[petsc-users] Matrix-free vs finite differenced Jacobian approximation

[Matthew Knepley]

On Tue, Dec 12, 2017 at 10:54 AM, Alexander Lindsay <
alexlindsay239 at gmail.com> wrote:

> I'm working with a relatively new set of physics (new to me) and the
> Jacobians are bad. While debugging the Jacobians, I've been running with
> different finite difference approximations. I've found in general that
> matrix-free approximation of the Jacobian action leads to much better
> convergence than explicitly forming the Jacobian matrix using finite
> differences. Should I be surprised by this or is this something that's
> known? It would be great if anyone has a reference they could point me to.
>

They should give the same answer. Maybe I do not understand what you are
doing. First, use -pc_type lu so that the linear solver is not a factor.

   Matt


> Just to illustrate the different solves:
>
> Matrix-free (preconditioner formed from finite-differenced approximation
> of Jacobian):
>
>  0 Nonlinear |R| = 2.259203e-02
>       0 Linear |R| = 2.259203e-02
>       1 Linear |R| = 2.259203e-02
>       2 Linear |R| = 1.682777e-02
>       3 Linear |R| = 9.274378e-09
>  1 Nonlinear |R| = 1.744830e-02
>       0 Linear |R| = 1.744830e-02
>       1 Linear |R| = 2.335817e-08
>  2 Nonlinear |R| = 2.704512e-08
>       0 Linear |R| = 2.704512e-08
>       1 Linear |R| = 1.265577e-14
>  3 Nonlinear |R| = 1.478929e-10
>  Solve Converged!
>
> Explicit formation of Jacobian using finite-differences (preconditioner
> formed from same matrix):
>
>  0 Nonlinear |R| = 2.259203e-02
>       0 Linear |R| = 2.259203e-02
>       1 Linear |R| = 2.259203e-02
>       2 Linear |R| = 1.481452e-12
>  1 Nonlinear |R| = 2.258733e-02
>       0 Linear |R| = 2.258733e-02
>       1 Linear |R| = 2.258520e-02
>       2 Linear |R| = 1.594456e-07
>  2 Nonlinear |R| = 2.258733e-02
>       0 Linear |R| = 2.258733e-02
>       1 Linear |R| = 2.258520e-02
>       2 Linear |R| = 1.869913e-07
> Nonlinear solve did not converge due to DIVERGED_LINE_SEARCH iterations 2
>
>
>
>


-- 
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/~mk51/>
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