Searching my mail archive for something else turned up this unanswered question, so I'll reply.
Patrick Farrell <[email protected]> writes: > Hi, > > As I mentioned in a question > > (http://fenicsproject.org/qa/321/is-it-possible-to-solve-a-nonlinear-problem-matrix-free) > > I'd like to solve a problem where the Jacobian of my nonlinear problem > is dense, but its action is easy to compute. This is the setting where classical optimization methods like nonlinear CG, nonlinear GMRES, and quasi-Newton/BFGS are commonly used. If your nonlinear problem is actually the first-order optimality conditions for an optimization problem, then all the theory applies. If you don't have an objective functional or your residual is not the gradient of something, then I would recommend starting with nonlinear GMRES. There are a bunch of examples in PETSc, but it could be as simple as "-snes_type ngmres". How to expose this through Dolfin is something for you all to decide. If the spectrum of your operator decays rapidly, then the above should converge well, similar to an unpreconditioned Krylov method. This is common for many regularized inverse problems because they have a spectrum that is spectrally equivalent to a second kind Fredholm integral operator. But if you do not have such decay or if you have a great deal of geometric complexity, you may have to put some effort into preconditioning. I'd have to know a lot more about your problem to suggest preconditioning strategies, but you should try the unpreconditioned method first.
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