In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton's method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and some standard benchmark examples.

翻译：本文针对非线性系统中雅可比矩阵在解处奇异的情况，研究了安德森加速方法应用于牛顿法的收敛性与加速理论。对于此类问题，标准牛顿算法在解附近区域呈线性收敛；此前已有研究表明，安德森加速无需额外先验知识且计算成本极低，即可显著提升收敛速度。我们在此背景下对牛顿-安德森算法进行了分析，并提出一种具有理论支撑的新型保护策略。结合钱德拉塞卡H方程及若干标准基准算例，验证了收敛性理论结果。