The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We do this by developing an adaptive safeguarding scheme, which we call gamma-safeguarding, that one can apply to Anderson accelerated Newton's method when solving problems near singular points. The key features of adaptive gamma-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. This leads to faster local convergence compared to both Newton's method and Newton-Anderson without safeguarding, at 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：本文旨在开发一种在奇异点附近加速牛顿法的实用策略。我们提出了一种自适应防护方案（称为γ-防护），可应用于求解奇异点附近问题时采用的安德森加速牛顿法。自适应γ-防护的关键特性在于：对于奇异问题具有局部收敛性，并可检测非奇异问题——此时牛顿-安德森迭代会向标准牛顿步长方向缩放。与标准牛顿法及无防护的牛顿-安德森算法相比，该方法在不增加计算成本的前提下实现了更快的局部收敛。我们展示了三种实现策略，用于在求解奇异点附近的参数依赖问题时应用牛顿-安德森算法及γ-防护牛顿-安德森算法。基准测试选取两个参数依赖的不可压缩流动系统：槽道流动与瑞利-贝纳德对流。