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形，发展了安德森加速应用于牛顿法的收敛性与加速理论。在这类问题中，标准牛顿算法在解附近区域呈线性收敛；已有研究表明，安德森加速可在无需额外先验知识且仅增加极少计算成本的前提下显著提升收敛性能。我们在此背景下对牛顿-安德森算法进行了分析，并提出一种新颖且具有理论保障的安全防护策略。钱德拉塞卡H方程及若干标准基准算例验证了该收敛结果。