Robust Matrix-Free Newton-Krylov Solvers via Automatic Differentiation

2026-05-13

AI Summary

This work addresses the susceptibility of traditional finite difference methods to floating-point round-off errors when computing Jacobian-vector products in Jacobian-Free Newton-Krylov (JFNK) solvers, which compromises linearization accuracy and solver robustness. The study proposes, for the first time, a systematic replacement of finite differences with forward-mode automatic differentiation to compute Gâteaux derivatives exactly, thereby significantly enhancing numerical precision and stability while preserving the original JFNK framework. This approach achieves a unified balance between computational efficiency and accuracy, substantially improving solver robustness—particularly in strongly nonlinear regimes or low-precision environments. Experimental results across multiple benchmark nonlinear partial differential equations demonstrate up to two to three orders of magnitude speedup in convergence and an increase in success rate from 42% to over 95%.

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.

Problem

• Jacobian-Free Newton-Krylov
• Automatic Differentiation
• Finite Differences
• Gateaux derivatives
• solver robustness

Innovation

• Automatic Differentiation
• Jacobian-Free Newton-Krylov
• Gateaux derivative
• robustness
• finite-precision