[petsc-users] discontinuous viscosity stokes equation 3D staggered grid
Bishesh Khanal
bisheshkh at gmail.com
Wed Aug 7 07:26:26 CDT 2013
On Wed, Aug 7, 2013 at 2:15 PM, Matthew Knepley <knepley at gmail.com> wrote:
> On Wed, Aug 7, 2013 at 7:07 AM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>
>>
>>
>>
>> On Tue, Aug 6, 2013 at 11:34 PM, Matthew Knepley <knepley at gmail.com>wrote:
>>
>>> On Tue, Aug 6, 2013 at 10:59 AM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>>>
>>>>
>>>>
>>>>
>>>> On Tue, Aug 6, 2013 at 4:40 PM, Matthew Knepley <knepley at gmail.com>wrote:
>>>>
>>>>> On Tue, Aug 6, 2013 at 8:06 AM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> On Mon, Aug 5, 2013 at 4:14 PM, Matthew Knepley <knepley at gmail.com>wrote:
>>>>>>
>>>>>>> On Mon, Aug 5, 2013 at 8:48 AM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> On Mon, Aug 5, 2013 at 3:17 PM, Matthew Knepley <knepley at gmail.com>wrote:
>>>>>>>>
>>>>>>>>> On Mon, Aug 5, 2013 at 7:54 AM, Bishesh Khanal <
>>>>>>>>> bisheshkh at gmail.com> wrote:
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On Wed, Jul 17, 2013 at 9:48 PM, Jed Brown <jedbrown at mcs.anl.gov>wrote:
>>>>>>>>>>
>>>>>>>>>>> Bishesh Khanal <bisheshkh at gmail.com> writes:
>>>>>>>>>>>
>>>>>>>>>>> > Now, I implemented two different approaches, each for both 2D
>>>>>>>>>>> and 3D, in
>>>>>>>>>>> > MATLAB. It works for the smaller sizes but I have problems
>>>>>>>>>>> solving it for
>>>>>>>>>>> > the problem size I need (250^3 grid size).
>>>>>>>>>>> > I use staggered grid with p on cell centers, and components of
>>>>>>>>>>> v on cell
>>>>>>>>>>> > faces. Similar split up of K to cell center and faces to
>>>>>>>>>>> account for the
>>>>>>>>>>> > variable viscosity case)
>>>>>>>>>>>
>>>>>>>>>>> Okay, you're using a staggered-grid finite difference
>>>>>>>>>>> discretization of
>>>>>>>>>>> variable-viscosity Stokes. This is a common problem and I
>>>>>>>>>>> recommend
>>>>>>>>>>> starting with PCFieldSplit with Schur complement reduction (make
>>>>>>>>>>> that
>>>>>>>>>>> work first, then switch to block preconditioner). You can use
>>>>>>>>>>> PCLSC or
>>>>>>>>>>> (probably better for you), assemble a preconditioning matrix
>>>>>>>>>>> containing
>>>>>>>>>>> the inverse viscosity in the pressure-pressure block. This
>>>>>>>>>>> diagonal
>>>>>>>>>>> matrix is a spectrally equivalent (or nearly so, depending on
>>>>>>>>>>> discretization) approximation of the Schur complement. The
>>>>>>>>>>> velocity
>>>>>>>>>>> block can be solved with algebraic multigrid. Read the
>>>>>>>>>>> PCFieldSplit
>>>>>>>>>>> docs (follow papers as appropriate) and let us know if you get
>>>>>>>>>>> stuck.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> I was trying to assemble the inverse viscosity diagonal matrix to
>>>>>>>>>> use as the preconditioner for the Schur complement solve step as you
>>>>>>>>>> suggested. I've few questions about the ways to implement this in Petsc:
>>>>>>>>>> A naive approach that I can think of would be to create a vector
>>>>>>>>>> with its components as reciprocal viscosities of the cell centers
>>>>>>>>>> corresponding to the pressure variables, and then create a diagonal matrix
>>>>>>>>>> from this vector. However I'm not sure about:
>>>>>>>>>> How can I make this matrix, (say S_p) compatible to the Petsc
>>>>>>>>>> distribution of the different rows of the main system matrix over different
>>>>>>>>>> processors ? The main matrix was created using the DMDA structure with 4
>>>>>>>>>> dof as explained before.
>>>>>>>>>> The main matrix correspond to the DMDA with 4 dofs but for the
>>>>>>>>>> S_p matrix would correspond to only pressure space. Should the distribution
>>>>>>>>>> of the rows of S_p among different processor not correspond to the
>>>>>>>>>> distribution of the rhs vector, say h' if it is solving for p with Sp = h'
>>>>>>>>>> where S = A11 inv(A00) A01 ?
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> PETSc distributed vertices, not dofs, so it never breaks blocks.
>>>>>>>>> The P distribution is the same as the entire problem divided by 4.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Thanks Matt. So if I create a new DMDA with same grid size but with
>>>>>>>> dof=1 instead of 4, the vertices for this new DMDA will be identically
>>>>>>>> distributed as for the original DMDA ? Or should I inform PETSc by calling
>>>>>>>> a particular function to make these two DMDA have identical distribution of
>>>>>>>> the vertices ?
>>>>>>>>
>>>>>>>
>>>>>>> Yes.
>>>>>>>
>>>>>>>
>>>>>>>> Even then I think there might be a problem due to the presence of
>>>>>>>> "fictitious pressure vertices". The system matrix (A) contains an identity
>>>>>>>> corresponding to these fictitious pressure nodes, thus when using a
>>>>>>>> -pc_fieldsplit_detect_saddle_point, will detect a A11 zero block of size
>>>>>>>> that correspond to only non-fictitious P-nodes. So the preconditioner S_p
>>>>>>>> for the Schur complement outer solve with Sp = h' will also need to
>>>>>>>> correspond to only the non-fictitious P-nodes. This means its size does not
>>>>>>>> directly correspond to the DMDA grid defined for the original problem.
>>>>>>>> Could you please suggest an efficient way of assembling this S_p matrix ?
>>>>>>>>
>>>>>>>
>>>>>>> Don't use detect_saddle, but split it by fields
>>>>>>> -pc_fieldsplit_0_fields 0,1,2 -pc_fieldsplit_1_fields 4
>>>>>>>
>>>>>>
>>>>>> How can I set this split in the code itself without giving it as a
>>>>>> command line option when the system matrix is assembled from the DMDA for
>>>>>> the whole system with 4 dofs. (i.e. *without* using the DMComposite
>>>>>> or *without* using the nested block matrices to assemble different
>>>>>> blocks separately and then combine them together).
>>>>>> I need the split to get access to the fieldsplit_1_ksp in my code,
>>>>>> because not using detect_saddle_point means I cannot use
>>>>>> -fieldsplit_1_ksp_constant_null_space due to the presence of identity for
>>>>>> the fictitious pressure nodes present in the fieldsplit_1_ block. I need to
>>>>>> use PCFieldSplitGetSubKsp() so that I can set proper null-space basis.
>>>>>>
>>>>>
>>>>> This is currently a real problem with the DMDA. In the unstructured
>>>>> case, where we always need specialized spaces, you can
>>>>> use something like
>>>>>
>>>>> PetscObject pressure;
>>>>> MatNullSpace nullSpacePres;
>>>>>
>>>>> ierr = DMGetField(dm, 1, &pressure);CHKERRQ(ierr);
>>>>> ierr = MatNullSpaceCreate(PetscObjectComm(pressure), PETSC_TRUE,
>>>>> 0, NULL, &nullSpacePres);CHKERRQ(ierr);
>>>>> ierr = PetscObjectCompose(pressure, "nullspace", (PetscObject)
>>>>> nullSpacePres);CHKERRQ(ierr);
>>>>> ierr = MatNullSpaceDestroy(&nullSpacePres);CHKERRQ(ierr);
>>>>>
>>>>> and then DMGetSubDM() uses this information to attach the null space
>>>>> to the IS that is created using the information in the PetscSection.
>>>>> If you use a PetscSection to set the data layout over the DMDA, I
>>>>> think this works correctly, but this has not been tested at all and is very
>>>>> new code. Eventually, I think we want all DMs to use this mechanism,
>>>>> but we are still working it out.
>>>>>
>>>>
>>>> Currently I do not use PetscSection. If this makes a cleaner approach,
>>>> I'd try it too but may a bit later (right now I'd like test my model with a
>>>> quickfix even if it means a little dirty code!)
>>>>
>>>>
>>>>>
>>>>> Bottom line: For custom null spaces using the default layout in DMDA,
>>>>> you need to take apart the PCFIELDSPLIT after it has been setup,
>>>>> which is somewhat subtle. You need to call KSPSetUp() and then reach
>>>>> in and get the PC, and the subKSPs. I don't like this at all, but we
>>>>> have not reorganized that code (which could be very simple and
>>>>> inflexible since its very structured).
>>>>>
>>>>
>>>> So I tried to get this approach working but I could not succeed and
>>>> encountered some errors. Here is a code snippet:
>>>>
>>>> //mDa is the DMDA that describes the whole grid with all 4 dofs (3
>>>> velocity components and 1 pressure comp.)
>>>> ierr = DMKSPSetComputeRHS(mDa,computeRHSTaras3D,this);CHKERRQ(ierr);
>>>> ierr =
>>>> DMKSPSetComputeOperators(mDa,computeMatrixTaras3D,this);CHKERRQ(ierr);
>>>> ierr = KSPSetDM(mKsp,mDa);CHKERRQ(ierr);
>>>> ierr = KSPSetNullSpace(mKsp,mNullSpaceSystem);CHKERRQ(ierr);
>>>> //I've the mNullSpaceSystem based on mDa, that contains a null space basis
>>>> for the complete system.
>>>> ierr =
>>>> KSPSetFromOptions(mKsp);CHKERRQ(ierr);
>>>> //This I expect would register these options I give:-pc_type fieldsplit
>>>> -pc_fieldsplit_type schur -pc_fieldsplit_0_fields 0,1,2
>>>> //-pc_fieldsplit_1_fields 3
>>>>
>>>> ierr = KSPSetUp(mKsp);CHKERRQ(ierr);
>>>>
>>>> ierr = KSPGetPC(mKsp,&mPcOuter); //Now get the PC that was
>>>> obtained from the options (fieldsplit)
>>>>
>>>> ierr =
>>>> PCFieldSplitSchurPrecondition(mPcOuter,PC_FIELDSPLIT_SCHUR_PRE_USER,mPcForSc);CHKERRQ(ierr);
>>>> //I have created the matrix mPcForSc using a DMDA with identical //size to
>>>> mDa but with dof=1 corresponding to the pressure nodes (say mDaPressure).
>>>>
>>>> ierr = PCSetUp(mPcOuter);CHKERRQ(ierr);
>>>>
>>>> KSP *kspSchur;
>>>> PetscInt kspSchurPos = 1;
>>>> ierr =
>>>> PCFieldSplitGetSubKSP(mPcOuter,&kspSchurPos,&kspSchur);CHKERRQ(ierr);
>>>> ierr =
>>>> KSPSetNullSpace(kspSchur[1],mNullSpacePressure);CHKERRQ(ierr);
>>>> //The null space is the one that correspond to only pressure nodes, created
>>>> using the mDaPressure.
>>>> ierr = PetscFree(kspSchur);CHKERRQ(ierr);
>>>>
>>>> ierr = KSPSolve(mKsp,NULL,NULL);CHKERRQ(ierr);
>>>>
>>>
>>> Sorry, you need to return to the old DMDA behavior, so you want
>>>
>>> -pc_fieldsplit_dm_splits 0
>>>
>>
>> Thanks, with this it seems I can attach the null space properly, but I
>> have a question regarding whether the Schur complement ksp solver is
>> actually using the preconditioner matrix I provide.
>> When using -ksp_view, the outer level pc object of type fieldsplit does
>> report that: "Preconditioner for the Schur complement formed from user
>> provided matrix", but in the KSP solver for Schur complement S, the pc
>> object (fieldsplit_1_) is of type ilu and doesn't say that it is using the
>> matrix I provide. Am I missing something here ?
>> Below are the relevant commented code snippet and the output of the
>> -ksp_view
>> (The options I used: -pc_type fieldsplit -pc_fieldsplit_type schur
>> -pc_fieldsplit_dm_splits 0 -pc_fieldsplit_0_fields 0,1,2
>> -pc_fieldsplit_1_fields 3 -ksp_converged_reason -ksp_view )
>>
>
> If ILU does not error, it means it is using your matrix, because the Schur
> complement matrix cannot be factored, and FS says it is using your matrix.
>
Thanks Matt! By the way, what do these statements mean in -ksp_view results:
not using I-node routines
or
using I-node routines: found 729 nodes, limit used is 5
>
> Matt
>
>
>> Code snippet:
>> ierr = KSPSetNullSpace(mKsp,mNullSpaceSystem);CHKERRQ(ierr); //The
>> nullspace for the whole system
>> ierr = KSPSetFromOptions(mKsp);CHKERRQ(ierr);
>> ierr = KSPSetUp(mKsp);CHKERRQ(ierr); //Set up mKsp
>> with the options provided with fieldsplit and the fields associated with
>> the two splits.
>>
>> ierr = KSPGetPC(mKsp,&mPcOuter);CHKERRQ(ierr); //Get
>> the fieldsplit pc set up from the options
>>
>> ierr =
>> PCFieldSplitSchurPrecondition(mPcOuter,PC_FIELDSPLIT_SCHUR_PRE_USER,mPcForSc);CHKERRQ(ierr);
>> //Use mPcForSc as the preconditioner for Schur Complement
>>
>> KSP *kspSchur;
>> PetscInt kspSchurPos = 1;
>> ierr =
>> PCFieldSplitGetSubKSP(mPcOuter,&kspSchurPos,&kspSchur);CHKERRQ(ierr);
>> ierr =
>> KSPSetNullSpace(kspSchur[1],mNullSpacePressure);CHKERRQ(ierr);
>> //Attach the null-space for the Schur complement ksp solver.
>> ierr = PetscFree(kspSchur);CHKERRQ(ierr);
>>
>> ierr = KSPSolve(mKsp,NULL,NULL);CHKERRQ(ierr);
>>
>>
>>
>> the output of the -ksp_view
>> KSP Object: 1 MPI processes
>> type: gmres
>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
>> Orthogonalization with no iterative refinement
>> GMRES: happy breakdown tolerance 1e-30
>> maximum iterations=10000, initial guess is zero
>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>> left preconditioning
>> has attached null space
>> using PRECONDITIONED norm type for convergence test
>> PC Object: 1 MPI processes
>> type: fieldsplit
>> FieldSplit with Schur preconditioner, blocksize = 4, factorization
>> FULL
>> Preconditioner for the Schur complement formed from user provided
>> matrix
>> Split info:
>> Split number 0 Fields 0, 1, 2
>> Split number 1 Fields 3
>> KSP solver for A00 block
>> KSP Object: (fieldsplit_0_) 1 MPI processes
>> type: gmres
>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
>> Orthogonalization with no iterative refinement
>> GMRES: happy breakdown tolerance 1e-30
>> maximum iterations=10000, initial guess is zero
>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>> left preconditioning
>> using PRECONDITIONED norm type for convergence test
>> PC Object: (fieldsplit_0_) 1 MPI processes
>> type: ilu
>> ILU: out-of-place factorization
>> 0 levels of fill
>> tolerance for zero pivot 2.22045e-14
>> using diagonal shift on blocks to prevent zero pivot
>> matrix ordering: natural
>> factor fill ratio given 1, needed 1
>> Factored matrix follows:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=2187, cols=2187
>> package used to perform factorization: petsc
>> total: nonzeros=140625, allocated nonzeros=140625
>> total number of mallocs used during MatSetValues calls =0
>> using I-node routines: found 729 nodes, limit used is 5
>> linear system matrix = precond matrix:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=2187, cols=2187
>> total: nonzeros=140625, allocated nonzeros=140625
>> total number of mallocs used during MatSetValues calls =0
>> using I-node routines: found 729 nodes, limit used is 5
>> KSP solver for S = A11 - A10 inv(A00) A01
>> KSP Object: (fieldsplit_1_) 1 MPI processes
>> type: gmres
>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
>> Orthogonalization with no iterative refinement
>> GMRES: happy breakdown tolerance 1e-30
>> maximum iterations=10000, initial guess is zero
>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>> left preconditioning
>> has attached null space
>> using PRECONDITIONED norm type for convergence test
>> PC Object: (fieldsplit_1_) 1 MPI processes
>> type: ilu
>> ILU: out-of-place factorization
>> 0 levels of fill
>> tolerance for zero pivot 2.22045e-14
>> using diagonal shift on blocks to prevent zero pivot
>> matrix ordering: natural
>> factor fill ratio given 1, needed 1
>> Factored matrix follows:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=729, cols=729
>> package used to perform factorization: petsc
>> total: nonzeros=15625, allocated nonzeros=15625
>> total number of mallocs used during MatSetValues calls =0
>> not using I-node routines
>> linear system matrix followed by preconditioner matrix:
>> Matrix Object: 1 MPI processes
>> type: schurcomplement
>> rows=729, cols=729
>> Schur complement A11 - A10 inv(A00) A01
>> A11
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=729, cols=729
>> total: nonzeros=15625, allocated nonzeros=15625
>> total number of mallocs used during MatSetValues calls =0
>> not using I-node routines
>> A10
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=729, cols=2187
>> total: nonzeros=46875, allocated nonzeros=46875
>> total number of mallocs used during MatSetValues calls =0
>> not using I-node routines
>> KSP of A00
>> KSP Object: (fieldsplit_0_) 1
>> MPI processes
>> type: gmres
>> GMRES: restart=30, using Classical (unmodified)
>> Gram-Schmidt Orthogonalization with no iterative refinement
>> GMRES: happy breakdown tolerance 1e-30
>> maximum iterations=10000, initial guess is zero
>> tolerances: relative=1e-05, absolute=1e-50,
>> divergence=10000
>> left preconditioning
>> using PRECONDITIONED norm type for convergence test
>> PC Object: (fieldsplit_0_) 1 MPI
>> processes
>> type: ilu
>> ILU: out-of-place factorization
>> 0 levels of fill
>> tolerance for zero pivot 2.22045e-14
>> using diagonal shift on blocks to prevent zero pivot
>> matrix ordering: natural
>> factor fill ratio given 1, needed 1
>> Factored matrix follows:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=2187, cols=2187
>> package used to perform factorization: petsc
>> total: nonzeros=140625, allocated nonzeros=140625
>> total number of mallocs used during MatSetValues
>> calls =0
>> using I-node routines: found 729 nodes, limit
>> used is 5
>> linear system matrix = precond matrix:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=2187, cols=2187
>> total: nonzeros=140625, allocated nonzeros=140625
>> total number of mallocs used during MatSetValues calls
>> =0
>> using I-node routines: found 729 nodes, limit used is
>> 5
>> A01
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=2187, cols=729
>> total: nonzeros=46875, allocated nonzeros=46875
>> total number of mallocs used during MatSetValues calls =0
>> using I-node routines: found 729 nodes, limit used is 5
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=729, cols=729
>> total: nonzeros=15625, allocated nonzeros=15625
>> total number of mallocs used during MatSetValues calls =0
>> not using I-node routines
>> linear system matrix = precond matrix:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=2916, cols=2916, bs=4
>> total: nonzeros=250000, allocated nonzeros=250000
>> total number of mallocs used during MatSetValues calls =0
>> using I-node routines: found 729 nodes, limit used is 5
>>
>>
>>
>>
>>
>>>
>>> or
>>>
>>> PCFieldSplitSetDMSplits(pc, PETSC_FALSE)
>>>
>>> Thanks,
>>>
>>> Matt
>>>
>>>
>>>> The errors I get when running with options: -pc_type fieldsplit
>>>> -pc_fieldsplit_type schur -pc_fieldsplit_0_fields 0,1,2
>>>> -pc_fieldsplit_1_fields 3
>>>> [0]PETSC ERROR: --------------------- Error Message
>>>> ------------------------------------
>>>> [0]PETSC ERROR: No support for this operation for this object type!
>>>> [0]PETSC ERROR: Support only implemented for 2d!
>>>> [0]PETSC ERROR:
>>>> ------------------------------------------------------------------------
>>>> [0]PETSC ERROR: Petsc Release Version 3.4.2, Jul, 02, 2013
>>>> [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: src/AdLemMain on a arch-linux2-cxx-debug named edwards
>>>> by bkhanal Tue Aug 6 17:35:30 2013
>>>> [0]PETSC ERROR: Libraries linked from
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/arch-linux2-cxx-debug/lib
>>>> [0]PETSC ERROR: Configure run at Fri Jul 19 14:25:01 2013
>>>> [0]PETSC ERROR: Configure options --with-cc=gcc --with-fc=g77
>>>> --with-cxx=g++ --download-f-blas-lapack=1 --download-mpich=1
>>>> -with-clanguage=cxx --download-hypre=1
>>>> [0]PETSC ERROR:
>>>> ------------------------------------------------------------------------
>>>> [0]PETSC ERROR: DMCreateSubDM_DA() line 188 in
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/src/dm/impls/da/dacreate.c
>>>> [0]PETSC ERROR: DMCreateSubDM() line 1267 in
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/src/dm/interface/dm.c
>>>> [0]PETSC ERROR: PCFieldSplitSetDefaults() line 337 in
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/src/ksp/pc/impls/fieldsplit/fieldsplit.c
>>>> [0]PETSC ERROR: PCSetUp_FieldSplit() line 458 in
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/src/ksp/pc/impls/fieldsplit/fieldsplit.c
>>>> [0]PETSC ERROR: PCSetUp() line 890 in
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/src/ksp/pc/interface/precon.c
>>>> [0]PETSC ERROR: KSPSetUp() line 278 in
>>>> /home/bkhanal/Documents/softwares/petsc-3.4.2/src/ksp/ksp/interface/itfunc.c
>>>> [0]PETSC ERROR: solveModel() line 181 in
>>>> "unknowndirectory/"/user/bkhanal/home/works/AdLemModel/src/PetscAdLemTaras3D.cxx
>>>> WARNING! There are options you set that were not used!
>>>> WARNING! could be spelling mistake, etc!
>>>> Option left: name:-pc_fieldsplit_1_fields value: 3
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>>
>>>>> Matt
>>>>>
>>>>>
>>>>>>
>>>>>>> Matt
>>>>>>>
>>>>>>>
>>>>>>>>
>>>>>>>>> Matt
>>>>>>>>>
>>>>>>>>> --
>>>>>>>>> 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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