[petsc-users] How to find the map between the high order coordinates of DMPlex and vertex numbering?
Matthew Knepley
knepley at gmail.com
Mon May 15 04:24:44 CDT 2023
On Sun, May 14, 2023 at 7:23 PM Zongze Yang <yangzongze at gmail.com> wrote:
> Could you try to project the coordinates into the continuity space by
> enabling the option
> `-dm_plex_gmsh_project_petscdualspace_lagrange_continuity true`?
>
There is a comment in the code about that:
/* XXX FIXME Requires DMPlexSetClosurePermutationLexicographic() */
So what is currently done is you project into the discontinuous space from
the GMsh coordinates,
and then we get the continuous coordinates from those later. This is why we
get the right answer.
Thanks,
Matt
> Best wishes,
> Zongze
>
>
> On Mon, 15 May 2023 at 04:24, Matthew Knepley <knepley at gmail.com> wrote:
>
>> On Sun, May 14, 2023 at 12:27 PM Zongze Yang <yangzongze at gmail.com>
>> wrote:
>>
>>>
>>>
>>>
>>> On Sun, 14 May 2023 at 23:54, Matthew Knepley <knepley at gmail.com> wrote:
>>>
>>>> On Sun, May 14, 2023 at 9:21 AM Zongze Yang <yangzongze at gmail.com>
>>>> wrote:
>>>>
>>>>> Hi, Matt,
>>>>>
>>>>> The issue has been resolved while testing on the latest version of
>>>>> PETSc. It seems that the problem has been fixed in the following merge
>>>>> request: https://gitlab.com/petsc/petsc/-/merge_requests/5970
>>>>>
>>>>
>>>> No problem. Glad it is working.
>>>>
>>>>
>>>>> I sincerely apologize for any inconvenience caused by my previous
>>>>> message. However, I would like to provide you with additional information
>>>>> regarding the test files. Attached to this email, you will find two Gmsh
>>>>> files: "square_2rd.msh" and "square_3rd.msh." These files contain
>>>>> high-order triangulated mesh data for the unit square.
>>>>>
>>>>> ```
>>>>> $ ./ex33 -coord_space 0 -dm_plex_filename square_2rd.msh
>>>>> -dm_plex_gmsh_project
>>>>> -dm_plex_gmsh_project_petscdualspace_lagrange_continuity true
>>>>> -dm_plex_gmsh_project_fe_view -volume 1
>>>>> PetscFE Object: P2 1 MPI process
>>>>> type: basic
>>>>> Basic Finite Element in 2 dimensions with 2 components
>>>>> PetscSpace Object: P2 1 MPI process
>>>>> type: sum
>>>>> Space in 2 variables with 2 components, size 12
>>>>> Sum space of 2 concatenated subspaces (all identical)
>>>>> PetscSpace Object: sum component (sumcomp_) 1 MPI process
>>>>> type: poly
>>>>> Space in 2 variables with 1 components, size 6
>>>>> Polynomial space of degree 2
>>>>> PetscDualSpace Object: P2 1 MPI process
>>>>> type: lagrange
>>>>> Dual space with 2 components, size 12
>>>>> Continuous Lagrange dual space
>>>>> Quadrature on a triangle of order 5 on 9 points (dim 2)
>>>>> Volume: 1.
>>>>> $ ./ex33 -coord_space 0 -dm_plex_filename square_3rd.msh
>>>>> -dm_plex_gmsh_project
>>>>> -dm_plex_gmsh_project_petscdualspace_lagrange_continuity true
>>>>> -dm_plex_gmsh_project_fe_view -volume 1
>>>>> PetscFE Object: P3 1 MPI process
>>>>> type: basic
>>>>> Basic Finite Element in 2 dimensions with 2 components
>>>>> PetscSpace Object: P3 1 MPI process
>>>>> type: sum
>>>>> Space in 2 variables with 2 components, size 20
>>>>> Sum space of 2 concatenated subspaces (all identical)
>>>>> PetscSpace Object: sum component (sumcomp_) 1 MPI process
>>>>> type: poly
>>>>> Space in 2 variables with 1 components, size 10
>>>>> Polynomial space of degree 3
>>>>> PetscDualSpace Object: P3 1 MPI process
>>>>> type: lagrange
>>>>> Dual space with 2 components, size 20
>>>>> Continuous Lagrange dual space
>>>>> Quadrature on a triangle of order 7 on 16 points (dim 2)
>>>>> Volume: 1.
>>>>> ```
>>>>>
>>>>> Thank you for your attention and understanding. I apologize once again
>>>>> for my previous oversight.
>>>>>
>>>>
>>>> Great! If you make an MR for this, you will be included on the next
>>>> list of PETSc contributors. Otherwise, I can do it.
>>>>
>>>>
>>> I appreciate your offer to handle the MR. Please go ahead and take care
>>> of it. Thank you!
>>>
>>
>> I have created the MR with your tests. They are working for me:
>>
>> https://gitlab.com/petsc/petsc/-/merge_requests/6463
>>
>> Thanks,
>>
>> Matt
>>
>>
>>> Best Wishes,
>>> Zongze
>>>
>>>
>>>> Thanks,
>>>>
>>>> Matt
>>>>
>>>>
>>>>> Best wishes,
>>>>> Zongze
>>>>>
>>>>>
>>>>> On Sun, 14 May 2023 at 16:44, Matthew Knepley <knepley at gmail.com>
>>>>> wrote:
>>>>>
>>>>>> On Sat, May 13, 2023 at 6:08 AM Zongze Yang <yangzongze at gmail.com>
>>>>>> wrote:
>>>>>>
>>>>>>> Hi, Matt,
>>>>>>>
>>>>>>> There seem to be ongoing issues with projecting high-order
>>>>>>> coordinates from a gmsh file to other spaces. I would like to inquire
>>>>>>> whether there are any plans to resolve this problem.
>>>>>>>
>>>>>>> Thank you for your attention to this matter.
>>>>>>>
>>>>>>
>>>>>> Yes, I will look at it. The important thing is to have a good test.
>>>>>> Here are the higher order geometry tests
>>>>>>
>>>>>>
>>>>>> https://gitlab.com/petsc/petsc/-/blob/main/src/dm/impls/plex/tests/ex33.c
>>>>>>
>>>>>> I take shapes with known volume, mesh them with higher order
>>>>>> geometry, and look at the convergence to the true volume. Could you add a
>>>>>> GMsh test, meaning the .msh file and known volume, and I will fix it?
>>>>>>
>>>>>> Thanks,
>>>>>>
>>>>>> Matt
>>>>>>
>>>>>>
>>>>>>>
>>>>>>> Best wishes,
>>>>>>> Zongze
>>>>>>>
>>>>>>>
>>>>>>> On Sat, 18 Jun 2022 at 20:31, Zongze Yang <yangzongze at gmail.com>
>>>>>>> wrote:
>>>>>>>
>>>>>>>> Thank you for your reply. May I ask for some references on the
>>>>>>>> order of the dofs on PETSc's FE Space (especially high order elements)?
>>>>>>>>
>>>>>>>> Thanks,
>>>>>>>>
>>>>>>>> Zongze
>>>>>>>>
>>>>>>>> Matthew Knepley <knepley at gmail.com> 于2022年6月18日周六 20:02写道：
>>>>>>>>
>>>>>>>>> On Sat, Jun 18, 2022 at 2:16 AM Zongze Yang <yangzongze at gmail.com>
>>>>>>>>> wrote:
>>>>>>>>>
>>>>>>>>>> In order to check if I made mistakes in the python code, I try to
>>>>>>>>>> use c code to show the issue on DMProjectCoordinates. The code and mesh
>>>>>>>>>> file is attached.
>>>>>>>>>> If the code is correct, there must be something wrong with
>>>>>>>>>> `DMProjectCoordinates` or `DMPlexCreateGmshFromFile` for high-order mesh.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Something is definitely wrong with high order, periodic simplices
>>>>>>>>> from Gmsh. We had not tested that case. I am at a conference and cannot
>>>>>>>>> look at it for a week.
>>>>>>>>> My suspicion is that the space we make when reading in the Gmsh
>>>>>>>>> coordinates does not match the values (wrong order).
>>>>>>>>>
>>>>>>>>> Thanks,
>>>>>>>>>
>>>>>>>>> Matt
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>> The command and the output are listed below: (Obviously the
>>>>>>>>>> bounding box is changed.)
>>>>>>>>>> ```
>>>>>>>>>> $ ./test_gmsh_load_2rd -filename cube-p2.msh -old_fe_view
>>>>>>>>>> -new_fe_view
>>>>>>>>>> Old Bounding Box:
>>>>>>>>>> 0: lo = 0. hi = 1.
>>>>>>>>>> 1: lo = 0. hi = 1.
>>>>>>>>>> 2: lo = 0. hi = 1.
>>>>>>>>>> PetscFE Object: OldCoordinatesFE 1 MPI processes
>>>>>>>>>> type: basic
>>>>>>>>>> Basic Finite Element in 3 dimensions with 3 components
>>>>>>>>>> PetscSpace Object: P2 1 MPI processes
>>>>>>>>>> type: sum
>>>>>>>>>> Space in 3 variables with 3 components, size 30
>>>>>>>>>> Sum space of 3 concatenated subspaces (all identical)
>>>>>>>>>> PetscSpace Object: sum component (sumcomp_) 1 MPI processes
>>>>>>>>>> type: poly
>>>>>>>>>> Space in 3 variables with 1 components, size 10
>>>>>>>>>> Polynomial space of degree 2
>>>>>>>>>> PetscDualSpace Object: P2 1 MPI processes
>>>>>>>>>> type: lagrange
>>>>>>>>>> Dual space with 3 components, size 30
>>>>>>>>>> Discontinuous Lagrange dual space
>>>>>>>>>> Quadrature of order 5 on 27 points (dim 3)
>>>>>>>>>> PetscFE Object: NewCoordinatesFE 1 MPI processes
>>>>>>>>>> type: basic
>>>>>>>>>> Basic Finite Element in 3 dimensions with 3 components
>>>>>>>>>> PetscSpace Object: P2 1 MPI processes
>>>>>>>>>> type: sum
>>>>>>>>>> Space in 3 variables with 3 components, size 30
>>>>>>>>>> Sum space of 3 concatenated subspaces (all identical)
>>>>>>>>>> PetscSpace Object: sum component (sumcomp_) 1 MPI processes
>>>>>>>>>> type: poly
>>>>>>>>>> Space in 3 variables with 1 components, size 10
>>>>>>>>>> Polynomial space of degree 2
>>>>>>>>>> PetscDualSpace Object: P2 1 MPI processes
>>>>>>>>>> type: lagrange
>>>>>>>>>> Dual space with 3 components, size 30
>>>>>>>>>> Continuous Lagrange dual space
>>>>>>>>>> Quadrature of order 5 on 27 points (dim 3)
>>>>>>>>>> New Bounding Box:
>>>>>>>>>> 0: lo = 2.5624e-17 hi = 8.
>>>>>>>>>> 1: lo = -9.23372e-17 hi = 7.
>>>>>>>>>> 2: lo = 2.72091e-17 hi = 8.5
>>>>>>>>>> ```
>>>>>>>>>>
>>>>>>>>>> Thanks,
>>>>>>>>>> Zongze
>>>>>>>>>>
>>>>>>>>>> Zongze Yang <yangzongze at gmail.com> 于2022年6月17日周五 14:54写道：
>>>>>>>>>>
>>>>>>>>>>> I tried the projection operation. However, it seems that the
>>>>>>>>>>> projection gives the wrong solution. After projection, the bounding box is
>>>>>>>>>>> changed! See logs below.
>>>>>>>>>>>
>>>>>>>>>>> First, I patch the petsc4py by adding `DMProjectCoordinates`:
>>>>>>>>>>> ```
>>>>>>>>>>> diff --git a/src/binding/petsc4py/src/PETSc/DM.pyx
>>>>>>>>>>> b/src/binding/petsc4py/src/PETSc/DM.pyx
>>>>>>>>>>> index d8a58d183a..dbcdb280f1 100644
>>>>>>>>>>> --- a/src/binding/petsc4py/src/PETSc/DM.pyx
>>>>>>>>>>> +++ b/src/binding/petsc4py/src/PETSc/DM.pyx
>>>>>>>>>>> @@ -307,6 +307,12 @@ cdef class DM(Object):
>>>>>>>>>>> PetscINCREF(c.obj)
>>>>>>>>>>> return c
>>>>>>>>>>>
>>>>>>>>>>> + def projectCoordinates(self, FE fe=None):
>>>>>>>>>>> + if fe is None:
>>>>>>>>>>> + CHKERR( DMProjectCoordinates(self.dm, NULL) )
>>>>>>>>>>> + else:
>>>>>>>>>>> + CHKERR( DMProjectCoordinates(self.dm, fe.fe) )
>>>>>>>>>>> +
>>>>>>>>>>> def getBoundingBox(self):
>>>>>>>>>>> cdef PetscInt i,dim=0
>>>>>>>>>>> CHKERR( DMGetCoordinateDim(self.dm, &dim) )
>>>>>>>>>>> diff --git a/src/binding/petsc4py/src/PETSc/petscdm.pxi
>>>>>>>>>>> b/src/binding/petsc4py/src/PETSc/petscdm.pxi
>>>>>>>>>>> index 514b6fa472..c778e39884 100644
>>>>>>>>>>> --- a/src/binding/petsc4py/src/PETSc/petscdm.pxi
>>>>>>>>>>> +++ b/src/binding/petsc4py/src/PETSc/petscdm.pxi
>>>>>>>>>>> @@ -90,6 +90,7 @@ cdef extern from * nogil:
>>>>>>>>>>> int DMGetCoordinateDim(PetscDM,PetscInt*)
>>>>>>>>>>> int DMSetCoordinateDim(PetscDM,PetscInt)
>>>>>>>>>>> int DMLocalizeCoordinates(PetscDM)
>>>>>>>>>>> + int DMProjectCoordinates(PetscDM, PetscFE)
>>>>>>>>>>>
>>>>>>>>>>> int
>>>>>>>>>>> DMCreateInterpolation(PetscDM,PetscDM,PetscMat*,PetscVec*)
>>>>>>>>>>> int DMCreateInjection(PetscDM,PetscDM,PetscMat*)
>>>>>>>>>>> ```
>>>>>>>>>>>
>>>>>>>>>>> Then in python, I load a mesh and project the coordinates to P2:
>>>>>>>>>>> ```
>>>>>>>>>>> import firedrake as fd
>>>>>>>>>>> from firedrake.petsc import PETSc
>>>>>>>>>>>
>>>>>>>>>>> # plex = fd.mesh._from_gmsh('test-fd-load-p2.msh')
>>>>>>>>>>> plex = fd.mesh._from_gmsh('test-fd-load-p2-rect.msh')
>>>>>>>>>>> print('old bbox:', plex.getBoundingBox())
>>>>>>>>>>>
>>>>>>>>>>> dim = plex.getDimension()
>>>>>>>>>>> # (dim, nc, isSimplex, k,
>>>>>>>>>>> qorder, comm=None)
>>>>>>>>>>> fe_new = PETSc.FE().createLagrange(dim, dim, True, 2,
>>>>>>>>>>> PETSc.DETERMINE)
>>>>>>>>>>> plex.projectCoordinates(fe_new)
>>>>>>>>>>> fe_new.view()
>>>>>>>>>>>
>>>>>>>>>>> print('new bbox:', plex.getBoundingBox())
>>>>>>>>>>> ```
>>>>>>>>>>>
>>>>>>>>>>> The output is (The bounding box is changed!)
>>>>>>>>>>> ```
>>>>>>>>>>>
>>>>>>>>>>> old bbox: ((0.0, 1.0), (0.0, 1.0), (0.0, 1.0))
>>>>>>>>>>> PetscFE Object: P2 1 MPI processes
>>>>>>>>>>> type: basic
>>>>>>>>>>> Basic Finite Element in 3 dimensions with 3 components
>>>>>>>>>>> PetscSpace Object: P2 1 MPI processes
>>>>>>>>>>> type: sum
>>>>>>>>>>> Space in 3 variables with 3 components, size 30
>>>>>>>>>>> Sum space of 3 concatenated subspaces (all identical)
>>>>>>>>>>> PetscSpace Object: sum component (sumcomp_) 1 MPI processes
>>>>>>>>>>> type: poly
>>>>>>>>>>> Space in 3 variables with 1 components, size 10
>>>>>>>>>>> Polynomial space of degree 2
>>>>>>>>>>> PetscDualSpace Object: P2 1 MPI processes
>>>>>>>>>>> type: lagrange
>>>>>>>>>>> Dual space with 3 components, size 30
>>>>>>>>>>> Continuous Lagrange dual space
>>>>>>>>>>> Quadrature of order 5 on 27 points (dim 3)
>>>>>>>>>>> new bbox: ((-6.530133708576188e-17, 36.30670832662781), (-3.899962995254311e-17, 36.2406171632539), (-8.8036464152166e-17, 36.111577025012224))
>>>>>>>>>>>
>>>>>>>>>>> ```
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> By the way, for the original DG coordinates, where can I find the relation of the closure and the order of the dofs for the cell?
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> Thanks!
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> Zongze
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> Matthew Knepley <knepley at gmail.com> 于2022年6月17日周五 01:11写道：
>>>>>>>>>>>
>>>>>>>>>>>> On Thu, Jun 16, 2022 at 12:06 PM Zongze Yang <
>>>>>>>>>>>> yangzongze at gmail.com> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> 在 2022年6月16日，23:22，Matthew Knepley <knepley at gmail.com> 写道：
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> On Thu, Jun 16, 2022 at 11:11 AM Zongze Yang <
>>>>>>>>>>>>> yangzongze at gmail.com> wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Hi, if I load a `gmsh` file with second-order elements, the
>>>>>>>>>>>>>> coordinates will be stored in a DG-P2 space. After obtaining the
>>>>>>>>>>>>>> coordinates of a cell, how can I map the coordinates to vertex and edge?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> By default, they are stored as P2, not DG.
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> I checked the coordinates vector, and found the dogs only
>>>>>>>>>>>>> defined on cell other than vertex and edge, so I said they are stored as DG.
>>>>>>>>>>>>> Then the function DMPlexVecGetClosure
>>>>>>>>>>>>> <https://petsc.org/main/docs/manualpages/DMPLEX/DMPlexVecGetClosure/> seems return
>>>>>>>>>>>>> the coordinates in lex order.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Some code in reading gmsh file reads that
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> 1756: if (isSimplex) continuity = PETSC_FALSE
>>>>>>>>>>>>> <https://petsc.org/main/docs/manualpages/Sys/PETSC_FALSE/>;
>>>>>>>>>>>>> /* XXX FIXME Requires DMPlexSetClosurePermutationLexicographic() */
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> 1758: GmshCreateFE(comm, NULL, isSimplex, continuity,
>>>>>>>>>>>>> nodeType, dim, coordDim, order, &fe)
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> The continuity is set to false for simplex.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Oh, yes. That needs to be fixed. For now, you can just project
>>>>>>>>>>>> it to P2 if you want using
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> https://petsc.org/main/docs/manualpages/DM/DMProjectCoordinates/
>>>>>>>>>>>>
>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>
>>>>>>>>>>>> Matt
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>> Zongze
>>>>>>>>>>>>>
>>>>>>>>>>>>> You can ask for the coordinates of a vertex or an edge
>>>>>>>>>>>>> directly using
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> https://petsc.org/main/docs/manualpages/DMPLEX/DMPlexPointLocalRead/
>>>>>>>>>>>>>
>>>>>>>>>>>>> by giving the vertex or edge point. You can get all the
>>>>>>>>>>>>> coordinates on a cell, in the closure order, using
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> https://petsc.org/main/docs/manualpages/DMPLEX/DMPlexVecGetClosure/
>>>>>>>>>>>>>
>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>>
>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Below is some code load the gmsh file, I want to know the
>>>>>>>>>>>>>> relation between `cl` and `cell_coords`.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> ```
>>>>>>>>>>>>>> import firedrake as fd
>>>>>>>>>>>>>> import numpy as np
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> # Load gmsh file (2rd)
>>>>>>>>>>>>>> plex = fd.mesh._from_gmsh('test-fd-load-p2-rect.msh')
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> cs, ce = plex.getHeightStratum(0)
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> cdm = plex.getCoordinateDM()
>>>>>>>>>>>>>> csec = dm.getCoordinateSection()
>>>>>>>>>>>>>> coords_gvec = dm.getCoordinates()
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> for i in range(cs, ce):
>>>>>>>>>>>>>> cell_coords = cdm.getVecClosure(csec, coords_gvec, i)
>>>>>>>>>>>>>> print(f'coordinates for cell {i}
>>>>>>>>>>>>>> :\n{cell_coords.reshape([-1, 3])}')
>>>>>>>>>>>>>> cl = dm.getTransitiveClosure(i)
>>>>>>>>>>>>>> print('closure:', cl)
>>>>>>>>>>>>>> break
>>>>>>>>>>>>>> ```
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Best wishes,
>>>>>>>>>>>>>> Zongze
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> --
>>>>>>>>>>>>> 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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