[petsc-users] DMPlex: a Mapping Array between Natural and Distributed Cell Index

Matthew Knepley knepley at gmail.com
Mon Jul 18 20:03:24 CDT 2022 

On Mon, Jul 18, 2022 at 7:35 PM Bora Jeong <boraj1021 at gmail.com> wrote:

> Thank you for the reply.
>
> I am actually putting 1dof on the vertex points.
> In the attached sample code, from line 63 to line 99, 1dof on all vertex
> points is defined for the section. But my problem is, if I print out
> Natural SF using the viewer, I only have 10 vertex points, instead of 15.
> Here 15 is the size of the global vector defined by the aforementioned
> section. The working example code was attached to the previous email with a
> small grid.
>

The SF is mapping roots to leaves. The number of roots is the number of
local dofs in the global vector. So there would not be 15, which you say is
the global number of unknowns, but rather the local number of unknowns,
which it seems here is 10.

  Thanks,

     Matt


> Best
>
> On Mon, Jul 18, 2022 at 6:50 PM Matthew Knepley <knepley at gmail.com> wrote:
>
>> On Mon, Jul 18, 2022 at 4:30 PM Bora Jeong <boraj1021 at gmail.com> wrote:
>>
>>> Actually, it seems that my problem is that I do not properly understand
>>> what Natural SF means.
>>>
>>> What I want to achieve is to extract natural vertex index from Natural
>>> SF, and make sure this natural vertex index is the same with what is
>>> written in a mesh file from Gmsh. The natural vertex index defined by Gmsh
>>> is attached as a snapshot with sample code and mesh. I want to reproduce
>>> this indexing based on the Natural SF that Petsc defines.
>>>
>>> I am struggling with that because the number of entries defined in the
>>> Natural SF that I can see through PetscSFView() is different from the
>>> number of vertex points that each processor actually owns. For example,
>>> with two processors and with the attached "square.msh", each processor
>>> might have 15 vertex points if I configure Petsc with ParMetis. However,
>>> the defined Natural SF for proc=0 only has 10 roots. I expected, for each
>>> processor, the Natural SF would show me 15 entries through PetscSFView(),
>>> if I define 1 degree of freedom on vertex when the Petsc Section is
>>> defined.
>>>
>>> When Petsc defines the Natural SF, are there any omitted components
>>> since they are trivial to save?
>>>
>>
>> The naturalSF is mapping the field defined by the Section, not the mesh.
>> You can get the effect you are asking for by putting one
>> degree of freedom (dof) on every vertex for the Section. How are you
>> defining your Section now?
>>
>>    Thanks,
>>
>>       Matt
>>
>>
>>> Best
>>>
>>> On Mon, Jul 18, 2022 at 3:50 PM Matthew Knepley <knepley at gmail.com>
>>> wrote:
>>>
>>>> On Mon, Jul 18, 2022 at 12:14 PM Bora Jeong <boraj1021 at gmail.com>
>>>> wrote:
>>>>
>>>>> Thank you for the corrections. It works.
>>>>>
>>>>> However, I still have a problem.
>>>>> I tested zero stratum for DMGetStratumIS() with the default depth
>>>>> label, since the graph of vertex is needed. Then, PetscSFGetGraph() from
>>>>> the Natural SF is crashed. I checked that pBcCompIS contains proper set of
>>>>> integers via ISView().
>>>>>
>>>>> On the other hand, it works okay if DMGetStratumIS() is with dim
>>>>> stratum, which is for cell stratum.
>>>>> The problem is that if the dim stratum is used, the number of entry in
>>>>> ileaves(i)% pointer does not match with the actual number of vertex that
>>>>> each processor has. That is why I tried to put the zero stratum on
>>>>> DMGetStratumIS(), instead of the stratum for cell (i.e., "dim") to see if
>>>>> any changes on the graph. Can I get comments?
>>>>>
>>>>
>>>> I cannot understand your question. I am not sure what you are using the
>>>> set of vertices or cell for.
>>>>
>>>>   Thanks,
>>>>
>>>>      Matt
>>>>
>>>>
>>>>> Best
>>>>>
>>>>> On Sun, Jul 17, 2022 at 9:59 AM Matthew Knepley <knepley at gmail.com>
>>>>> wrote:
>>>>>
>>>>>> On Fri, Jul 15, 2022 at 7:05 PM Bora Jeong <boraj1021 at gmail.com>
>>>>>> wrote:
>>>>>>
>>>>>>> I found that iroots() and ileaves() have size of nleaves and tested
>>>>>>> through the attached sample code with the grid previously shared. Then I
>>>>>>> get the results written in the attached monitor file.
>>>>>>>
>>>>>>> It seems ileaves(i)%rank and ileaves(i)%index at line 127 of the
>>>>>>> attached code have garbage values, different from displayed values by
>>>>>>> PetscSFView() at line 113.
>>>>>>>
>>>>>>> It is tough to get it why the garbage values are returned from
>>>>>>> PetscSFGetGraph(). Any comments will be very appreciated.
>>>>>>>
>>>>>>
>>>>>> Unfortunately, Fortran is not very good at reporting declaration
>>>>>> errors. The problem is that you did not include or use the Vec module. I
>>>>>> have done this and your example runs for me. I have included the modified
>>>>>> code.
>>>>>>
>>>>>>   Thanks,
>>>>>>
>>>>>>      Matt
>>>>>>
>>>>>>
>>>>>>> Best
>>>>>>>
>>>>>>> On Fri, Jul 15, 2022 at 3:53 PM Matthew Knepley <knepley at gmail.com>
>>>>>>> wrote:
>>>>>>>
>>>>>>>> On Fri, Jul 15, 2022 at 2:25 PM Bora Jeong <boraj1021 at gmail.com>
>>>>>>>> wrote:
>>>>>>>>
>>>>>>>>> Thank you for the comments. Updated sample code is attached here;
>>>>>>>>> DMPlexDistribute() is explicitly called and by defining a section before
>>>>>>>>> mesh distribution, a Natural SF was able to be defined as found from
>>>>>>>>> PetscSFView().
>>>>>>>>>
>>>>>>>>> However, I am still struggling to call PetscSFGetGraph()
>>>>>>>>> https://petsc.org/main/docs/manualpages/PetscSF/PetscSFGetGraph/
>>>>>>>>> due to data type definition of ilocal and iremote. What is the
>>>>>>>>> proper size allocation for those two variables? Does this size for the
>>>>>>>>> number of processors? or the number of vertex?
>>>>>>>>>
>>>>>>>>
>>>>>>>> Since we need to pass back arrays, you need to pass us in F90
>>>>>>>> pointers. Here is an example of doing such a thing:
>>>>>>>>
>>>>>>>>
>>>>>>>> https://gitlab.com/petsc/petsc/-/blob/main/src/vec/is/sf/tutorials/ex1f.F90#L94
>>>>>>>>
>>>>>>>>   Thanks,
>>>>>>>>
>>>>>>>>       Matt
>>>>>>>>
>>>>>>>>
>>>>>>>>> Best
>>>>>>>>>
>>>>>>>>> On Fri, Jul 15, 2022 at 9:09 AM Matthew Knepley <knepley at gmail.com>
>>>>>>>>> wrote:
>>>>>>>>>
>>>>>>>>>> On Fri, Jul 15, 2022 at 8:46 AM Bora Jeong <boraj1021 at gmail.com>
>>>>>>>>>> wrote:
>>>>>>>>>>
>>>>>>>>>>> Okay, I got it what's happening. First, this very bright
>>>>>>>>>>> functionality of petsc (the natural-global converting) needs to be
>>>>>>>>>>> documented in a better way for sure. Currently, it is very difficult to
>>>>>>>>>>> use/follow this features as an external user. Hope this will move forward
>>>>>>>>>>> in a better way.
>>>>>>>>>>>
>>>>>>>>>>> Then next question is if I load/distribute mesh just like I am
>>>>>>>>>>> doing right now shown in the sample code, can I assume that my code does
>>>>>>>>>>> not create natural sf during the distribution(always)? In other words, can
>>>>>>>>>>> I always get the natural order of each node by simply stacking the
>>>>>>>>>>> preceding processor's number of node? For example, for proc=0, natural node
>>>>>>>>>>> ID might be just 1 to nnode_proc_0,
>>>>>>>>>>> for proc=1, it might be {nnode_proc_0 + 1 to nnode_proc_0 +
>>>>>>>>>>> nnode_proc_1} and so on.
>>>>>>>>>>>
>>>>>>>>>>> Does that always make sense in this case?
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> No, but if you call DMPlexDistribute() yourself, rather than
>>>>>>>>>> having it called automatically by DMSetFromOptions(), you can
>>>>>>>>>> preserve the mapping:
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> https://petsc.org/main/docs/manualpages/DMPLEX/DMPlexDistribute/
>>>>>>>>>>
>>>>>>>>>> The SF returned maps the original point distribution, which is in
>>>>>>>>>> the same order as the file, to the redistributed points, which are
>>>>>>>>>> determined
>>>>>>>>>> by the mesh partitioner.
>>>>>>>>>>
>>>>>>>>>>   Thanks,
>>>>>>>>>>
>>>>>>>>>>       Matt
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>> Best
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> On Fri, Jul 15, 2022 at 8:07 AM Matthew Knepley <
>>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> On Fri, Jul 15, 2022 at 7:17 AM Bora Jeong <boraj1021 at gmail.com>
>>>>>>>>>>>> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>> A sample code for loading dm and declaring Natural SF from a
>>>>>>>>>>>>> sphere mesh is attached here. PetscSFView() returns NULL from sf_nat.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> The Global-to-Natural mapping relates global dofs to natural
>>>>>>>>>>>> dofs. Thus, in order to compute this mapping the DM has to know
>>>>>>>>>>>> about the dof layout before distribution. This means you need
>>>>>>>>>>>> to setup a local section before distributing, as we do for example in
>>>>>>>>>>>> Plex test ex15. This makes things more complicated since
>>>>>>>>>>>> everything cannot be packaged up into DMSetFromOptions(), but we need
>>>>>>>>>>>> user input so I do not see a simpler way to do things.
>>>>>>>>>>>>
>>>>>>>>>>>>   Thanks,
>>>>>>>>>>>>
>>>>>>>>>>>>     Matt
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>> Best
>>>>>>>>>>>>>
>>>>>>>>>>>>> On Fri, Jul 15, 2022 at 6:39 AM Matthew Knepley <
>>>>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> On Thu, Jul 14, 2022 at 8:25 PM Bora Jeong <
>>>>>>>>>>>>>> boraj1021 at gmail.com> wrote:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Okay, I checked it and you are correct. In my case, simply,
>>>>>>>>>>>>>>> natural node index can be identified by stacking all the preceding
>>>>>>>>>>>>>>> processor's numbers of nodes for a particular processor, which is good due
>>>>>>>>>>>>>>> to simplicity. However, one serious question is why this is happening in my
>>>>>>>>>>>>>>> code? In other words, why the natural SF is not created during the mesh
>>>>>>>>>>>>>>> distribution? My code wants to have consistency in dealing with this
>>>>>>>>>>>>>>> natural indexing for several different kinds of mesh files. So there is a
>>>>>>>>>>>>>>> necessity to guarantee of consistency in this weird behavior.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> I can't tell what is going on in your code unless I can run
>>>>>>>>>>>>>> it. Do you have a simple example?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>   Thanks,
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>      Matt
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Best,
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> On Thu, Jul 14, 2022 at 6:43 PM Matthew Knepley <
>>>>>>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> On Thu, Jul 14, 2022 at 5:47 PM Bora Jeong <
>>>>>>>>>>>>>>>> boraj1021 at gmail.com> wrote:
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Thank you for the comments. I have these errors when I
>>>>>>>>>>>>>>>>> call PetscSFView() after DMGetNaturalSF()
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: --------------------- Error Message
>>>>>>>>>>>>>>>>> --------------------------------------------------------------
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: Null argument, when expecting valid pointer
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: Null Pointer: Parameter # 1
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: See https://petsc.org/release/faq/ for
>>>>>>>>>>>>>>>>> trouble shooting.
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: Petsc Release Version 3.17.0, unknown
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: #1 PetscSFView() at
>>>>>>>>>>>>>>>>> [2]PETSC ERROR: #2 User provided function() at User file:0
>>>>>>>>>>>>>>>>> Abort(85) on node 2 (rank 0 in comm 16): application
>>>>>>>>>>>>>>>>> called MPI_Abort(MPI_COMM_SELF, 85) - process 0
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Clearly NULL was returned, which means no natural SF was
>>>>>>>>>>>>>>>> created.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Below is the structure to load a mesh file from gmsh;
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>   call DMCreate(PETSC_COMM_WORLD, dm, ierr);CHKERRA(ierr)
>>>>>>>>>>>>>>>>>   call DMSetType(dm, plex, ierr);CHKERRA(ierr)
>>>>>>>>>>>>>>>>>   call DMSetUseNatural(dm, PETSC_TRUE, ierr);CHKERRA(ierr)
>>>>>>>>>>>>>>>>>   call DMSetFromOptions(dm, ierr);CHKERRA(ierr)
>>>>>>>>>>>>>>>>>   call DMGetNaturalSF(dm, sf_nat, ierr);CHKERRA(ierr)
>>>>>>>>>>>>>>>>>   call PetscSFView(sf_nat, PETSC_VIEWER_STDOUT_WORLD,
>>>>>>>>>>>>>>>>> ierr);CHKERRA(ierr)
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> The natural SF is created during mesh distribution. That
>>>>>>>>>>>>>>>> has not happened here. This means that
>>>>>>>>>>>>>>>> the order of cells is identical to the file it was read
>>>>>>>>>>>>>>>> from.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>   Thanks,
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>       Matt
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Best
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> On Thu, Jul 14, 2022 at 10:49 AM Matthew Knepley <
>>>>>>>>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> On Wed, Jul 13, 2022 at 10:17 PM Bora Jeong <
>>>>>>>>>>>>>>>>>> boraj1021 at gmail.com> wrote:
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Dear petsc team,
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> I am a user of DMPlex for a finite volume code and there
>>>>>>>>>>>>>>>>>>> is a necessity to know global index of each cell. Here the global index
>>>>>>>>>>>>>>>>>>> means the indexing that can be found from a mesh file itself without
>>>>>>>>>>>>>>>>>>> distribution over processors. It seems petsc community denotes this
>>>>>>>>>>>>>>>>>>> indexing term as "natural".
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> What I want to do is to create a local array (not petsc
>>>>>>>>>>>>>>>>>>> vector but just an array variable in the program) to map distributed cell
>>>>>>>>>>>>>>>>>>> ID to natual cell ID, for example, an array "A";
>>>>>>>>>>>>>>>>>>> A(distributed_node_ID) = natural_node_ID
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> There are some petsc functions to support mapping
>>>>>>>>>>>>>>>>>>> between global and natural vectors. However, I just need to define the
>>>>>>>>>>>>>>>>>>> array "A" as above example. To achieve this, what is a proper/smart way? In
>>>>>>>>>>>>>>>>>>> other words, how can I extract the natural_cell_ID from a distributed
>>>>>>>>>>>>>>>>>>> local_cell_ID?
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> I turned on DMSetUseNatural(DM, PETSC_TRUE) before
>>>>>>>>>>>>>>>>>>> distribution, but after this, defining all the required section and star
>>>>>>>>>>>>>>>>>>> forest objects to get natural and global vectors seems not that direct way
>>>>>>>>>>>>>>>>>>> for my purpose, which is just to extract the above mapping array "A". Can I
>>>>>>>>>>>>>>>>>>> get any comments about it?
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> There is only one thing created, the sfNatural PetscSF
>>>>>>>>>>>>>>>>>> object, which you can get with DMGetNaturalSF(). The roots of this SF are
>>>>>>>>>>>>>>>>>> the global numbers of dofs stored in PETSc vectors, and
>>>>>>>>>>>>>>>>>> the leaves are natural numbers for these dofs. Thus, when we map global
>>>>>>>>>>>>>>>>>> vectors to natural vectors
>>>>>>>>>>>>>>>>>> in DMPlexGlobalToNaturalBegin/End(), we call PetscSFBcastBegin/End().
>>>>>>>>>>>>>>>>>> Mapping natural to global we call
>>>>>>>>>>>>>>>>>> PetscSFReduceBegin/End(). You could pull the information
>>>>>>>>>>>>>>>>>> out of the SF using PetscSFGetGraph() if you want.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>   Thanks,
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>     Matt
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Regards
>>>>>>>>>>>>>>>>>>> Mo
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> --
>>>>>>>>>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> --
>>>>>>>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> --
>>>>>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> --
>>>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>>>
>>>>>>>
>>>>>>
>>>>>> --
>>>>>> 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.cse.buffalo.edu/~knepley/>
>>>>>>
>>>>>
>>>>
>>>> --
>>>> 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.cse.buffalo.edu/~knepley/>
>>>>
>>>
>>
>> --
>> 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.cse.buffalo.edu/~knepley/>
>>
>

-- 
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.cse.buffalo.edu/~knepley/>
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