Solving Nonlinear Equations with Newton-Krylov Method Based on Automatic Differentiation

The Jacobian-free Newton-Krylov method is widely used in solving nonlinear equations arising in many applications. However, an effective preconditioner is required each iteration and determining such may be hard or expensive. In this paper, we propose an efficient two-sided bicoloring method to determine the lower triangular half of the sparse Jacobian matrix via automatic differentiation. Then, with this lower triangular matrix, an effective preconditioner is constructed to accelerate the convergence of the Newton-Krylov method. We demonstrate that the proposed bicoloring approach is significantly more effective than the one-sided coloring method proposed in [13] and yields an effective preconditioning strategy.