[petsc-users] Simultaneously compute Residual+Jacobian in SNES

[Jed Brown]

Patrick Sanan <patrick.sanan at gmail.com> writes:
> Maybe a good way to proceed is to have the "both at once computation"
> explicitly available in PETSc's Newton or other solvers that "know"
> that they can gain efficiency by computing both at once. That is, the
> user is always required to register a callback to form the residual,
> and may optionally register a function to compute (only) the Jacobian
> and/or another function to compute the combined Jacobian/Residual.
> Newton can look for the combined function at the times when it would
> help, and if nothing was registered would fall back to the current
> technique of calling separate residual and Jacobian routines. This
> would preserve the benefits of the current approach, allow
> optimizations of the kind mentioned, and hopefully remain
> somewhat-maintainable; in particular, this doesn't require the user to
> be promised anything about the order in which Jacobians and residuals
> are computed.

How would the solver decide?  It requires a model for the relative costs
of {residual, Jacobian, residual+Jacobian}.  Where would the SNES learn
that?

I'm not saying the above is wrong, but it adds complexity and wouldn't
be useful without the cost model.  An alternative would be for the
user's residual to be able to query the SNES's estimated probability
that a Jacobian will be needed.  So if the SNES has observed quadratic
convergence, it might be quite confident (unless it expects this
residual to meet the convergence criteria).  The advantage of this
scheme is that it applies to partial representations of the Jacobian
(like storing the linearization at quadrature points, versus assembling
the sparse matrix).
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