSwarmMigrate
>>>>> several times (27 times for 3D cells), during the migration process.
>>>>>
>>>>
>>>> That is terrible. Let's just fix DMSwarmMigrate to have a mode that
>>>> sends the particle to all overlapping neighbors at once. It can't be that
>>>> hard.
>>>>
>>>>
>>>>> -2º You need to set DMSWARM_MIGRATE_BASIC. Otherwise the proposed
>>>>> algorithm will not work at all!
>>>>>
>>>>
>>>> Oh, I should have thought of that. Sorry.
>>>>
>>>> I can help code up that extension. Can you take a quick look at the
>>>> BASIC code? Right now, we just use the rank attached to the particle
>>>> to send it. We could have an arrays of ranks, but that seems crazy, and
>>>> would blow up particle storage. How about just adding new particles
>>>> with the other ranks right before migration?
>>>>
>>>>    Thanks,
>>>>
>>>>      Matt
>>>>
>>>>
>>>>> Hope this helps to other folks!
>>>>>
>>>>> I have a follow-up question about periodic bcc on this context, should
>>>>> I open a new thread of keep posting here?
>>>>>
>>>>> Thanks,
>>>>> Miguel
>>>>>
>>>>> On Aug 7, 2024, at 4:22 AM, MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>>>>> wrote:
>>>>>
>>>>> Thanks Matt, I think I'll start by making a small program as a proof
>>>>> of concept. Then, if it works I'll implement it in my code and I'll be
>>>>> happy to share it too :-)
>>>>>
>>>>> Miguel
>>>>>
>>>>> On Aug 4, 2024, at 3:30 AM, Matthew Knepley <knepley at gmail.com> wrote:
>>>>>
>>>>> On Fri, Aug 2, 2024 at 7:15 PM MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>>>>> wrote:
>>>>>
>>>>>> Thanks again Matt, that makes a lot more sense !!
>>>>>>
>>>>>> Just to check that we are on the same page. You are saying:
>>>>>>
>>>>>> 1. create a field define a field called "owner rank" for each
>>>>>> particle.
>>>>>>
>>>>>> 2. Identify the phantom particles and modify the internal variable
>>>>>> defined by the DMSwarmField_rank variable.
>>>>>>
>>>>>> 3. Call DMSwarmMigrate(*,PETSC_FALSE), do the calculations using the
>>>>>> new local vector including the ghost particles.
>>>>>>
>>>>>> 4. Then, once the calculations are done, rename the DMSwarmField_rank
>>>>>> variable using the "owner rank" variable and call
>>>>>> DMSwarmMigrate(*,PETSC_FALSE) once again.
>>>>>>
>>>>>
>>>>> I don't think we need this last step. We can just remove those ghost
>>>>> particles for the next step I think.
>>>>>
>>>>>   Thanks,
>>>>>
>>>>>      Matt
>>>>>
>>>>>
>>>>>> Thank you,
>>>>>> Miguel
>>>>>>
>>>>>>
>>>>>> On Aug 2, 2024, at 5:33 PM, Matthew Knepley <knepley at gmail.com>
>>>>>> wrote:
>>>>>>
>>>>>> On Fri, Aug 2, 2024 at 11:15 AM MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>>>>>> wrote:
>>>>>>
>>>>>>> Thank you Matt for your time,
>>>>>>>
>>>>>>> What you describe seems to me the ideal approach.
>>>>>>>
>>>>>>> 1) Add a particle field 'ghost' that identifies ghost vs owned
>>>>>>> particles. I think it needs options OWNED, OVERLAP, and GHOST
>>>>>>>
>>>>>>> This means, locally, I need to allocate Nlocal + ghost particles
>>>>>>> (duplicated) for my model?
>>>>>>>
>>>>>>
>>>>>> I would do it another way. I would allocate the particles with no
>>>>>> overlap and set them up. Then I would identify the halo particles, mark
>>>>>> them as OVERLAP, call DMSwarmMigrate(), and mark the migrated particles as
>>>>>> GHOST, then unmark the OVERLAP particles. Shoot! That marking will not work
>>>>>> since we cannot tell the difference between particles we received and
>>>>>> particles we sent. Okay, instead of the `ghost` field we need an `owner
>>>>>> rank` field. So then we
>>>>>>
>>>>>> 1) Setup the non-overlapping particles
>>>>>>
>>>>>> 2) Identify the halo particles
>>>>>>
>>>>>> 3) Change the `rank`, but not the `owner rank`
>>>>>>
>>>>>> 4) Call DMSwarmMigrate()
>>>>>>
>>>>>> Now we can identify ghost particles by the `owner rank`
>>>>>>
>>>>>>
>>>>>>> If that so, how to do the communication between the ghost particles
>>>>>>> living in the rank i and their “real” counterpart in the rank j.
>>>>>>>
>>>>>>> Algo, as an alternative, what about:
>>>>>>> 1) Use an IS tag which contains, for each rank, a list of the
>>>>>>> global index of the neighbors particles outside of the rank.
>>>>>>> 2) Use VecCreateGhost to create a new vector which contains extra
>>>>>>> local space for the ghost components of the vector.
>>>>>>> 3) Use VecScatterCreate, VecScatterBegin, and VecScatterEnd to do
>>>>>>> the transference of data between a vector obtained with
>>>>>>> DMSwarmCreateGlobalVectorFromField
>>>>>>> 4) Do necessary computations using the vectors created with
>>>>>>> VecCreateGhost.
>>>>>>>
>>>>>>
>>>>>> This is essentially what Migrate() does. I was trying to reuse the
>>>>>> code.
>>>>>>
>>>>>>   Thanks,
>>>>>>
>>>>>>      Matt
>>>>>>
>>>>>>
>>>>>>> Thanks,
>>>>>>> Miguel
>>>>>>>
>>>>>>> On Aug 2, 2024, at 8:58 AM, Matthew Knepley <knepley at gmail.com>
>>>>>>> wrote:
>>>>>>>
>>>>>>> On Thu, Aug 1, 2024 at 4:40 PM MIGUEL MOLINOS PEREZ <mmolinos at us.es>
>>>>>>> wrote:
>>>>>>>
>>>>>>>> This Message Is From an External Sender
>>>>>>>> This message came from outside your organization.
>>>>>>>>
>>>>>>>>
>>>>>>>> Dear all,
>>>>>>>>
>>>>>>>> I am implementing a Molecular Dynamics (MD) code using the DMSWARM interface. In the MD simulations we evaluate on each particle (atoms) some kind of scalar functional using data from the neighbouring atoms. My problem lies in the parallel implementation of the model, because sometimes, some of these neighbours lie on a different processor.
>>>>>>>>
>>>>>>>> This is usually solved by using ghost particles.  A similar approach (with nodes instead) is already implemented for other PETSc mesh structures like DMPlexConstructGhostCells. Unfortunately, I don't see this kind of constructs for DMSWARM. Am I missing something?
>>>>>>>>
>>>>>>>> I this could be done by applying a buffer region by exploiting the background DMDA mesh that I already use to do domain decomposition. Then using the buffer region of each cell to locate the ghost particles and finally using VecCreateGhost. Is this feasible? Or is there an easier approach using other PETSc functions.
>>>>>>>>
>>>>>>>>
>>>>>>> This is feasible, but it would be good to develop a set of best
>>>>>>> practices, since we have been mainly focused on the case of non-redundant
>>>>>>> particles. Here is how I think I would do what you want.
>>>>>>>
>>>>>>> 1) Add a particle field 'ghost' that identifies ghost vs owned
>>>>>>> particles. I think it needs options OWNED, OVERLAP, and GHOST
>>>>>>>
>>>>>>> 2) At some interval identify particles that should be sent to other
>>>>>>> processes as ghosts. I would call these "overlap particles". The
>>>>>>> determination
>>>>>>>     seems application specific, so I would leave this determination
>>>>>>> to the user right now. We do two things to these particles
>>>>>>>
>>>>>>>     a) Mark chosen particles as OVERLAP
>>>>>>>
>>>>>>>     b) Change rank to process we are sending to
>>>>>>>
>>>>>>> 3) Call DMSwarmMigrate with PETSC_FALSE for the particle deletion
>>>>>>> flag
>>>>>>>
>>>>>>> 4) Mark OVERLAP particles as GHOST when they arrive
>>>>>>>
>>>>>>> There is one problem in the above algorithm. It does not allow
>>>>>>> sending particles to multiple ranks. We would have to do this
>>>>>>> in phases right now, or make a small adjustment to the interface
>>>>>>> allowing replication of particles when a set of ranks is specified.
>>>>>>>
>>>>>>>   THanks,
>>>>>>>
>>>>>>>      Matt
>>>>>>>
>>>>>>>
>>>>>>>> Thank you,
>>>>>>>> Miguel
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> --
>>>>>>> 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://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63NPQ60c$ 
>>>>>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk62oSLJR_$ >
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> --
>>>>>> 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://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63NPQ60c$ 
>>>>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk62oSLJR_$ >
>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>> --
>>>>> 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://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63NPQ60c$ 
>>>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk62oSLJR_$ >
>>>>>
>>>>>
>>>>>
>>>>>
>>>>
>>>> --
>>>> 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://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63NPQ60c$ 
>>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk62oSLJR_$ >
>>>>
>>>>
>>>>
>>>
>>> --
>>> 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://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63NPQ60c$ 
>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk62oSLJR_$ >
>>>
>>>
>>>
>>
>> --
>> 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://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk63NPQ60c$ 
>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!aIhw4GigszsZG9HmvzDp8Lxt-OKLh4BxuMAUcfn8901TVodKwWeu7X3DFBjBZ5X0IGbICPb6T0PIj0GDVOAk62oSLJR_$ >
>>
>>
>
>
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