Adaptive Residual-Driven Newton Solver for Nonlinear Systems of Equations

TL;DR

This paper introduces an adaptive residual-driven strategy for Newton-type solvers that improves efficiency and robustness in solving complex nonlinear systems with unbalanced nonlinearities by dynamically adjusting component weights.

Contribution

The paper proposes a novel residual-driven adaptive weighting strategy that enhances Newton solver performance on challenging nonlinear systems with minimal additional computational cost.

Findings

Outperforms existing solvers in efficiency on benchmark problems

Demonstrates superior robustness with highly imbalanced nonlinearities

Seamlessly integrates with existing Newton-type methods

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.