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

[Smith, Barry F.]

> On Sep 7, 2018, at 5:39 AM, Yingjie Wu <yjwu16 at gmail.com> wrote:
> 
> 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? 

   SNESMonitorSet()

> 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/