[petsc-users] using petsc tools to solve isolated irregular domains with finite difference
Matthew Knepley
knepley at gmail.com
Fri Nov 1 07:30:51 CDT 2013
On Fri, Nov 1, 2013 at 12:08 AM, Bishesh Khanal <bisheshkh at gmail.com> wrote:
> On Thu, Oct 31, 2013 at 11:34 PM, Matthew Knepley <knepley at gmail.com>wrote:
>
>> On Thu, Oct 31, 2013 at 5:02 PM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>>
>>> On Thu, Oct 31, 2013 at 7:43 PM, Matthew Knepley <knepley at gmail.com>wrote:
>>>
>>>> On Thu, Oct 31, 2013 at 1:25 PM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>>>>
>>>>> I did not call DMCreateMatrix() before when using just dmda (and it is
>>>>> working). In any case, I added a call to DMCreateMatrix() now but still
>>>>> there are problems. Here are the code snippets and the error I get:
>>>>>
>>>>
>>>> Then you were allowing it to be called automatically by PETSc, and it
>>>> would have been this time as well.
>>>>
>>>>
>>>>> //Setting up the section and linking it to DM:
>>>>> ierr = PetscSectionCreate(PETSC_COMM_WORLD, &s);CHKERRQ(ierr);
>>>>> ierr = DMDAGetNumCells(dm, NULL, NULL, NULL, &nC);CHKERRQ(ierr);
>>>>> ierr = PetscSectionSetChart(s, 0, nC);CHKERRQ(ierr);
>>>>> for (PetscInt j = info.ys; j < info.ys+info.ym; ++j) {
>>>>> for (PetscInt i = info.xs; i < info.xs+info.xm; ++i) {
>>>>> PetscInt point;
>>>>> if(isPosInDomain(&testPoisson,i,j,0)) {
>>>>> ierr = DMDAGetCellPoint(dm, i, j, 0,
>>>>> &point);CHKERRQ(ierr);
>>>>> ierr = PetscSectionSetDof(s, point, testPoisson.mDof);
>>>>> // No. of dofs associated with the point.
>>>>> }
>>>>> }
>>>>> }
>>>>> ierr = PetscSectionSetUp(s);CHKERRQ(ierr);
>>>>> ierr = DMSetDefaultSection(dm, s);CHKERRQ(ierr);
>>>>> ierr = PetscSectionDestroy(&s);CHKERRQ(ierr);
>>>>> ierr = DMCreateMatrix(dm,&A);CHKERRQ(ierr);
>>>>>
>>>>> //Set up KSP:
>>>>> ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);
>>>>> ierr = KSPSetDM(ksp,dm);CHKERRQ(ierr);
>>>>> ierr =
>>>>> KSPSetComputeOperators(ksp,computeMatrix2dSection,(void*)&testPoisson);CHKERRQ(ierr);
>>>>> ierr =
>>>>> KSPSetComputeRHS(ksp,computeRhs2dSection,(void*)&testPoisson);CHKERRQ(ierr);
>>>>> ierr = KSPSetFromOptions(ksp);CHKERRQ(ierr);
>>>>> ierr = KSPSolve(ksp,NULL,NULL);CHKERRQ(ierr);
>>>>>
>>>>> ------------------------------------------------------
>>>>> The computeMatrix2dSection function has:
>>>>>
>>>>> ierr = KSPGetDM(ksp,&da);CHKERRQ(ierr);
>>>>> ierr = DMDAGetLocalInfo(da,&info);CHKERRQ(ierr);
>>>>> ierr = DMGetDefaultGlobalSection(da,&gs);CHKERRQ(ierr);
>>>>> for(PetscInt j = info.ys; j<info.ys+info.ym; ++j) {
>>>>> for(PetscInt i = info.xs; i<info.xs+info.xm; ++i) {
>>>>> if (isPosInDomain(ctx,i,j)) {
>>>>> ierr = DMDAGetCellPoint(da,i,j,0,&point);CHKERRQ(ierr);
>>>>> ierr = PetscSectionGetOffset(gs,point,&row);
>>>>> ierr = PetscSectionGetDof(gs,point,&dof);
>>>>> for(PetscInt cDof = 0; cDof < dof; ++cDof) {
>>>>> num = 0;
>>>>> row+=cDof;
>>>>> col[num] = row; //(i,j) position
>>>>> v[num++] = -4;
>>>>> if(isPosInDomain(ctx,i+1,j)) {
>>>>> ierr =
>>>>> DMDAGetCellPoint(da,i+1,j,0,&point);CHKERRQ(ierr);
>>>>> ierr =
>>>>> PetscSectionGetOffset(gs,point,&col[num]);
>>>>> col[num] += cDof;
>>>>> v[num++] = 1;
>>>>> }
>>>>> if(isPosInDomain(ctx,i-1,j)) {
>>>>> ierr =
>>>>> DMDAGetCellPoint(da,i-1,j,0,&point);CHKERRQ(ierr);
>>>>> ierr =
>>>>> PetscSectionGetOffset(gs,point,&col[num]);
>>>>> col[num] += cDof;
>>>>> v[num++] = 1;
>>>>> }
>>>>> if(isPosInDomain(ctx,i,j+1)) {
>>>>> ierr =
>>>>> DMDAGetCellPoint(da,i,j+1,0,&point);CHKERRQ(ierr);
>>>>> ierr =
>>>>> PetscSectionGetOffset(gs,point,&col[num]);
>>>>> col[num] += cDof;
>>>>> v[num++] = 1;
>>>>> }
>>>>> if(isPosInDomain(ctx,i,j-1)) {
>>>>> ierr =
>>>>> DMDAGetCellPoint(da,i,j-1,0,&point);CHKERRQ(ierr);
>>>>> ierr =
>>>>> PetscSectionGetOffset(gs,point,&col[num]);
>>>>> col[num] += cDof;
>>>>> v[num++] = 1;
>>>>> }
>>>>> ierr =
>>>>> MatSetValues(A,1,&row,num,col,v,INSERT_VALUES);CHKERRQ(ierr);
>>>>> }
>>>>> }
>>>>> }
>>>>> }
>>>>> ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
>>>>> ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
>>>>>
>>>>> But this is not working. For e.g. for a 6X6 grid size I get the
>>>>> following error:
>>>>>
>>>>
>>>> Okay, here is how to debug this:
>>>>
>>>> 0) Go to a single scalar equations to make things easier
>>>>
>>>> 1) Use MatView(A, PETSC_VIEWER_STDOUT_WORLD), which should output a
>>>> 36 row matrix with
>>>> the preallocated nonzero pattern, filled with zeros
>>>>
>>>> 2) Make sure this is the pattern you want
>>>>
>>>> 3) Run in the debugger with -start_in_debugger
>>>>
>>>> 4) When you get the error, see
>>>>
>>>> a) If the (i, j) is one that should be setting a value
>>>>
>>>> b) Why this (i, j) was not preallocated
>>>>
>>>
>>> Up to 4 (a), it is correct. There is a problem in the way
>>> DMDAGetCellPoint and PetscSectionGetOffset works in this case scenario. I
>>> will try to explain it below for the case of 4X4 grid.
>>> First case:
>>> If I set the computational domain to be all the points of the dmda grid,
>>> (i.e. isPosInDomain(..,i,j,..) returns true for all i and j in the dmda
>>> grid), the program runs fine and does not give any error.
>>>
>>> Second case:
>>> I want the computational domain to be some part of the whole grid. There
>>> is a problem in this case.
>>> The following test is in a case where,
>>> isPosInDomain(..,i,j,..) returns true ONLY for (i,j) pairs of (2,1)
>>> (1,2) (2,2) (3,2) and (3,2). The grid with its corresponding point number
>>> in the petscsection is shown below:
>>>
>>> 12 13 14 15
>>> 8 9 10 11
>>> 4 5 6 7
>>> 0 1 2 3
>>>
>>> of which 6, 9, 10, 11 and 14 correspond to the (i,j) points that returns
>>> true for isPosInDomain(..,i,j,..)
>>> MatView gives me the 16-row matrix with the star stencil non-zero
>>> structure as expected.
>>> The error I get is: new non-zero at (0,2) caused a malloc.
>>>
>>> This error is for the value of (i,j) = (2,1), i.e. point 6.
>>> The loop to set values in the matrix starts as:
>>> for(PetscInt j = info.ys; j<info.ys+info.ym; ++j) {
>>> for(PetscInt i = info.xs; i<info.xs+info.xm; ++i) {
>>> if (isPosInDomain(ctx,i,j)) {
>>> ierr = DMDAGetCellPoint(da,i,j,0,&point);CHKERRQ(ierr);
>>> ierr = PetscSectionGetOffset(gs,point,&row);
>>>
>>> Now, what I wanted from the beginning is to create the matrix containing
>>> the rows corresponding to only those points (i,j) which has true values for
>>> isPosInDomain(ctx,i,j), that means in this case 5 row matrix. In the
>>> current test, the first (i,j) that reaches DMDAGetCellPoint is (2,1), which
>>> gives sets point variable to 6.
>>> The line PetscSectionGetOffset(gs,point,&row) sets the value of row to 0.
>>> So I think this where the inconsistency lies.
>>> MatSetValues(A,1,&row,num,col,v,INSERT_VALUES) expects row variable and
>>> col[] array to correspond to (i,j) based on DMDA, but PetsSectionGetOffset
>>> gives result based on how we masked away some of the points from dmda grid.
>>> Or am I missing something very basic here ?
>>>
>>
>> You are missing something basic. The row is correctly identified as 0,
>> and that means the 2 is
>> point 10.
>>
> The problem must be preallocation. First, you should send the result of
>> MatView().
>>
>
> MatView shows that the preallocation and the matrix created is for the
> full dmda grid which is probably not what we want here, right ? Here is the
> matview result:
>
Thats right. Did you call
DMSetDefaultSection(da, s)
after you made the new section and before you called DMGetMatrix()?
Matt
> Mat Object: 1 MPI processes
>
> type: seqaij
>
> row 0: (0, 0) (1, 0) (4, 0)
>
> row 1: (0, 0) (1, 0) (2, 0) (5, 0)
>
> row 2: (1, 0) (2, 0) (3, 0) (6, 0)
>
> row 3: (2, 0) (3, 0) (7, 0)
>
> row 4: (0, 0) (4, 0) (5, 0) (8, 0)
>
> row 5: (1, 0) (4, 0) (5, 0) (6, 0) (9, 0)
>
> row 6: (2, 0) (5, 0) (6, 0) (7, 0) (10, 0)
>
> row 7: (3, 0) (6, 0) (7, 0) (11, 0)
>
> row 8: (4, 0) (8, 0) (9, 0) (12, 0)
>
> row 9: (5, 0) (8, 0) (9, 0) (10, 0) (13, 0)
>
> row 10: (6, 0) (9, 0) (10, 0) (11, 0) (14, 0)
>
> row 11: (7, 0) (10, 0) (11, 0) (15, 0)
>
> row 12: (8, 0) (12, 0) (13, 0)
>
> row 13: (9, 0) (12, 0) (13, 0) (14, 0)
>
> row 14: (10, 0) (13, 0) (14, 0) (15, 0)
>
> row 15: (11, 0) (14, 0) (15, 0)
>
> [0]PETSC ERROR: --------------------- Error Message
> ------------------------------------
>
> [0]PETSC ERROR: Argument out of range!
>
> [0]PETSC ERROR: New nonzero at (0,2) caused a malloc!
>
>
>
>> Matt
>>
>>
>>>
>>>> Thanks,
>>>>
>>>> Matt
>>>>
>>>>
>>>>> [0]PETSC ERROR: --------------------- Error Message
>>>>> ------------------------------------
>>>>> [0]PETSC ERROR: Argument out of range!
>>>>> [0]PETSC ERROR: New nonzero at (0,6) caused a malloc!
>>>>> [0]PETSC ERROR:
>>>>> ------------------------------------------------------------------------
>>>>> [0]PETSC ERROR: Petsc Development GIT revision:
>>>>> 61e5e40bb2c5bf2423e94b71a15fef47e411b0da GIT Date: 2013-10-25 21:47:45
>>>>> -0500
>>>>> [0]PETSC ERROR: See docs/changes/index.html for recent updates.
>>>>> [0]PETSC ERROR: See docs/faq.html for hints about trouble shooting.
>>>>> [0]PETSC ERROR: See docs/index.html for manual pages.
>>>>> [0]PETSC ERROR:
>>>>> ------------------------------------------------------------------------
>>>>> [0]PETSC ERROR: build/poissonIrregular on a arch-linux2-cxx-debug
>>>>> named edwards by bkhanal Thu Oct 31 19:23:58 2013
>>>>> [0]PETSC ERROR: Libraries linked from
>>>>> /home/bkhanal/Documents/softwares/petsc/arch-linux2-cxx-debug/lib
>>>>> [0]PETSC ERROR: Configure run at Sat Oct 26 16:35:15 2013
>>>>> [0]PETSC ERROR: Configure options --download-mpich
>>>>> -download-f-blas-lapack=1 --download-metis --download-parmetis
>>>>> --download-superlu_dist --download-scalapack --download-mumps
>>>>> --download-hypre --with-clanguage=cxx
>>>>> [0]PETSC ERROR:
>>>>> ------------------------------------------------------------------------
>>>>> [0]PETSC ERROR: MatSetValues_SeqAIJ() line 413 in
>>>>> src/mat/impls/aij/seq/aij.c
>>>>> [0]PETSC ERROR: MatSetValues() line 1130 in src/mat/interface/matrix.c
>>>>> [0]PETSC ERROR: computeMatrix2dSection() line 318 in
>>>>> /user/bkhanal/home/works/poissonIrregular/poissonIrregular.cxx
>>>>> [0]PETSC ERROR: KSPSetUp() line 228 in src/ksp/ksp/interface/itfunc.c
>>>>> [0]PETSC ERROR: KSPSolve() line 399 in src/ksp/ksp/interface/itfunc.c
>>>>> [0]PETSC ERROR: main() line 598 in
>>>>> /user/bkhanal/home/works/poissonIrregular/poissonIrregular.cxx
>>>>> application called MPI_Abort(MPI_COMM_WORLD, 63) - process 0
>>>>> [unset]: aborting job:
>>>>> application called MPI_Abort(MPI_COMM_WORLD, 63) - process 0
>>>>>
>>>>>
>>>>>> Matt
>>>>>>
>>>>>>
>>>>>>> However, now since I do not want the rows in the matrix
>>>>>>> corresponding to the points in the grid which do not lie in the
>>>>>>> computational domain. This means the size of the matrix will not
>>>>>>> necessarily equal the total number of cells in DMDA. So in the following
>>>>>>> code:
>>>>>>>
>>>>>>> ierr = DMDAGetLocalInfo(dm,&info);CHKERRQ(ierr);
>>>>>>> ierr = PetscSectionCreate(PETSC_COMM_WORLD, &s);CHKERRQ(ierr);
>>>>>>> ierr = DMDAGetNumCells(dm, NULL, NULL, NULL, &nC);CHKERRQ(ierr);
>>>>>>> ierr = PetscSectionSetChart(s, 0, nC);CHKERRQ(ierr);
>>>>>>> for (PetscInt j = info.ys; j < info.ys+info.ym; ++j) {
>>>>>>> for (PetscInt i = info.xs; i < info.xs+info.xm; ++i) {
>>>>>>> PetscInt point;
>>>>>>> if(isPosInDomain(&testPoisson,i,j,0)) {
>>>>>>> ierr = DMDAGetCellPoint(dm, i, j, 0,
>>>>>>> &point);CHKERRQ(ierr);
>>>>>>> ierr = PetscSectionSetDof(s, point,
>>>>>>> testPoisson.mDof); // No. of dofs associated with the point.
>>>>>>> }
>>>>>>>
>>>>>>> }
>>>>>>> }
>>>>>>> ierr = PetscSectionSetUp(s);CHKERRQ(ierr);
>>>>>>> ierr = DMSetDefaultSection(dm, s);CHKERRQ(ierr);
>>>>>>> ierr = PetscSectionDestroy(&s);CHKERRQ(ierr);
>>>>>>>
>>>>>>> should I myself compute proper nC (i.e. total no. of points for
>>>>>>> which I want the matrix to have a corresponding row) before calling
>>>>>>> PetscSectionSetChart() or is the masking out of the points inside the loop
>>>>>>> with if(isPosInDomain(&testPoisson,i,j,0)) sufficient ?
>>>>>>> And, when you write:
>>>>>>> PetscSectionGetOffset(gs, p, &ind);
>>>>>>> row = ind < 0 ? -(ind+1) : ind;
>>>>>>>
>>>>>>> it seems you allow the possibility of getting a -ve ind when using
>>>>>>> PetscSectionGetOffset. When would an offset to a particular dof and point?
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>>
>>>>>>>>> for(PetscInt j = info.ys; j<info.ys+info.ym; ++j) {
>>>>>>>>> for(PetscInt i = info.xs; i<info.xs+info.xm; ++i) {
>>>>>>>>> num = 0;
>>>>>>>>> /*ierr =
>>>>>>>>> DMDAGetCellPoint(da,i,j,0,&point);CHKERRQ(ierr);*/ /*Get the PetscPoint*/
>>>>>>>>> /*But I did now understand how I would emulate the
>>>>>>>>> row.c and col.c info with PetscSection*/
>>>>>>>>> row.i = i; row.j = j;
>>>>>>>>>
>>>>>>>>
>>>>>>>> Here you would call
>>>>>>>>
>>>>>>>> ierr = DMDAGetCellPoint(da, i, j, 0, &cell);CHKERRQ(ierr);
>>>>>>>> ierr = PetscSectionGetOffset(da, cell, &row);CHKERRQ(ierr);
>>>>>>>> ierr = PetscSectionGetDof(da, cell, &dof);CHKERRQ(ierr);
>>>>>>>>
>>>>>>>> This means that this cell has global rows = [row, row+dof).
>>>>>>>>
>>>>>>>>
>>>>>>>>> col[num].i = i; col[num].j = j;
>>>>>>>>> if (isPosInDomain(ctx,i,j)) { /*put the
>>>>>>>>> operator for only the values for only the points lying in the domain*/
>>>>>>>>> v[num++] = -4;
>>>>>>>>> if(isPosInDomain(ctx,i+1,j)) {
>>>>>>>>> col[num].i = i+1; col[num].j = j;
>>>>>>>>> v[num++] = 1;
>>>>>>>>> }
>>>>>>>>> if(isPosInDomain(ctx,i-1,j)) {
>>>>>>>>> col[num].i = i-1; col[num].j = j;
>>>>>>>>> v[num++] = 1;
>>>>>>>>> }
>>>>>>>>> if(isPosInDomain(ctx,i,j+1)) {
>>>>>>>>> col[num].i = i; col[num].j = j+1;
>>>>>>>>> v[num++] = 1;
>>>>>>>>> }
>>>>>>>>> if(isPosInDomain(ctx,i,j-1)) {
>>>>>>>>> col[num].i = i; col[num].j = j-1;
>>>>>>>>> v[num++] = 1;
>>>>>>>>> }
>>>>>>>>> } else {
>>>>>>>>> v[num++] = dScale; /*set Dirichlet identity
>>>>>>>>> rows for points in the rectangle but outside the computational domain*/
>>>>>>>>> }
>>>>>>>>>
>>>>>>>>
>>>>>>>> You do the same thing for the columns, and then call
>>>>>>>>
>>>>>>>> ierr = MatSetValues(A, dof, rows, num*dof, cols, v,
>>>>>>>> INSERT_VALUES);CHKERRQ(ierr);
>>>>>>>>
>>>>>>>>
>>>>>>>>> ierr =
>>>>>>>>> MatSetValuesStencil(A,1,&row,num,col,v,INSERT_VALUES);CHKERRQ(ierr);
>>>>>>>>> }
>>>>>>>>> }
>>>>>>>>> }
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>> Thanks,
>>>>>>>>>>
>>>>>>>>>> Matt
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>> Matt
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> If not, just put the identity into
>>>>>>>>>>>>>>>>>>>> the rows you do not use on the full cube. It will not
>>>>>>>>>>>>>>>>>>>> hurt scalability or convergence.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> In the case of Poisson with Dirichlet condition this
>>>>>>>>>>>>>>>>>>> might be the case. But is it always true that having identity rows in the
>>>>>>>>>>>>>>>>>>> system matrix will not hurt convergence ? I thought otherwise for the
>>>>>>>>>>>>>>>>>>> following reasons:
>>>>>>>>>>>>>>>>>>> 1) Having read Jed's answer here :
>>>>>>>>>>>>>>>>>>> http://scicomp.stackexchange.com/questions/3426/why-is-pinning-a-point-to-remove-a-null-space-bad/3427#3427
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Jed is talking about a constraint on a the pressure at a
>>>>>>>>>>>>>>>>>> point. This is just decoupling these unknowns from the rest
>>>>>>>>>>>>>>>>>> of the problem.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> 2) Some observation I am getting (but I am still doing
>>>>>>>>>>>>>>>>>>> more experiments to confirm) while solving my staggered-grid 3D stokes flow
>>>>>>>>>>>>>>>>>>> with schur complement and using -pc_type gamg for A00 matrix. Putting the
>>>>>>>>>>>>>>>>>>> identity rows for dirichlet boundaries and for ghost cells seemed to have
>>>>>>>>>>>>>>>>>>> effects on its convergence. I'm hoping once I know how to use PetscSection,
>>>>>>>>>>>>>>>>>>> I can get rid of using ghost cells method for the staggered grid and get
>>>>>>>>>>>>>>>>>>> rid of the identity rows too.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> It can change the exact iteration, but it does not make
>>>>>>>>>>>>>>>>>> the matrix conditioning worse.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Anyway please provide me with some pointers so that I
>>>>>>>>>>>>>>>>>>> can start trying with petscsection on top of a dmda, in the beginning for
>>>>>>>>>>>>>>>>>>> non-staggered case.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>>>>>>>> Bishesh
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>>>>>>>>>> Bishesh
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>>>>>>> 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
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>>> 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
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> --
>>>>>>>>>>>> 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
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> --
>>>>>>>> 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
>>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>> --
>>>> 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
>>
>
>
--
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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