For large system typically derived from nonlinear PDE with ill-conditonned Jacobian (due to a near symmetry or close to fold bifurcation), using an iterative solver (even with a good precondtionner), can cause the Newton method to diverge, as the near zero mode are amplified too much at each Newton iteration.
A solution is to to adapt the tolerance of the iterative solver where the tolerance is increased far from the solution and reduce near to it, and combine it with linearsearch to avoid overcorrecting along near-zero eigenmode.
This issue is more problematic for the initial solve before starting the continuation, as in the latter, the initial guess is often sufficiently good to ensure quadratic convergence.
This would require a slight modification of the API to allow passing a tolerance to the solver call.
Then a dedicated InexactNewton(prob, par::NewtonPar, ...) could be added.
For large system typically derived from nonlinear PDE with ill-conditonned Jacobian (due to a near symmetry or close to fold bifurcation), using an iterative solver (even with a good precondtionner), can cause the Newton method to diverge, as the near zero mode are amplified too much at each Newton iteration.
A solution is to to adapt the tolerance of the iterative solver where the tolerance is increased far from the solution and reduce near to it, and combine it with linearsearch to avoid overcorrecting along near-zero eigenmode.
This issue is more problematic for the initial solve before starting the continuation, as in the latter, the initial guess is often sufficiently good to ensure quadratic convergence.
This would require a slight modification of the API to allow passing a tolerance to the solver call.
Then a dedicated
InexactNewton(prob, par::NewtonPar, ...)could be added.