Robust Matrix-Free Newton-Krylov Solvers via Automatic Differentiation

TL;DR

This paper demonstrates that using Automatic Differentiation (AD) for Jacobian-vector products in Jacobian-Free Newton-Krylov methods significantly enhances robustness and computational efficiency across various nonlinear PDE benchmarks.

Contribution

It evaluates the impact of AD as a replacement for finite difference Jacobian-vector products in JFNK, showing substantial improvements in speed and solver robustness.

Findings

AD accelerates computation by 2-3 orders of magnitude.

AD improves minimum solver completion rate to 95%.

AD prevents degradation of Krylov operator performance.

Abstract

Jacobian-Free Newton-Krylov (JFNK) methods avoid forming the full Jacobian, but still require Jacobian-vector products, i.e., Gateaux derivatives of the nonlinear residual along Krylov directions. In standard Finite Differences (FD) formulations, these products are obtained by perturbing the Newton state and differencing residuals, making the linearization sensitive to round-off error and floating-point precision. This work evaluates the global impact of forward-mode Automatic Differentiation (AD) as a replacement for FD Jacobian-vector product in finite-precision JFNK solvers. The comparison keeps the discretization, Newton iteration, line search, Krylov methods, tolerances, and CPU/GPU backend fixed, only varying linearization strategy. Benchmarks include Burgers dynamics, Su-Olson radiation diffusion, reaction-diffusion, and nonlinear time-harmonic Maxwell equations, each evaluated in different nonlinear regimes. By preventing degradation of the Krylov operator, AD accelerates computation by 2-3 orders of magnitude across both CPU and GPU architectures. More importantly, it drastically improves global solver robustness, achieving a minimum completion rate of 95%, compared to just 42% for FD. Ultimately, accurate Gateaux derivatives unify performance and accuracy in JFNK methods, making AD the optimal choice for stiff nonlinear and reduced-precision environments.