标题： 固定点迭代的加速方法 显示英文标题

标题： Acceleration methods for fixed point iterations

摘要： 在科学计算中，一种普遍的方法是将给定问题的解表示为向量或其他数学对象序列的极限。 在许多情况下，这些序列是由收敛缓慢的迭代过程生成的，这促使从业者寻求更快的替代方法以达到极限。 “加速技术”包括一系列广泛的方法，专门为此目标而设计。 它们最初是通过各种形式的“极限外推”来改善一般标量序列的收敛性，即通过线性组合将最近的迭代外推到极限。 这类外推方法，其中最著名的是艾特肯的Δ²过程，只需要向量序列作为输入。 然而，仅使用迭代值来限制方法过于严格。 通过利用迭代值和固定点映射本身来加速由固定点迭代生成的序列已被证明在物理学的各个领域非常成功。 这些固定点加速器（FP-加速器）的一个显著例子是D.安德森于1965年开发的方法，现在广为人知的安德森加速（AA）。 此外，拟牛顿法和不精确牛顿法也可以归入此类。 本文综述了这些方法——重点介绍如AA这样旨在加速固定点迭代的方法。 显示英文摘要

摘要： A pervasive approach in scientific computing is to express the solution to a given problem as the limit of a sequence of vectors or other mathematical objects. In many situations these sequences are generated by slowly converging iterative procedures and this led practitioners to seek faster alternatives to reach the limit. ``Acceleration techniques'' comprise a broad array of methods specifically designed with this goal in mind. They started as a means of improving the convergence of general scalar sequences by various forms of ``extrapolation to the limit'', i.e., by extrapolating the most recent iterates to the limit via linear combinations. Extrapolation methods of this type, the best known example of which is Aitken's Delta-squared process, require only the sequence of vectors as input. However, limiting methods to only use the iterates is too restrictive. Accelerating sequences generated by fixed-point iterations by utilizing both the iterates and the fixed-point mapping itself has proven highly successful across various areas of physics. A notable example of these Fixed-Point accelerators (FP-Accelerators) is a method developed by D. Anderson in 1965 and now widely known as Anderson Acceleration (AA). Furthermore, Quasi-Newton and Inexact Newton methods can also be placed in this category as well. This paper presents an overview of these methods -- with an emphasis on those, such as AA, that are geared toward accelerating fixed point iterations.