[petsc-users] Question concerning ilu and bcgs
Dave May
dave.mayhem23 at gmail.com
Wed Feb 18 13:04:12 CST 2015
(Thanks to Matt for taking the words out of my mouth :D).
If using LU on the splits isn't possible as the problem is too large, you
will have to use an iterative method.
However, in your initial tests you need to remove all the junk which causes
early termination of the Krylov solves (e.g. -fieldsplit_0_ksp_max_it 10
-fieldsplit_1_ksp_max_it 10).
I would also monitor the residual history of the outer method AND the
splits.
If your system is badly conditioned, I recommend examining the true
residual in your experiments until you understand what FS is really doing.
Cheers
Dave
On 18 February 2015 at 19:51, Sun, Hui <hus003 at ucsd.edu> wrote:
> Thank you Dave. In fact I have tried fieldsplit several months ago, and
> today I go back to the previous code and ran it again. How can I tell it is
> doing what I want it to do? Here are the options:
>
> -pc_type fieldsplit -fieldsplit_0_pc_type jacobi -fieldsplit_1_pc_type
> jacobi -pc_fieldsplit_type SC\
>
> HUR -ksp_monitor_short -ksp_converged_reason -ksp_rtol 1e-4
> -fieldsplit_1_ksp_rtol 1e-2 -fieldsplit_0_ksp_rtol 1e-4
> -fieldsplit_1_ksp_max_it 10 -fieldsplit_0_ksp_max_it 10 -ksp_type fgmres
> -ksp_max_it 10 -ksp_view
>
> And here is the output:
>
> Starting...
>
> 0 KSP Residual norm 17.314
>
> 1 KSP Residual norm 10.8324
>
> 2 KSP Residual norm 10.8312
>
> 3 KSP Residual norm 10.7726
>
> 4 KSP Residual norm 10.7642
>
> 5 KSP Residual norm 10.7634
>
> 6 KSP Residual norm 10.7399
>
> 7 KSP Residual norm 10.7159
>
> 8 KSP Residual norm 10.6602
>
> 9 KSP Residual norm 10.5756
>
> 10 KSP Residual norm 10.5224
>
> Linear solve did not converge due to DIVERGED_ITS iterations 10
>
> KSP Object: 1 MPI processes
>
> type: fgmres
>
> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
> Orthogonalization with no iterative refinement
>
> GMRES: happy breakdown tolerance 1e-30
>
> maximum iterations=10, initial guess is zero
>
> tolerances: relative=0.0001, absolute=1e-50, divergence=10000
>
> right preconditioning
>
> using UNPRECONDITIONED norm type for convergence test
>
> PC Object: 1 MPI processes
>
> type: fieldsplit
>
> FieldSplit with Schur preconditioner, factorization FULL
>
> Preconditioner for the Schur complement formed from A11
>
> Split info:
>
> Split number 0 Defined by IS
>
> Split number 1 Defined by IS
>
> 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=10, initial guess is zero
>
> tolerances: relative=0.0001, absolute=1e-50, divergence=10000
>
> left preconditioning
>
> using PRECONDITIONED norm type for convergence test
>
> PC Object: (fieldsplit_0_) 1 MPI processes
>
> type: jacobi
>
> linear system matrix = precond matrix:
>
> Mat Object: (fieldsplit_0_) 1 MPI processes
>
> type: mpiaij
>
> rows=20000, cols=20000
>
> total: nonzeros=85580, allocated nonzeros=760000
>
> total number of mallocs used during MatSetValues calls =0
>
> not using I-node (on process 0) routines
>
> 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=10, initial guess is zero
>
> tolerances: relative=0.01, absolute=1e-50, divergence=10000
>
> left preconditioning
>
> using PRECONDITIONED norm type for convergence test
>
> PC Object: (fieldsplit_1_) 1 MPI processes
>
> type: jacobi
>
> linear system matrix followed by preconditioner matrix:
>
> Mat Object: (fieldsplit_1_) 1 MPI processes
>
> type: schurcomplement
>
> rows=10000, cols=10000
>
> Schur complement A11 - A10 inv(A00) A01
>
> A11
>
> Mat Object: (fieldsplit_1_) 1 MPI
> processes
>
> type: mpiaij
>
> rows=10000, cols=10000
>
> total: nonzeros=2110, allocated nonzeros=80000
>
> total number of mallocs used during MatSetValues calls =0
>
> using I-node (on process 0) routines: found 3739 nodes,
> limit used is 5
>
> A10
>
> Mat Object: (a10_) 1 MPI processes
>
> type: mpiaij
>
> rows=10000, cols=20000
>
> total: nonzeros=31560, allocated nonzeros=80000
>
> total number of mallocs used during MatSetValues calls =0
>
> not using I-node (on process 0) 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=10, initial guess is zero
>
> tolerances: relative=0.0001, absolute=1e-50,
> divergence=10000
>
> left preconditioning
>
> using PRECONDITIONED norm type for convergence test
>
> PC Object: (fieldsplit_0_) 1 MPI
> processes
>
> type: jacobi
>
> linear system matrix = precond matrix:
>
> Mat Object: (fieldsplit_0_)
> 1 MPI processes
>
> type: mpiaij
>
> rows=20000, cols=20000
>
> total: nonzeros=85580, allocated nonzeros=760000
>
> total number of mallocs used during MatSetValues calls =0
>
> not using I-node (on process 0) routines
>
> A01
>
> Mat Object: (a01_) 1 MPI processes
>
> type: mpiaij
>
> rows=20000, cols=10000
>
> total: nonzeros=32732, allocated nonzeros=240000
>
> total number of mallocs used during MatSetValues calls =0
>
> not using I-node (on process 0) routines
>
> Mat Object: (fieldsplit_1_) 1 MPI processes
>
> type: mpiaij
>
> rows=10000, cols=10000
>
> total: nonzeros=2110, allocated nonzeros=80000
>
> total number of mallocs used during MatSetValues calls =0
>
> using I-node (on process 0) routines: found 3739 nodes, limit
> used is 5
>
> linear system matrix = precond matrix:
>
> Mat Object: 1 MPI processes
>
> type: nest
>
> rows=30000, cols=30000
>
> Matrix object:
>
> type=nest, rows=2, cols=2
>
> MatNest structure:
>
> (0,0) : prefix="fieldsplit_0_", type=mpiaij, rows=20000,
> cols=20000
>
> (0,1) : prefix="a01_", type=mpiaij, rows=20000, cols=10000
>
> (1,0) : prefix="a10_", type=mpiaij, rows=10000, cols=20000
>
> (1,1) : prefix="fieldsplit_1_", type=mpiaij, rows=10000,
> cols=10000
>
> residual u = 10.3528
>
> residual p = 1.88199
>
> residual [u,p] = 10.5224
>
> L^2 discretization error u = 0.698386
>
> L^2 discretization error p = 1.0418
>
> L^2 discretization error [u,p] = 1.25423
>
> number of processors = 1 0
>
> Time cost for creating solver context 0.100217 s, and for solving 3.78879
> s, and for printing 0.0908558 s.
>
>
> ------------------------------
> *From:* Dave May [dave.mayhem23 at gmail.com]
> *Sent:* Wednesday, February 18, 2015 10:00 AM
> *To:* Sun, Hui
> *Cc:* Matthew Knepley; petsc-users at mcs.anl.gov; hong at aspiritech.org
>
> *Subject:* Re: [petsc-users] Question concerning ilu and bcgs
>
>
> Fieldsplit will not work if you just set pc_type fieldsplit and you have
> an operator with a block size if 1. In this case, you will need to define
> the splits using index sets.
>
> I cannot believe that defining all the v and p dofs is really hard.
> Certainly it is far easier than trying to understand the difference between
> the petsc, matlab and the hypre implementations of ilut. Even if you did
> happen to find one implemtation of ilu you were "happy" with, as soon as
> you refine the mesh a couple of times the iterations will increase.
>
> I second Matt's opinion - forget about ilu and focus time on trying
> to make fieldsplit work. Fieldsplit will generate spectrally equivalent
> operators of your flow problem, ilu won't
>
> Cheers
> Dave
>
>
> On Wednesday, 18 February 2015, Sun, Hui <hus003 at ucsd.edu> wrote:
>
>> I tried fieldsplitting several months ago, it didn't work due to the
>> complicated coupled irregular bdry conditions. So I tried direct solver and
>> now I modified the PDE system a little bit so that the ILU/bcgs works in
>> MATLAB. But thank you for the suggestions, although I doubt it would work,
>> maybe I will still try fieldsplitting with my new system.
>>
>>
>> ------------------------------
>> *From:* Matthew Knepley [knepley at gmail.com <http://UrlBlockedError.aspx>]
>> *Sent:* Wednesday, February 18, 2015 8:54 AM
>> *To:* Sun, Hui
>> *Cc:* hong at aspiritech.org <http://UrlBlockedError.aspx>;
>> petsc-users at mcs.anl.gov <http://UrlBlockedError.aspx>
>> *Subject:* Re: [petsc-users] Question concerning ilu and bcgs
>>
>> On Wed, Feb 18, 2015 at 10:47 AM, Sun, Hui <hus003 at ucsd.edu
>> <http://UrlBlockedError.aspx>> wrote:
>>
>>> The matrix is from a 3D fluid problem, with complicated irregular
>>> boundary conditions. I've tried using direct solvers such as UMFPACK,
>>> SuperLU_dist and MUMPS. It seems that SuperLU_dist does not solve for my
>>> linear system; UMFPACK solves the system but would run into memory issue
>>> even with small size matrices and it cannot parallelize; MUMPS does solve
>>> the system but it also fails when the size is big and it takes much time.
>>> That's why I'm seeking an iterative method.
>>>
>>> I guess the direct method is faster than an iterative method for a
>>> small A, but that may not be true for bigger A.
>>>
>>
>> If this is a Stokes flow, you should use PCFIELDSPLIT and multigrid. If
>> it is advection dominated, I know of nothing better
>> than sparse direct or perhaps Block-Jacobi with sparse direct blocks.
>> Since MUMPS solved your system, I would consider
>> using BJacobi/ASM and MUMPS or UMFPACK as the block solver.
>>
>> Thanks,
>>
>> Matt
>>
>>
>>>
>>> ------------------------------
>>> *From:* Matthew Knepley [knepley at gmail.com <http://UrlBlockedError.aspx>
>>> ]
>>> *Sent:* Wednesday, February 18, 2015 8:33 AM
>>> *To:* Sun, Hui
>>> *Cc:* hong at aspiritech.org <http://UrlBlockedError.aspx>;
>>> petsc-users at mcs.anl.gov <http://UrlBlockedError.aspx>
>>> *Subject:* Re: [petsc-users] Question concerning ilu and bcgs
>>>
>>> On Wed, Feb 18, 2015 at 10:31 AM, Sun, Hui <hus003 at ucsd.edu
>>> <http://UrlBlockedError.aspx>> wrote:
>>>
>>>> So far I just try around, I haven't looked into literature yet.
>>>>
>>>> However, both MATLAB's ilu+gmres and ilu+bcgs work. Is it possible
>>>> that some parameter or options need to be tuned in using PETSc's ilu or
>>>> hypre's ilu? Besides, is there a way to view how good the performance of
>>>> the pc is and output the matrices L and U, so that I can do some test in
>>>> MATLAB?
>>>>
>>>
>>> 1) Its not clear exactly what Matlab is doing
>>>
>>> 2) PETSc uses ILU(0) by default (you can set it to use ILU(k))
>>>
>>> 3) I don't know what Hypre's ILU can do
>>>
>>> I would really discourage from using ILU. I cannot imagine it is
>>> faster than sparse direct factorization
>>> for your system, such as from SuperLU or MUMPS.
>>>
>>> Thanks,
>>>
>>> Matt
>>>
>>>
>>>> Hui
>>>>
>>>>
>>>> ------------------------------
>>>> *From:* Matthew Knepley [knepley at gmail.com
>>>> <http://UrlBlockedError.aspx>]
>>>> *Sent:* Wednesday, February 18, 2015 8:09 AM
>>>> *To:* Sun, Hui
>>>> *Cc:* hong at aspiritech.org <http://UrlBlockedError.aspx>;
>>>> petsc-users at mcs.anl.gov <http://UrlBlockedError.aspx>
>>>> *Subject:* Re: [petsc-users] Question concerning ilu and bcgs
>>>>
>>>> On Wed, Feb 18, 2015 at 10:02 AM, Sun, Hui <hus003 at ucsd.edu
>>>> <http://UrlBlockedError.aspx>> wrote:
>>>>
>>>>> Yes I've tried other solvers, gmres/ilu does not work, neither does
>>>>> bcgs/ilu. Here are the options:
>>>>>
>>>>> -pc_type ilu -pc_factor_nonzeros_along_diagonal -pc_factor_levels 0
>>>>> -pc_factor_reuse_ordering -ksp_ty\
>>>>>
>>>>> pe bcgs -ksp_rtol 1e-6 -ksp_max_it 10 -ksp_monitor_short -ksp_view
>>>>>
>>>>
>>>> Note here that ILU(0) is an unreliable and generally crappy
>>>> preconditioner. Have you looked in the
>>>> literature for the kinds of preconditioners that are effective for your
>>>> problem?
>>>>
>>>> Thanks,
>>>>
>>>> Matt
>>>>
>>>>
>>>>> Here is the output:
>>>>>
>>>>> 0 KSP Residual norm 211292
>>>>>
>>>>> 1 KSP Residual norm 13990.2
>>>>>
>>>>> 2 KSP Residual norm 9870.08
>>>>>
>>>>> 3 KSP Residual norm 9173.9
>>>>>
>>>>> 4 KSP Residual norm 9121.94
>>>>>
>>>>> 5 KSP Residual norm 7386.1
>>>>>
>>>>> 6 KSP Residual norm 6222.55
>>>>>
>>>>> 7 KSP Residual norm 7192.94
>>>>>
>>>>> 8 KSP Residual norm 33964
>>>>>
>>>>> 9 KSP Residual norm 33960.4
>>>>>
>>>>> 10 KSP Residual norm 1068.54
>>>>>
>>>>> KSP Object: 1 MPI processes
>>>>>
>>>>> type: bcgs
>>>>>
>>>>> maximum iterations=10, initial guess is zero
>>>>>
>>>>> tolerances: relative=1e-06, absolute=1e-50, divergence=10000
>>>>>
>>>>> left preconditioning
>>>>>
>>>>> using PRECONDITIONED norm type for convergence test
>>>>>
>>>>> PC Object: 1 MPI processes
>>>>>
>>>>> type: ilu
>>>>>
>>>>> ILU: out-of-place factorization
>>>>>
>>>>> ILU: Reusing reordering from past factorization
>>>>>
>>>>> 0 levels of fill
>>>>>
>>>>> tolerance for zero pivot 2.22045e-14
>>>>>
>>>>> using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
>>>>>
>>>>> matrix ordering: natural
>>>>>
>>>>> factor fill ratio given 1, needed 1
>>>>>
>>>>> Factored matrix follows:
>>>>>
>>>>> Mat Object: 1 MPI processes
>>>>>
>>>>> type: seqaij
>>>>>
>>>>> rows=62500, cols=62500
>>>>>
>>>>> package used to perform factorization: petsc
>>>>>
>>>>> total: nonzeros=473355, allocated nonzeros=473355
>>>>>
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>>
>>>>> not using I-node routines
>>>>>
>>>>> linear system matrix = precond matrix:
>>>>>
>>>>> Mat Object: 1 MPI processes
>>>>>
>>>>> type: seqaij
>>>>>
>>>>> rows=62500, cols=62500
>>>>>
>>>>> total: nonzeros=473355, allocated nonzeros=7.8125e+06
>>>>>
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>>
>>>>> not using I-node routines
>>>>>
>>>>> Time cost: 0.307149, 0.268402, 0.0990018
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> ------------------------------
>>>>> *From:* hong at aspiritech.org <http://UrlBlockedError.aspx> [
>>>>> hong at aspiritech.org <http://UrlBlockedError.aspx>]
>>>>> *Sent:* Wednesday, February 18, 2015 7:49 AM
>>>>> *To:* Sun, Hui
>>>>> *Cc:* Matthew Knepley; petsc-users at mcs.anl.gov
>>>>> <http://UrlBlockedError.aspx>
>>>>> *Subject:* Re: [petsc-users] Question concerning ilu and bcgs
>>>>>
>>>>> Have you tried other solvers, e.g., PETSc default gmres/ilu,
>>>>> bcgs/ilu etc.
>>>>> The matrix is small. If it is ill-conditioned, then pc_type lu would
>>>>> work the best.
>>>>>
>>>>> Hong
>>>>>
>>>>> On Wed, Feb 18, 2015 at 9:34 AM, Sun, Hui <hus003 at ucsd.edu
>>>>> <http://UrlBlockedError.aspx>> wrote:
>>>>>
>>>>>> With options:
>>>>>>
>>>>>> -pc_type hypre -pc_hypre_type pilut -pc_hypre_pilut_maxiter 1000
>>>>>> -pc_hypre_pilut_tol 1e-3 -ksp_type bcgs -ksp_rtol 1e-10 -ksp_max_it
>>>>>> 10 -ksp_monitor_short -ksp_converged_reason -ksp_view
>>>>>>
>>>>>> Here is the full output:
>>>>>>
>>>>>> 0 KSP Residual norm 1404.62
>>>>>>
>>>>>> 1 KSP Residual norm 88.9068
>>>>>>
>>>>>> 2 KSP Residual norm 64.73
>>>>>>
>>>>>> 3 KSP Residual norm 71.0224
>>>>>>
>>>>>> 4 KSP Residual norm 69.5044
>>>>>>
>>>>>> 5 KSP Residual norm 455.458
>>>>>>
>>>>>> 6 KSP Residual norm 174.876
>>>>>>
>>>>>> 7 KSP Residual norm 183.031
>>>>>>
>>>>>> 8 KSP Residual norm 650.675
>>>>>>
>>>>>> 9 KSP Residual norm 79.2441
>>>>>>
>>>>>> 10 KSP Residual norm 84.1985
>>>>>>
>>>>>> Linear solve did not converge due to DIVERGED_ITS iterations 10
>>>>>>
>>>>>> KSP Object: 1 MPI processes
>>>>>>
>>>>>> type: bcgs
>>>>>>
>>>>>> maximum iterations=10, initial guess is zero
>>>>>>
>>>>>> tolerances: relative=1e-10, absolute=1e-50, divergence=10000
>>>>>>
>>>>>> left preconditioning
>>>>>>
>>>>>> using PRECONDITIONED norm type for convergence test
>>>>>>
>>>>>> PC Object: 1 MPI processes
>>>>>>
>>>>>> type: hypre
>>>>>>
>>>>>> HYPRE Pilut preconditioning
>>>>>>
>>>>>> HYPRE Pilut: maximum number of iterations 1000
>>>>>>
>>>>>> HYPRE Pilut: drop tolerance 0.001
>>>>>>
>>>>>> HYPRE Pilut: default factor row size
>>>>>>
>>>>>> linear system matrix = precond matrix:
>>>>>>
>>>>>> Mat Object: 1 MPI processes
>>>>>>
>>>>>> type: seqaij
>>>>>>
>>>>>> rows=62500, cols=62500
>>>>>>
>>>>>> total: nonzeros=473355, allocated nonzeros=7.8125e+06
>>>>>>
>>>>>> total number of mallocs used during MatSetValues calls =0
>>>>>>
>>>>>> not using I-node routines
>>>>>>
>>>>>> Time cost: 0.756198, 0.662984, 0.105672
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> ------------------------------
>>>>>> *From:* Matthew Knepley [knepley at gmail.com
>>>>>> <http://UrlBlockedError.aspx>]
>>>>>> *Sent:* Wednesday, February 18, 2015 3:30 AM
>>>>>> *To:* Sun, Hui
>>>>>> *Cc:* petsc-users at mcs.anl.gov <http://UrlBlockedError.aspx>
>>>>>> *Subject:* Re: [petsc-users] Question concerning ilu and bcgs
>>>>>>
>>>>>> On Wed, Feb 18, 2015 at 12:33 AM, Sun, Hui <hus003 at ucsd.edu
>>>>>> <http://UrlBlockedError.aspx>> wrote:
>>>>>>
>>>>>>> I have a matrix system Ax = b, A is of type MatSeqAIJ or
>>>>>>> MatMPIAIJ, depending on the number of cores.
>>>>>>>
>>>>>>> I try to solve this problem by pc_type ilu and ksp_type bcgs, it
>>>>>>> does not converge. The options I specify are:
>>>>>>>
>>>>>>> -pc_type hypre -pc_hypre_type pilut -pc_hypre_pilut_maxiter 1000
>>>>>>> -pc_hypre_pilut_tol 1e-3 -ksp_type b\
>>>>>>>
>>>>>>> cgs -ksp_rtol 1e-10 -ksp_max_it 1000 -ksp_monitor_short
>>>>>>> -ksp_converged_reason
>>>>>>>
>>>>>>
>>>>>> 1) Run with -ksp_view, so we can see exactly what was used
>>>>>>
>>>>>> 2) ILUT is unfortunately not a well-defined algorithm, and I
>>>>>> believe the parallel version makes different decisions
>>>>>> than the serial version.
>>>>>>
>>>>>> Thanks,
>>>>>>
>>>>>> Matt
>>>>>>
>>>>>>
>>>>>>> The first a few lines of the output are:
>>>>>>>
>>>>>>> 0 KSP Residual norm 1404.62
>>>>>>>
>>>>>>> 1 KSP Residual norm 88.9068
>>>>>>>
>>>>>>> 2 KSP Residual norm 64.73
>>>>>>>
>>>>>>> 3 KSP Residual norm 71.0224
>>>>>>>
>>>>>>> 4 KSP Residual norm 69.5044
>>>>>>>
>>>>>>> 5 KSP Residual norm 455.458
>>>>>>>
>>>>>>> 6 KSP Residual norm 174.876
>>>>>>>
>>>>>>> 7 KSP Residual norm 183.031
>>>>>>>
>>>>>>> 8 KSP Residual norm 650.675
>>>>>>>
>>>>>>> 9 KSP Residual norm 79.2441
>>>>>>>
>>>>>>> 10 KSP Residual norm 84.1985
>>>>>>>
>>>>>>>
>>>>>>> This clearly indicates non-convergence. However, I output the
>>>>>>> sparse matrix A and vector b to MATLAB, and run the following command:
>>>>>>>
>>>>>>> [L,U] = ilu(A,struct('type','ilutp','droptol',1e-3));
>>>>>>>
>>>>>>> [ux1,fl1,rr1,it1,rv1] = bicgstab(A,b,1e-10,1000,L,U);
>>>>>>>
>>>>>>>
>>>>>>> And it converges in MATLAB, with flag fl1=0, relative residue
>>>>>>> rr1=8.2725e-11, and iteration it1=89.5. I'm wondering how can I figure out
>>>>>>> what's wrong.
>>>>>>>
>>>>>>>
>>>>>>> Best,
>>>>>>>
>>>>>>> Hui
>>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> --
>>>>>> 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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