[petsc-users] TS failed due to diverged_step_rejected

Matthew Knepley knepley at gmail.com
Fri Feb 17 14:38:54 CST 2023 

On Fri, Feb 17, 2023 at 3:32 PM Tu, Jiannan <Jiannan_Tu at uml.edu> wrote:

> The ts_type arkimex is used. There is right hand-side function RHSFunction
> set by TSSetRHSFunction() and also stiff function set by TSSetIFunction().
>
>
>
> With adaptivity shut off, TS can finish its first time step after the 3rd
> “Nonlinear solve converged due to …”. The solution gives negative electron
> and neutral temperatures at the bottom boundary. I need to fix the negative
> temperatures and see how the code works.
>
>
>
> BTW, what is this ts_adapt? Is it by default on?
>

It is a controller for adaptive timestepping. It is on by default, but I
suspect that you step into an unphysical regime, which makes
its estimates unreliable.

  Thanks,

     Matt


> Thank you,
>
> Jiannan
>
>
>
> *From: *Matthew Knepley <knepley at gmail.com>
> *Sent: *Friday, February 17, 2023 3:15 PM
> *To: *Tu, Jiannan <Jiannan_Tu at uml.edu>
> *Cc: *Barry Smith <bsmith at petsc.dev>; petsc-users
> <petsc-users at mcs.anl.gov>
> *Subject: *Re: [petsc-users] TS failed due to diverged_step_rejected
>
>
>
> *CAUTION:* This email was sent from outside the UMass Lowell network.
>
>
>
> I am not sure what TS you are using, but the estimate of the local
> truncation error is 91.4, and does not seem
>
> to change when you make the step smaller, so something is off. You can
> shut off the adaptivity using
>
>
>
>  -ts_adapt_type none
>
>
>
>   Thanks,
>
>
>
>       Matt
>
>
>
> On Fri, Feb 17, 2023 at 3:01 PM Tu, Jiannan <Jiannan_Tu at uml.edu> wrote:
>
> These are what I got with the options you suggested.
>
>
>
> Thank you,
>
> Jiannan
>
>
>
>
> -------------------------------------------------------------------------------
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 1.673091274668e-03
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 8.715428433630e-02
>
>     1 SNES Function norm 4.995727626692e-04
>
>     2 SNES Function norm 5.498018152230e-08
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>     0 SNES Function norm 3.237461568254e-01
>
>     1 SNES Function norm 7.988531005091e-04
>
>     2 SNES Function norm 1.280948196292e-07
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 2.189e-02
> dt=4.374e-03  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 4.881903203545e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>    0 SNES Function norm 7.562592690785e-02
>
>     1 SNES Function norm 1.143078818923e-04
>
>     2 SNES Function norm 9.834547907735e-09
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>     0 SNES Function norm 2.683968949758e-01
>
>     1 SNES Function norm 1.838028436639e-04
>
>     2 SNES Function norm 9.470813523140e-09
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-03
> dt=4.374e-04  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 1.821562431175e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 1.005443458812e-01
>
>     1 SNES Function norm 3.633336946661e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 1.515368382715e-01
>
>     1 SNES Function norm 3.389298316830e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-04
> dt=4.374e-05  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 4.541003359206e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 1.713800906043e-01
>
>     1 SNES Function norm 1.179958172167e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.020265094117e-01
>
>     1 SNES Function norm 1.513971290464e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-05
> dt=4.374e-06  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 6.090269704320e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.136603895703e-01
>
>     1 SNES Function norm 1.877474016012e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 3.127812462507e-01
>
>     1 SNES Function norm 2.713146825704e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-06
> dt=4.374e-07  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 2.793512213059e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.205196267430e-01
>
>     1 SNES Function norm 2.572653773308e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 3.260057361977e-01
>
>     1 SNES Function norm 2.705816087598e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-07
> dt=4.374e-08  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 2.764855860446e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.212505522844e-01
>
>     1 SNES Function norm 2.958996472386e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 3.273222034162e-01
>
>     1 SNES Function norm 2.994512887620e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-08
> dt=4.374e-09  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 3.317240589134e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.213246532918e-01
>
>     1 SNES Function norm 2.799468604767e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.274570888397e-01
>
>     1 SNES Function norm 3.066048050994e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-09
> dt=4.374e-10  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072189e+03
>
>     1 SNES Function norm 2.653507278572e-03
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.213869585841e-01
>
>     1 SNES Function norm 2.177156902895e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.275136370365e-01
>
>     1 SNES Function norm 1.962849131557e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-10
> dt=4.374e-11  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473072218e+03
>
>     1 SNES Function norm 5.664907315679e-03
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.223208399368e-01
>
>     1 SNES Function norm 5.688863091415e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.287121218919e-01
>
>     1 SNES Function norm 4.085338521320e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-11
> dt=4.374e-12  wlte= 91.4  wltea=   -1 wlter=   -1
>
>     0 SNES Function norm 2.274473071968e+03
>
>     1 SNES Function norm 4.694691905235e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.211786508657e-01
>
>     1 SNES Function norm 1.503497433939e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.272667798977e-01
>
>     1 SNES Function norm 2.176132327279e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>       TSAdapt basic arkimex 0:3 step   0 rejected t=42960      + 4.374e-12
> dt=4.374e-13  wlte= 91.4  wltea=   -1 wlter=   -1
>
> [0]PETSC ERROR: --------------------- Error Message
> --------------------------------------------------------------
>
> [0]PETSC ERROR:
>
> [0]PETSC ERROR: TSStep has failed due to DIVERGED_STEP_REJECTED
>
> [0]PETSC ERROR: See https://petsc.org/release/faq/
> <https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fpetsc.org%2Frelease%2Ffaq%2F&data=05%7C01%7CJiannan_Tu%40uml.edu%7C22c17ae3c57841a2318a08db1123b692%7C4c25b8a617f746f983f054734ab81fb1%7C0%7C0%7C638122617338217328%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=8eFv1wG%2FSeP0lMGNMTKt582ERpT%2FBSE5wKSIecQlPkU%3D&reserved=0>
> for trouble shooting.
>
> [0]PETSC ERROR: Petsc Release Version 3.16.6, Mar 30, 2022
>
> [0]PETSC ERROR: ./iditm3d on a  named office by jtu Fri Feb 17 14:54:22
> 2023
>
> [0]PETSC ERROR: Configure options --prefix=/usr/local
> --with-mpi-dir=/usr/local --with-fc=0 --with-openmp
> --with-hdf5-dir=/usr/local --download-f2cblaslapack=1
>
> [0]PETSC ERROR: #1 TSStep() at
> /home/jtu/Downloads/petsc-3.16.6/src/ts/interface/ts.c:3583
>
>
>
>
>
>
>
> *From: *Barry Smith <bsmith at petsc.dev>
> *Sent: *Friday, February 17, 2023 12:58 PM
> *To: *Tu, Jiannan <Jiannan_Tu at uml.edu>
> *Cc: *petsc-users <petsc-users at mcs.anl.gov>
> *Subject: *Re: [petsc-users] TS failed due to diverged_step_rejected
>
>
>
> *CAUTION:* This email was sent from outside the UMass Lowell network.
>
>
>
>
>
>   Can you please run with also the options -ts_monitor -ts_adapt_monitor ?
>
>
>
>   The output is confusing because it prints that the Nonlinear solve has
> converged but then TSStep has failed due to DIVERGED_STEP_REJECTED which
> seems contradictory
>
>
>
>
>
>
>
> On Feb 17, 2023, at 12:09 PM, Tu, Jiannan <Jiannan_Tu at uml.edu> wrote:
>
>
>
> My code uses TS to solve a set of multi-fluid MHD equations. The jacobian
> is provided with function F(t, u, u'). Both linear and nonlinear solvers
> converge but snes repeats itself until gets "TSStep has failed due to
> diverged_step_rejected."
>
>
>
> Is it because I used TSStep rather than TSSolve? I have checked the
> condition number. The condition number with pc_type asm is about 1 (without
> precondition it is about 4x10^4). The maximum ratio of off-diagonal
> jacobian element over diagonal element is about 21.
>
>
>
> Could you help me to identify what is going wrong?
>
>
>
> Thank you very much!
>
>
>
> Jiannan
>
>
>
>
> ---------------------------------------------------------------------------------------------------
>
> Run command with options
>
>
>
> mpiexec -n $1 ./iditm3d -ts_type arkimex -snes_tyep ngmres -ksp_type
> gmres -pc_type asm \
>
> -ts_rtol 1.0e-4 -ts_atol 1.0e-4 -snes_monitor -snes_rtol 1.0e-4 -snes_atol
> 1.0e-4 \
>
> -snes_converged_reason
>
>
>
> The output message is
>
>
>
> Start time advancing ...
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 1.673091274668e-03
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 8.715428433630e-02
>
>     1 SNES Function norm 4.995727626692e-04
>
>     2 SNES Function norm 5.498018152230e-08
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>     0 SNES Function norm 3.237461568254e-01
>
>     1 SNES Function norm 7.988531005091e-04
>
>     2 SNES Function norm 1.280948196292e-07
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 4.881903203545e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 7.562592690785e-02
>
>     1 SNES Function norm 1.143078818923e-04
>
>     2 SNES Function norm 9.834547907735e-09
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>     0 SNES Function norm 2.683968949758e-01
>
>     1 SNES Function norm 1.838028436639e-04
>
>     2 SNES Function norm 9.470813523140e-09
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 2
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 1.821562431175e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 1.005443458812e-01
>
>     1 SNES Function norm 3.633336946661e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 1.515368382715e-01
>
>     1 SNES Function norm 3.389298316830e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 4.541003359206e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 1.713800906043e-01
>
>     1 SNES Function norm 1.179958172167e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.020265094117e-01
>
>     1 SNES Function norm 1.513971290464e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 6.090269704320e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.136603895703e-01
>
>     1 SNES Function norm 1.877474016012e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 3.127812462507e-01
>
>     1 SNES Function norm 2.713146825704e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 2.793512213059e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.205196267430e-01
>
>     1 SNES Function norm 2.572653773308e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 3.260057361977e-01
>
>     1 SNES Function norm 2.705816087598e-06
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 2.764855860446e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.212505522844e-01
>
>     1 SNES Function norm 2.958996472386e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 3.273222034162e-01
>
>     1 SNES Function norm 2.994512887620e-05
>
>   Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
>
>     0 SNES Function norm 2.274473072186e+03
>
>     1 SNES Function norm 3.317240589134e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.213246532918e-01
>
>     1 SNES Function norm 2.799468604767e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.274570888397e-01
>
>     1 SNES Function norm 3.066048050994e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.274473072189e+03
>
>     1 SNES Function norm 2.653507278572e-03
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.213869585841e-01
>
>     1 SNES Function norm 2.177156902895e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.275136370365e-01
>
>     1 SNES Function norm 1.962849131557e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.274473072218e+03
>
>     1 SNES Function norm 5.664907315679e-03
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.223208399368e-01
>
>     1 SNES Function norm 5.688863091415e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.287121218919e-01
>
>     1 SNES Function norm 4.085338521320e-03
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.274473071968e+03
>
>     1 SNES Function norm 4.694691905235e-04
>
>   Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 2.211786508657e-01
>
>     1 SNES Function norm 1.503497433939e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
>     0 SNES Function norm 3.272667798977e-01
>
>     1 SNES Function norm 2.176132327279e-04
>
>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 1
>
> [0]PETSC ERROR: --------------------- Error Message
> --------------------------------------------------------------
>
> [0]PETSC ERROR:
>
> [0]PETSC ERROR: TSStep has failed due to DIVERGED_STEP_REJECTED
>
> [0]PETSC ERROR: See https://petsc.org/release/faq/
> <https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fpetsc.org%2Frelease%2Ffaq%2F&data=05%7C01%7CJiannan_Tu%40uml.edu%7C22c17ae3c57841a2318a08db1123b692%7C4c25b8a617f746f983f054734ab81fb1%7C0%7C0%7C638122617338217328%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=8eFv1wG%2FSeP0lMGNMTKt582ERpT%2FBSE5wKSIecQlPkU%3D&reserved=0>
>  for trouble shooting.
>
> [0]PETSC ERROR: Petsc Release Version 3.16.6, Mar 30, 2022
>
> [0]PETSC ERROR: ./iditm3d on a  named office by jtu Fri Feb 17 11:59:43
> 2023
>
> [0]PETSC ERROR: Configure options --prefix=/usr/local
> --with-mpi-dir=/usr/local --with-fc=0 --with-openmp
> --with-hdf5-dir=/usr/local --download-f2cblaslapack=1
>
> [0]PETSC ERROR: #1 TSStep() at
> /home/jtu/Downloads/petsc-3.16.6/src/ts/interface/ts.c:3583
>
>
>
>
>
>
>
>
> --
>
> 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/
> <https://nam10.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.cse.buffalo.edu%2F~knepley%2F&data=05%7C01%7CJiannan_Tu%40uml.edu%7C22c17ae3c57841a2318a08db1123b692%7C4c25b8a617f746f983f054734ab81fb1%7C0%7C0%7C638122617338217328%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=SxBlKPP1FyoO5Vkz66viMmC5kmX1MHG8XTQYKSZTiIo%3D&reserved=0>
>
>
>


-- 
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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