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Re: Matrix free example snes/ex20.c


  Vijay,

    This is a very cool problem.

Because of the exact symmetry of the domain the EXACT Jacobian
at each step has exactly 9 different eigenvalues. This means the
GMRES will take exactly 9 iterations (and "completely" converge in
the ninth iteration) if the "exact" Jacobian is used. You can run with
-pc_type none -snes_mf -ksp_monitor_singular_value - ksp_plot_eigenvalues -display :0.0 -draw_pause -1
to see the 9 eigenvalues.

Now run without the -snes_mf option. You will see the first Newton iteration's
eigenvalues still look like 9; but starting at the second Newton iteration the
"identical" eigenvalues are now not all identically placed so GMRES needs
more iterations.

The question then becomes how come the matrix-free application of
the Jacobian is more accurate than actually computing it as a sparse
matrix then applying it? Here is my non-rigorous answer; the multiplication
of the sparse matrix values (even if very accurate) against the vector introduces
some rounding error that screws up the eigenvalues slightly. For some reason
for this problem the matrix-free application is accurate enough not to
perturb the eigenvalues.

  Barry





On Dec 30, 2007, at 2:19 PM, Vijay M wrote:

I ran both the cases with -ksp_monitor on and have attached the output in
two different files. 1.txt is the Jfree case and 2.txt is the analytical
case.

Barry, the example problem is ex20 from the snes tutorial directory. The
petsc version is 2.3.3-p7 if that helps to clear things a little. Now I
haven't yet completely checked for a bug in the analytical Jacobian but I
would imagine that if it were incorrect, wouldn't that affect only how the
nonlinear iteration converges and not the linear iteration since the matrix
sparsity structure is still the same (well assuming the condition number is
not very different from the exact Jacobian !). Just my 2 cents.

Anyway, I will look into the code for ex20 and then see if something is
messed up. Let me know if you find out the problem from the output.

Thanks,
Vijay

I understand that both the methods will not give me the same number
of total
linear iterations but a factor of 2 seems a little odd to me.


   Yes, this is surprising.

   Run with -ksp_monitor how are the linear convergence different?

This leads to
another question whether the user can actually change the epsilon
used for
computing the perturbation in J-free scheme or is this fixed in
PETSc ?


   Yes, see the manual page for MatMFFDSetFromOptions() and related
manual
pages.



If not, then what do you think is the reason for this ? 

   Bug in your analytic Jacobian? Run with -snes_monitor and -
ksp_monitor and
send all output.

   Barry

Do let me know your
comments when you get some time. Thanks.

Vijay

-----Original Message-----
From: owner-petsc-users@xxxxxxxxxxx [mailto:owner-petsc-users@xxxxxxxxxxx
]
On Behalf Of Matthew Knepley
Sent: Saturday, December 29, 2007 9:05 PM
To: petsc-users@xxxxxxxxxxx
Subject: Re: Matrix free example snes/ex20.c

On Dec 29, 2007 8:07 PM, Vijay M <vijay.m@xxxxxxxxx> wrote: 
Hi all,

I was trying to compile and run the ex20.c example code in the
tutorial
section of SNES. Although it does not explicitly specify that -
snes_mf
option can be used, my understanding is that as long as a nonlinear
residual 
function is written correctly, PETSc will calculate via finite
difference
the action of the Jacobian on a given vector. Is that correct ?


Yes.

Now if that is the case, then please observe the discrepancy in the
number
of linear iterations taken with an analytical Jacobian and matrix-
free
option. What puzzles me is that the SNES function norm are quite
close for
both the methods but the linear iterations differ by a factor of 3.
Why
exactly is this ?


There is no PC when using -snes_mf whereas the default is ILU for the
analytic
Jacobian.

 Matt

Here's the output to make this clearer.

vijay :mpirun -np 1 ex20 -ksp_type gmres -snes_monitor

0 SNES Function norm 2.271442542876e-01

1 SNES Function norm 6.881516100891e-02

2 SNES Function norm 1.813939751552e-02

3 SNES Function norm 2.354176462207e-03

4 SNES Function norm 3.063728077362e-05

5 SNES Function norm 3.106106268946e-08

6 SNES Function norm 5.344742712545e-12

0 SNES Function norm 2.271442542876e-01

1 SNES Function norm 6.881516100891e-02

2 SNES Function norm 1.813939751552e-02

3 SNES Function norm 2.354176462207e-03

4 SNES Function norm 3.063728077362e-05

5 SNES Function norm 3.106106268946e-08

6 SNES Function norm 5.344742712545e-12

Number of Newton iterations = 6

Number of Linear iterations = 18

Average Linear its / Newton = 3.000000e+00



vijay :mpirun -np 1 ex20 -ksp_type gmres -snes_monitor -snes_mf

0 SNES Function norm 2.271442542876e-01

1 SNES Function norm 6.870629867542e-02

2 SNES Function norm 1.804335379848e-02

3 SNES Function norm 2.290074339682e-03

4 SNES Function norm 3.082384186373e-05

5 SNES Function norm 3.926396277038e-09

6 SNES Function norm 3.754922566585e-16

0 SNES Function norm 2.271442542876e-01

1 SNES Function norm 6.870629867542e-02

2 SNES Function norm 1.804335379848e-02

3 SNES Function norm 2.290074339682e-03

4 SNES Function norm 3.082384186373e-05

5 SNES Function norm 3.926396277038e-09

6 SNES Function norm 3.754922566585e-16

Number of Newton iterations = 6

Number of Linear iterations = 54

Average Linear its / Newton = 9.000000e+00



Thanks,

Vijay





--
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

<1.txt><2.txt>