[firedrake] Pass approximate Jacobian to derivative() call

[Lawrence Mitchell]

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On 11/11/15 07:33, Buesing, Henrik wrote:
> Dear all,
> 
> 
> 
> is it possible to just pass the approximate Jacobian to the
> derivative call? Since it’s so complicated in my case to get the
> exact Jacobian I would like to do sth. like
> 
> 
> 
> du = [(F(u + h) – F(u - h))/(2*h)]

So you can't do this by passing something to derivative (which instead
computes the jacobian for you given a residual).  However, you can ask
PETSc to compute a finite difference approximation to your Jacobian
using colouring, it, however, uses one-sided, rather than centred
differences:

J = (F(u + h) - F(u))/h


if you pass:

solver_parameters={'snes_fd_color': True}

Then PETSc will compute the Jacobian using finite differencing with
colouring.  This requires an additional Ncolours residual computations
per Newton iteration, where Ncolours depends on the sparsity structure
of the matrix: for 3D it will typically be around 20.


You can instead of course build an approximate, symbolic, Jacobian "by
hand" and provide that to the solver: you'll then end up doing
quasi-newton, rather than newton.

Cheers,

Lawrence

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