Anderson acceleration (AA) has a long history of use and a strong recent interest due to its potential ability to dramatically improve the linear convergence of the fixed-point iteration. Most authors are simply using and analyzing the stationary version of Anderson acceleration (sAA) with a constant damping factor or without damping. Little attention has been paid to nonstationary algorithms. However, damping can be useful and is sometimes crucial for simulations in which the underlying fixed-point operator is not globally contractive. The role of this damping factor has not been fully understood. In the present work, we consider the non-stationary Anderson acceleration algorithm with optimized damping (AAoptD) in each iteration to further speed up linear and nonlinear iterations by applying one extra inexpensive optimization. We analyze this procedure and develop an efficient and inexpensive implementation scheme. We also show that, compared with the stationary Anderson acceleration with fixed window size sAA(m), optimizing the damping factors is related to dynamically packaging sAA(m) and sAA(1) in each iteration (alternating window size $m$ is another direction of producing non-stationary AA). Moreover, we show by extensive numerical experiments that the proposed non-stationary Anderson acceleration with optimized damping procedure often converges much faster than stationary AA with constant damping or without damping.
翻译:安德森加速( AA) 使用历史悠久,最近兴趣也很大,原因是它有可能大幅度改善固定点迭代线性趋同。大多数作者只是使用和分析固定版本的安德森加速(sAA),使用一个固定的阻滞因素或没有阻滞因素。很少注意非固定的算法。但是,阻滞可能有用,有时对模拟中基础固定点操作员不是全球合同性的固定点操作员来说至关重要。这一阻滞因素的作用尚未得到充分理解。在目前的工作中,我们考虑在每个迭代中采用非静止的安德森加速算法(AoptD),通过应用一个额外的廉价优化来进一步加速线性和非线性迭代法。我们分析这一程序并开发一个高效和廉价的实施方案。我们还表明,与固定窗口大小的固定安德森加速相比,优化阻滞因素与动态包装SAA(m)和每循环中SA(1)有关。 我们认为,每个迭代号非静止加速算式加速算法(固定的窗口规模为美元或非直线性迭迭迭叠式加速式的自动加速式加速式快速快速快速快速进行AAAAVAL- 。