基于自适应正则化的Anderson加速非单调全局化方法
Nonmonotone Globalization for Anderson Acceleration via Adaptive Regularization
AI总结：
针对Anderson加速（AA）的不稳定与停滞问题，提出基于非单调信赖域框架、自适应二次正则化及定制接受机制的AA全局化方法，兼具全局与AA同等局部收敛性，经数值实验验证有效。
摘要：
Anderson加速（AA）是一种广泛应用的不动点迭代加速方法，但可能存在不稳定与停滞问题。我们提出一种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。现有AA全局化方法依赖防护操作，可能阻碍快速局部收敛，与之不同，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。我们证明了该方法的全局收敛性，并表明在适当假设下，算法能达到与AA相同的局部收敛效果。多项数值实验验证了方法的有效性。
英文摘要：
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.
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