Newton's method linearizes viscous Hamilton–Jacobi equations directly: geometric convergence for nonconvex Hamiltonians, quadratic near the solution

Newton's method is applied directly to the nonlinear operator of the viscous Hamilton–Jacobi equation, with each step solving a linear parabolic equation, and the Hamiltonian need not be convex. Earlier policy-iteration analyses relied on a control representation or the min–max structure of a differential game, whereas this route linearizes the equation itself and instead requires suitable smoothness of the Hamiltonian. Global geometric convergence holds from every admissible initial guess satisfying the initial condition, with constants depending on the Hamiltonian and on local bounds for the modified Hamiltonian.

The iteration converges at two rates: geometric contraction away from the solution, and quadratic convergence once inside a neighborhood of it. The global estimate does not rely on a small initial error; it applies a comparison argument to the error and introduces auxiliary heat potentials driven by the gradient errors that contract from one iteration to the next. The local quadratic estimate combines the integral Taylor formula for the Newton remainder with global parabolic Schauder estimates whose constants are uniform while the iterates stay in a fixed ball around the solution.

When the Hamiltonian admits a control representation, the Newton iterates and classical policy iteration produce the same value-function sequence. Policy iteration had quantitative convergence results in convex and nonconvex settings but no equivalence characterization with Newton's method. Under differentiability of the Hamiltonian at the relevant points and unique bounded classical solutions of the policy-evaluation problems, using the value function from the initial policy as the Newton initial guess makes the two value-function sequences coincide, so they inherit the global geometric and local quadratic estimates.

The analysis targets the viscous Hamilton–Jacobi Cauchy problem, requires the Hamiltonian to satisfy stated smoothness and derivative bounds, and obtains a global gradient bound through a smooth cutoff outside the gradient range; local quadratic convergence holds while iterates stay in a fixed ball around the solution. The small-viscosity case uses a Nash–Moser variant that smooths the iterate and the residual in space, and its convergence criterion assumes a smallness condition on the initial residual. The numerical part approximates Newton corrections with physics-informed neural networks on a periodic cylinder, covering a one-dimensional exact-solution benchmark and nonconvex constructed problems in dimensions 10 to 100.

The small-viscosity Nash–Moser criterion is conditional: residual decay and convergence require the initial residual to satisfy a smallness condition compatible with the viscosity and the smoothing scale. The residual certificate is a deterministic estimate in terms of continuous residuals; in computations these residuals are approximated by quadrature or sampling, and that sampling error lies outside the stability estimate. The high-dimensional experiments use manufactured forcing, timing follows a memory-adapted batch protocol, and the indicator is used empirically without scaling. The indicator-driven stopping rule reached its threshold with true error also below threshold at dimensions 10 and 20, but this is empirical calibration for one benchmark family, not a coverage proof; at dimension 20 with threshold 0.05, all three runs failed to reach the threshold.