[petsc-users] SNES: approximating the Jacobian with computed residuals?

[Fischer, Greg A.]

Hello PETSc-users,

I'm using the SNES component with the NGMRES method in my application. I'm using a matrix-free context for the Jacobian and the MatMFFDComputeJacobian() function in my FormJacobian routine. My understanding is that this effectively approximates the Jacobian using the equation at the bottom of Page 103 in the PETSc User's Manual. This works, but the expense of computing two function evaluations in each SNES iteration nearly wipes out the performance improvements over Picard iteration. 

Based on my (limited) understanding of the Oosterlee/Washio SIAM paper ("Krylov Subspace Acceleration of Nonlinear Multigrid..."), they seem to suggest that it's possible to approximate the Jacobian with a series of previously-computed residuals (eq 2.14), rather than additional function evaluations in each iteration. Is this correct? If so, could someone point me to a reference that demonstrates how to do this with PETSc?

Or, perhaps a better question to ask is: are there other ways of reducing the computing burden associated with estimating the Jacobian?

Thanks,
Greg