[petsc-users] Approximation of matrix vector product by finite difference method

[Yingjie Wu]

Thank you very much for your reply.

I'm a bit confused if I use what you recommend:  - snes_fd -pc_type lu
means that I explicitly construct the Jacobian matrix using the finite
difference method, construct the precondition matrix using the completely
LU decomposition, and solve the step size \ delta x with the GMRES method
(default).

In fact, what I want to use is to approximate the vector product of a
matrix with finite difference, so that the explicit construction of
Jacobian matrices can be avoided. If so, should I use MatrixFreeMethod? How
should I set it up? If I want to set up precondition, what do I need to
add?

In addition, I want to output variables in each nolinear step. What should
I add code to make SNES step by step?
There may be many problems, but they bother me very much. I am looking
forward to your reply.

Thanks,
Yingjie


Matthew Knepley <knepley at gmail.com> 于2018年9月6日周四 下午10:34写道：

> On Thu, Sep 6, 2018 at 4:47 AM Yingjie Wu <yjwu16 at gmail.com> wrote:
>
>> Dear Petsc developer:
>> Hi,
>> Thank you for your previous help.
>> I recently modeled on PETSc's SNES example and wrote a computer program
>> myself. This program is mainly for solving nonlinear equations of thermal
>> hydraulics.
>>
>> ∇·（λ∇T） - ∇_y（ρ*Cp*u） - T_source = 0
>>
>> w*ρ*u = ρg - ∇_y（P）
>>
>>      ∇·（ 1/w * ∇P ） =  - ∇（ ρg / w ）
>>
>> Where P, T and u are variables, the distribution represents pressure,
>> temperature and velocity. The rest are nonlinear physical parameters and
>> constants.
>>
>> Because the program is very preliminary, so I use - snes_mf so that I can
>> save the part of writing to calculate the Jacobian matrix.
>>
>> After compiling and passing, I found that the residual function had not
>> dropped to a small enough level, but the program stopped, as follows:
>>
>> Setting Up: -snes_mf  -snes_monitor  -draw_pause  10  -snes_view
>>
>>
> First, do not use -snes_mf. It is not for testing, but for sophisticated
> use. The first option you
> might try is
>
>    -snes_fd -pc_type lu
>
> That uses a full Jacobian and LU factorization for a direct solve. Always
> run the solve using
>
>   -snes_view -snes_converged_reason -snes_monitor -ksp_converged_reason
> -ksp_monitor_true_residual
>
> When that gets too expensive, you can try
>
>   -snes_fd_color -snes_fd_color_use_mat -mat_coloring_type greedy
>
> but that requires you to preallocate the Jacobian matrix correctly.
>
>   Thanks,
>
>     Matt
>
>> 0 SNES Function norm 3.724996516631e+09
>>
>>   1 SNES Function norm 2.194322909557e+09
>>
>>   2 SNES Function norm 1.352051559826e+09
>>
>>   3 SNES Function norm 1.522311916217e+08
>>
>> SNES Object: 1 MPI processes
>>
>>   type: newtonls
>>
>>   maximum iterations=50, maximum function evaluations=10000
>>
>>   tolerances: relative=1e-08, absolute=1e-50, solution=1e-08
>>
>>   total number of linear solver iterations=1298
>>
>>   total number of function evaluations=11679
>>
>>   norm schedule ALWAYS
>>
>>   SNESLineSearch Object: 1 MPI processes
>>
>>     type: bt
>>
>>       interpolation: cubic
>>
>>       alpha=1.000000e-04
>>
>>     maxstep=1.000000e+08, minlambda=1.000000e-12
>>
>>     tolerances: relative=1.000000e-08, absolute=1.000000e-15,
>> lambda=1.000000e-08
>>
>>     maximum iterations=40
>>
>>   KSP Object: 1 MPI processes
>>
>>     type: gmres
>>
>>       restart=30, using Classical (unmodified) Gram-Schmidt
>> Orthogonalization with no iterative refinement
>>
>>       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: 1 MPI processes
>>
>>     type: none
>>
>>     linear system matrix = precond matrix:
>>
>>     Mat Object: 1 MPI processes
>>
>>       type: mffd
>>
>>       rows=300, cols=300
>>
>>         Matrix-free approximation:
>>
>>           err=1.49012e-08 (relative error in function evaluation)
>>
>>           Using wp compute h routine
>>
>>               Does not compute normU
>>
>> I would like to know why the residual function can not continue to
>> decline, and why the program will stop before convergence.
>> I do not know much about the convergence criteria and convergence rules
>> of PETSc for solving nonlinear equations. I hope I can get your help.
>> I'm looking forward to your reply~
>>
>> Thanks,
>> Yingjie
>>
>
>
> --
> 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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