Anderson acceleration (AA) is a popular method for accelerating fixed-point iterations, but may suffer from instability and stagnation. We propose a globalization method for AA to improve stability and achieve unified global and local convergence. Unlike existing AA globalization approaches that rely on safeguarding operations and might hinder fast local convergence, we adopt a nonmonotone trust-region framework and introduce an adaptive quadratic regularization together with a tailored acceptance mechanism. We prove global convergence and show that our algorithm attains the same local convergence as AA under appropriate assumptions. The effectiveness of our method is demonstrated in several numerical experiments.

翻译：安德森加速（AA）是一种用于加速不动点迭代的流行方法，但可能面临不稳定性与停滞问题。我们提出一种针对AA的全局化方法，以提升稳定性并实现统一的全局与局部收敛。不同于现有依赖保护操作且可能阻碍快速局部收敛的AA全局化方法，我们采用非单调信赖域框架，引入自适应二次正则化及定制的接受机制。我们证明了全局收敛性，并表明在适当假设下，该算法能达到与AA相同的局部收敛速度。数值实验验证了该方法的有效性。