Newton's method linearizes viscous Hamilton–Jacobi equations directly: geometric convergence for nonconvex Hamiltonians, quadratic near the solution
For the Cauchy problem for viscous Hamilton–Jacobi equations with Hamiltonians that need not be convex, each Newton step solves a linear parabolic equation obtained by linearizing at the current iterate, without a control representation or min–max structure. Any admissible initial guess gives geometric convergence, and initial guesses close enough to the solution give quadratic convergence in the parabolic Hölder norm; when the Hamiltonian admits a control representation, the Newton iterates coincide with classical policy iteration, which therefore inherits both rates.