[petsc-users] Issue of mg_coarse_ksp not converge
Wenbo Zhao
zhaowenbo.npic at gmail.com
Mon Oct 2 10:56:13 CDT 2017
On Mon, Oct 2, 2017 at 11:49 PM, Mark Adams <mfadams at lbl.gov> wrote:
> Whenbo, do you build your PETSc?
>
> Yes.
My configure option is listed below
./configure --with-mpi=1 --with-shared-libraries=1 \
--with-64-bit-indices=1 --with-debugging=1
And I set PETSC_DIR, PETSC_ARCH and SLEPC_DIR in my ~/.bashrc.
The Makefile for my problem is listed below,
PETSC_ARCH = arch-linux2-c-debug
PETSC_DIR = /home/zhaowenbo/research/petsc/petsc_git
SLEPC_DIR = /home/zhaowenbo/research/slepc/slepc_git
#PETSC_DIR = /home/zhaowenbo/research/petsc/petsc-3.7.4
#SLEPC_DIR = /home/zhaowenbo/research/slepc/slepc-3.7.3
HYPRE_DIR = /usr/local/hypre
#
DEBUG_OPT = -g
COMP_FLAGS = -fPIC -Wall \
-I${SLEPC_DIR}/include -I${SLEPC_DIR}/${PETSC_ARCH}/include \
-I${PETSC_DIR}/include -I${PETSC_DIR}/${PETSC_ARCH}/include \
-Isrc
LINK_FLAGS = -fPIC -Wall \
-Wl,-rpath,${SLEPC_DIR}/${PETSC_ARCH}/lib
-L${SLEPC_DIR}/${PETSC_ARCH}/lib -lslepc \
-Wl,-rpath,${PETSC_DIR}/${PETSC_ARCH}/lib
-L${PETSC_DIR}/${PETSC_ARCH}/lib -lpetsc \
-llapack -lblas -lhwloc -lm -lgfortran -lquadmath
step-41: src/main.o src/readinp.o src/base.o src/sp3.o src/diffu.o
mpicxx -o step-41 $^ ${LINK_FLAGS} ${DEBUG_OPT}
src/main.o: src/main.c
mpicxx -o src/main.o -c $^ ${COMP_FLAGS} ${DEBUG_OPT}
src/readinp.o: src/readinp.c
mpicxx -o src/readinp.o -c $^ ${COMP_FLAGS} ${DEBUG_OPT}
src/sp3.o: src/sp3.c
mpicxx -o src/sp3.o -c $^ ${COMP_FLAGS} ${DEBUG_OPT}
src/diffu.o: src/diffu.c
mpicxx -o src/diffu.o -c $^ ${COMP_FLAGS} ${DEBUG_OPT}
src/base.o: src/base.c
mpicxx -o src/base.o -c $^ ${COMP_FLAGS} ${DEBUG_OPT}
clean:
rm step-41 src/main.o src/readinp.o src/sp3.o src/diffu.o src/base.o
runkr_smooth:
mpirun -n ${NCORE} ./step-41 \
-st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
-st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 1 \
-mata AMAT.dat -matb BMAT.dat \
-st_gamg_est_ksp_view -st_gamg_est_ksp_monitor \
-st_gamg_est_ksp_converged_reason \
-eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_smooth 2>&1
runkr_nonsmooth:
mpirun -n ${NCORE} ./step-41 \
-st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
-st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 0 \
-mata AMAT.dat -matb BMAT.dat \
-st_gamg_est_ksp_view -st_gamg_est_ksp_monitor \
-st_gamg_est_ksp_converged_reason \
-eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_nonsmooth 2>&1
Thanks,
Wenbo
> On Mon, Oct 2, 2017 at 11:45 AM, Mark Adams <mfadams at lbl.gov> wrote:
>
>> This is normal:
>>
>> Linear st_gamg_est_ solve did not converge due to DIVERGED_ITS iterations
>> 10
>>
>> It looks like ksp->errorifnotconverged got set somehow. If the default
>> changed in KSP then (SAGG) GAMG would not ever work.
>>
>> I assume you don't have a .petscrc file with more (crazy) options in it
>> ...
>>
>>
>> On Mon, Oct 2, 2017 at 11:39 AM, Wenbo Zhao <zhaowenbo.npic at gmail.com>
>> wrote:
>>
>>>
>>>
>>> On Mon, Oct 2, 2017 at 11:30 PM, Matthew Knepley <knepley at gmail.com>
>>> wrote:
>>>
>>>> On Mon, Oct 2, 2017 at 11:15 AM, Mark Adams <mfadams at lbl.gov> wrote:
>>>>
>>>>> non-smoothed aggregation is converging very fast. smoothed fails in
>>>>> the eigen estimator.
>>>>>
>>>>> Run this again with -st_gamg_est_ksp_view and
>>>>> -st_gamg_est_ksp_monitor, and see if you get more output (I'm not 100% sure
>>>>> about these args).
>>>>>
>>>>
>>>> I also want -st_gamg_est_ksp_converged_reason
>>>>
>>>> Thanks,
>>>>
>>>> Matt
>>>>
>>> $make NCORE=1 runkr_smooth
>>> mpirun -n 1 ./step-41 \
>>> -st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 1 \
>>> -mata AMAT.dat -matb BMAT.dat \
>>> -st_gamg_est_ksp_view -st_gamg_est_ksp_monitor \
>>> -st_gamg_est_ksp_converged_reason \
>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_smooth 2>&1
>>> makefile:43: recipe for target 'runkr_smooth' failed
>>> make: *** [runkr_smooth] Error 91
>>>
>>> Thanks
>>> Wenbo
>>>
>>>
>>>>
>>>>
>>>>>
>>>>> On Mon, Oct 2, 2017 at 11:06 AM, Wenbo Zhao <zhaowenbo.npic at gmail.com>
>>>>> wrote:
>>>>>
>>>>>> Matt,
>>>>>>
>>>>>> Test 1 nonsmooth
>>>>>> zhaowenbo at ubuntu:~/test_slepc/SPARK/spark$ make NCORE=1
>>>>>> runkr_nonsmooth
>>>>>> mpirun -n 1 ./step-41 \
>>>>>> -st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
>>>>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 0 \
>>>>>> -mata AMAT.dat -matb BMAT.dat \
>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_nonsmooth
>>>>>> 2>&1
>>>>>>
>>>>>> Test 2 smooth
>>>>>> zhaowenbo at ubuntu:~/test_slepc/SPARK/spark$ make NCORE=1 runkr_smooth
>>>>>> mpirun -n 1 ./step-41 \
>>>>>> -st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
>>>>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 1 \
>>>>>> -mata AMAT.dat -matb BMAT.dat \
>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_smooth 2>&1
>>>>>> makefile:43: recipe for target 'runkr_smooth' failed
>>>>>> make: *** [runkr_smooth] Error 91
>>>>>>
>>>>>>
>>>>>> Thanks,
>>>>>>
>>>>>> Wenbo
>>>>>>
>>>>>> On Mon, Oct 2, 2017 at 10:48 PM, Matthew Knepley <knepley at gmail.com>
>>>>>> wrote:
>>>>>>
>>>>>>> On Mon, Oct 2, 2017 at 10:43 AM, Wenbo Zhao <
>>>>>>> zhaowenbo.npic at gmail.com> wrote:
>>>>>>>
>>>>>>>> Mark,
>>>>>>>>
>>>>>>>> Thanks for your reply.
>>>>>>>>
>>>>>>>> On Mon, Oct 2, 2017 at 9:51 PM, Mark Adams <mfadams at lbl.gov> wrote:
>>>>>>>>
>>>>>>>>> Please send the output with -st_ksp_view and -st_ksp_monitor and
>>>>>>>>> we can start to debug it.
>>>>>>>>>
>>>>>>>>> Test 1 with nonsmooth and preonly is OK
>>>>>>>> zhaowenbo at ubuntu:~/test_slepc/SPARK/spark$ make NCORE=1
>>>>>>>> runkr_nonsmooth
>>>>>>>> mpirun -n 1 ./step-41 \
>>>>>>>> -st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 0
>>>>>>>> \
>>>>>>>> -st_ksp_view -mata AMAT.dat -matb BMAT.dat \
>>>>>>>> -st_mg_coarse_ksp_type preonly -st_mg_coarse_ksp_monitor \
>>>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_nonsmooth
>>>>>>>> 2>&1
>>>>>>>>
>>>>>>>> Test 2 smooth and preonly is not OK
>>>>>>>> zhaowenbo at ubuntu:~/test_slepc/SPARK/spark$ make NCORE=1
>>>>>>>> runkr_smooth
>>>>>>>> mpirun -n 1 ./step-41 \
>>>>>>>> -st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 1
>>>>>>>> \
>>>>>>>> -st_ksp_view -mata AMAT.dat -matb BMAT.dat \
>>>>>>>> -st_mg_coarse_ksp_type preonly -st_mg_coarse_ksp_monitor \
>>>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_smooth 2>&1
>>>>>>>> makefile:43: recipe for target 'runkr_smooth' failed
>>>>>>>> make: *** [runkr_smooth] Error 91
>>>>>>>>
>>>>>>>> Test 3 nonsmooth and gmres is not OK
>>>>>>>> zhaowenbo at ubuntu:~/test_slepc/SPARK/spark$ make NCORE=1
>>>>>>>> runkr_gmres
>>>>>>>> mpirun -n 1 ./step-41 \
>>>>>>>> -st_ksp_type gmres -st_ksp_view -st_ksp_monitor \
>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 0
>>>>>>>> \
>>>>>>>> -st_ksp_view -mata AMAT.dat -matb BMAT.dat \
>>>>>>>> -st_mg_coarse_ksp_type gmres -st_mg_coarse_ksp_monitor
>>>>>>>> -st_mg_coarse_ksp_rtol 1.0e-6 \
>>>>>>>>
>>>>>>>
>>>>>>> DO NOT DO THIS. Please send the output where you do NOTHING to the
>>>>>>> coarse solver.
>>>>>>>
>>>>>>> Thanks,
>>>>>>>
>>>>>>> Matt
>>>>>>>
>>>>>>>
>>>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view > log_gmres 2>&1
>>>>>>>> makefile:59: recipe for target 'runkr_gmres' failed
>>>>>>>> make: *** [runkr_gmres] Error 91
>>>>>>>>
>>>>>>>> log-files is attached.
>>>>>>>>
>>>>>>>>
>>>>>>>> You mentioned that B is not symmetric. I assume it is elliptic
>>>>>>>>> (diffusion). Where does the asymmetry come from?
>>>>>>>>>
>>>>>>>>>
>>>>>>>> It is a two-group diffusion equations, where group denotes neutron
>>>>>>>> enegry discretisation.
>>>>>>>> Matrix B consists of neutron diffusion/leakage term, removal term
>>>>>>>> and minus neutron scatter source term between different energies, when
>>>>>>>> matrix A denotes neutron fission source.
>>>>>>>>
>>>>>>>> Diffusion term(Laplace operator) is elliptic and symmetric. Removal
>>>>>>>> term is diagonal only. However scatter term is asymmetry since scatter term
>>>>>>>> from high energy to low energy is far greater than the term from low to
>>>>>>>> high.
>>>>>>>>
>>>>>>>>
>>>>>>>> Wenbo
>>>>>>>>
>>>>>>>>
>>>>>>>>> On Mon, Oct 2, 2017 at 9:39 AM, Wenbo Zhao <
>>>>>>>>> zhaowenbo.npic at gmail.com> wrote:
>>>>>>>>>
>>>>>>>>>> Matt,
>>>>>>>>>> Thanks for your reply.
>>>>>>>>>> For the defalt option doesnt work firstly( -st_ksp_type gmres
>>>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg -st_pc_gamg_agg_nsmooths 1), I tried
>>>>>>>>>> to test those options.
>>>>>>>>>>
>>>>>>>>>> Wenbo
>>>>>>>>>>
>>>>>>>>>> On Mon, Oct 2, 2017 at 9:08 PM, Matthew Knepley <
>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>
>>>>>>>>>>> On Mon, Oct 2, 2017 at 8:30 AM, Wenbo Zhao <
>>>>>>>>>>> zhaowenbo.npic at gmail.com> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> Matt
>>>>>>>>>>>>
>>>>>>>>>>>> Because I am not clear about what will happen using 'preonly'
>>>>>>>>>>>> for large scale problem.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> The size of the problem has nothing to do with 'preonly'. All it
>>>>>>>>>>> means is to apply a preconditioner without a Krylov solver.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>> It seems to use a direct solver from below,
>>>>>>>>>>>> http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/
>>>>>>>>>>>> KSP/KSPPREONLY.html
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> However, I still cannot understand why you would change the
>>>>>>>>>>> default?
>>>>>>>>>>>
>>>>>>>>>>> Matt
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Thanks!
>>>>>>>>>>>> Wenbo
>>>>>>>>>>>>
>>>>>>>>>>>> On Mon, Oct 2, 2017 at 5:09 PM, Matthew Knepley <
>>>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>> On Sun, Oct 1, 2017 at 9:53 PM, Wenbo Zhao <
>>>>>>>>>>>>> zhaowenbo.npic at gmail.com> wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Matt,
>>>>>>>>>>>>>> Thanks for your reply.
>>>>>>>>>>>>>> It DOES make no sense for this problem.
>>>>>>>>>>>>>> But I am not clear about the 'preonly' option. Which solver
>>>>>>>>>>>>>> is used in preonly? I wonder if 'preonly' is suitable for large scale
>>>>>>>>>>>>>> problem such as 400,000,000 unknowns.
>>>>>>>>>>>>>> So I tried 'gmres' option and found these error messages.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> I mean, why are you setting this at all. Just do not set the
>>>>>>>>>>>>> coarse solver. The default should work fine.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>>
>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Could you give me some suggestions?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Thanks.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Wenbo
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> On Mon, Oct 2, 2017 at 12:34 AM, Matthew Knepley <
>>>>>>>>>>>>>> knepley at gmail.com> wrote:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> On Sun, Oct 1, 2017 at 6:49 AM, Wenbo Zhao <
>>>>>>>>>>>>>>> zhaowenbo.npic at gmail.com> wrote:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Hi,
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> I met some questions when I use PETSC/SLEPC to solve
>>>>>>>>>>>>>>>> two-group neutron diffusion equations with finite difference method. The
>>>>>>>>>>>>>>>> grid is 3*3*3, when DOF on each points is 2. So the matrix size is 54*54.
>>>>>>>>>>>>>>>> It is generalized eigenvalue problem Ax=\lamda Bx, where B
>>>>>>>>>>>>>>>> is diagonally dominant matrix but not symmetry.
>>>>>>>>>>>>>>>> EPS is set as below,
>>>>>>>>>>>>>>>> ierr = EPSSetProblemType(eps,EPS_GNHEP);CHKERRQ(ierr);¬
>>>>>>>>>>>>>>>> ierr = EPSSetWhichEigenpairs(eps,EPS_
>>>>>>>>>>>>>>>> LARGEST_REAL);CHKERRQ(ierr);¬
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Krylovschur is used as eps sovler. GAMG is used as PC.
>>>>>>>>>>>>>>>> I tried agg_nsmooths and mg_coarse_ksp_type. Only
>>>>>>>>>>>>>>>> non-smooths and preonly is OK.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Why are you setting the coarse solver. This makes no sense.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Matt
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Test 1
>>>>>>>>>>>>>>>> $ make NCORE=1 runkr_nonsmooth
>>>>>>>>>>>>>>>> mpirun -n 1 ./step-41 \
>>>>>>>>>>>>>>>> -st_ksp_type gmres \
>>>>>>>>>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg
>>>>>>>>>>>>>>>> -st_pc_gamg_agg_nsmooths 0 \
>>>>>>>>>>>>>>>> -st_ksp_view -mata AMAT.dat -matb BMAT.dat \
>>>>>>>>>>>>>>>> -st_mg_coarse_ksp_type preonly
>>>>>>>>>>>>>>>> -st_mg_coarse_ksp_monitor \
>>>>>>>>>>>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view >
>>>>>>>>>>>>>>>> log_nonsmooth 2>&1
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Test 2
>>>>>>>>>>>>>>>> $ make NCORE=1 runkr_smooth
>>>>>>>>>>>>>>>> mpirun -n 1 ./step-41 \
>>>>>>>>>>>>>>>> -st_ksp_type gmres \
>>>>>>>>>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg
>>>>>>>>>>>>>>>> -st_pc_gamg_agg_nsmooths 1 \
>>>>>>>>>>>>>>>> -st_ksp_view -mata AMAT.dat -matb BMAT.dat \
>>>>>>>>>>>>>>>> -st_mg_coarse_ksp_type preonly
>>>>>>>>>>>>>>>> -st_mg_coarse_ksp_monitor \
>>>>>>>>>>>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view >
>>>>>>>>>>>>>>>> log_smooth 2>&1
>>>>>>>>>>>>>>>> makefile:43: recipe for target 'runkr_smooth' failed
>>>>>>>>>>>>>>>> make: *** [runkr_smooth] Error 91
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Test 3
>>>>>>>>>>>>>>>> $ make NCORE=1 runkr_gmres
>>>>>>>>>>>>>>>> mpirun -n 1 ./step-41 \
>>>>>>>>>>>>>>>> -st_ksp_type gmres \
>>>>>>>>>>>>>>>> -st_pc_type gamg -st_pc_gamg_type agg
>>>>>>>>>>>>>>>> -st_pc_gamg_agg_nsmooths 0 \
>>>>>>>>>>>>>>>> -st_ksp_view -mata AMAT.dat -matb BMAT.dat \
>>>>>>>>>>>>>>>> -st_mg_coarse_ksp_type gmres -st_mg_coarse_ksp_monitor
>>>>>>>>>>>>>>>> -st_mg_coarse_ksp_rtol 1.0e-6 \
>>>>>>>>>>>>>>>> -eps_nev 1 -eps_ncv 10 -eps_monitor -log_view >
>>>>>>>>>>>>>>>> log_gmres 2>&1
>>>>>>>>>>>>>>>> makefile:59: recipe for target 'runkr_gmres' failed
>>>>>>>>>>>>>>>> make: *** [runkr_gmres] Error 91
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Log files were attched.
>>>>>>>>>>>>>>>> The matrix file were also attched as AMAT.dat and BMAT.dat.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Is it correct? Or something wrong with my code or
>>>>>>>>>>>>>>>> commad-line?
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Thanks!
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Wenbo
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>> 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.caam.rice.edu/%7Emk51/>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> --
>>>>>>>>>>>>> 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.caam.rice.edu/%7Emk51/>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> --
>>>>>>>>>>> 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.caam.rice.edu/%7Emk51/>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> --
>>>>>>> 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.caam.rice.edu/%7Emk51/>
>>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>
>>>>
>>>> --
>>>> 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.caam.rice.edu/%7Emk51/>
>>>>
>>>
>>>
>>
>
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