The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive gamma-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. The key features of adaptive gamma-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems automatically, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. The result is a flexible algorithm that performs well for singular and nonsingular problems, and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. This leads to faster local convergence compared to both Newton's method, and Newton- Anderson without safeguarding, with effectively no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and gamma-safeguarded Newton-Anderson to solve parameter-dependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Benard convection.

翻译：本文旨在提出一种实用策略，用于在奇点附近加速牛顿法。我们提出了一种具有可调参数的自适应保护方案——称为自适应伽马保护机制，该机制可与安德森加速法结合使用，以提升牛顿法在求解奇点处或邻近奇点问题时的性能。自适应伽马保护机制的关键特性在于：对于奇异问题具有局部收敛性，并能自动检测非奇异问题——在此情况下，牛顿-安德森迭代会向标准牛顿步长方向缩放。由此得到一种灵活算法，在奇异与非奇异问题上均表现良好，且通过合适的参数选择，能够从标准牛顿法和牛顿-安德森法中恢复收敛性。与牛顿法及无保护机制的牛顿-安德森法相比，该方法能以基本无额外计算成本的代价实现更快的局部收敛。我们展示了在奇点附近求解参数依赖问题时，实施牛顿-安德森法及伽马保护型牛顿-安德森法的三种策略。基准测试采用两个参数依赖的不可压缩流动系统：通道流动与瑞利-贝纳德对流。