I have a nonlinear function
fun with n inputs and n outputs. I also have a function
jac which calculates the Jacobian, which is sparse. Which SciPy method, if any, can I use to find the root of
I am surprised that I can't find a method in
scipy.optimize that takes a Jacobian in sparse matrix format. But I believe this can be simply solved by the Newton's method, and so I am confused if such a method does not exist. There are many quasi-Newton methods in this package that estimate the Jacobian, but they do not seem quite right. The only thing that looks similar is
newton_krylov which documents something about sparse preconditioners. Is this what I should be looking at? If there is no such method, should I just do something like the following?
while not some_stopping_condition(x): x -= scipy.sparse.spsolve(jac(x), fun(x))
I would prefer to use an existing SciPy function in case I do something stupid. Notably, as it happens, I believe my particular problem is poorly conditioned, and I think I would benefit from being able to use
lsmr which takes a damping parameter.
Incidentally, the problem is solving large-displacement elasticity equations on a 2D membrane in 3D space with FEM. The problem is sparse because it is FEM. The problem is nonlinear because of the large displacements (and the corresponding Green-Lagrange strains). The poor conditioning comes from the fact that a small perturbation of any node in the direction perpendicular to the surface creates little change in strain energy (and the flatter the surface, the smaller the change).