Adaptive Residual-Driven Newton Solver for Nonlinear Systems of Equations
Renjie Ding, Dongling Wang
TL;DR
The paper tackles slow convergence of Newton-type solvers for nonlinear systems with unbalanced nonlinearities by introducing Adaptive Residual-Driven Newton (ARDN), which assigns adaptive component-wise weights to the residuals to balance nonlinear effects during line search. ARDN augments the standard inexact Newton framework with a weighted merit function, a residual-driven weight update, and optional integration with PCA-based preconditioning (PIN^L) to further accelerate convergence. Across chemical equilibrium, convection–diffusion, and several challenging test problems, ARDN consistently reduces iteration counts and stagnations, and often outperforms or complements existing acceleration strategies. The approach offers a practical, low-overhead improvement that can be combined with other solvers to boost robustness and efficiency in solving nonlinear systems.
Abstract
Newton-type solvers have been extensively employed for solving a variety of nonlinear system of algebraic equations. However, for some complex nonlinear system of algebraic equations, efficiently solving these systems remains a challenging task. The primary reason for this challenge arises from the unbalanced nonlinearities within the nonlinear system. Therefore, accurately identifying and balancing the unbalanced nonlinearities in the system is essential. In this work, we propose a residual-driven adaptive strategy to identify and balance the nonlinearities in the system. The fundamental idea behind this strategy is to assign an adaptive weight multiplier to each component of the nonlinear system, with these weight multipliers increasing according to a specific update rule as the residual components increase, thereby enabling the Newton-type solver to select a more appropriate step length, ensuring that each component in the nonlinear system experiences sufficient reduction rather than competing against each other. More importantly, our strategy yields negligible additional computational overhead and can be seamlessly integrated with other Newton-type solvers, contributing to the improvement of their efficiency and robustness. We test our algorithm on a variety of benchmark problems, including a chemical equilibrium system, a convective diffusion problem, and a series of challenging nonlinear systems. The experimental results demonstrate that our algorithm not only outperforms existing Newton-type solvers in terms of computational efficiency but also exhibits superior robustness, particularly in handling systems with highly imbalanced nonlinearities.