SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 860-885, January 2020.
A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, and enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the past century, the Koopman operator has provided a mathematical framework for the study of dynamical systems which facilitates conjugacy arguments and can provide efficient reduced descriptions. More recently, numerical approximations of the operator have enabled the analysis of a large number of deterministic and stochastic dynamical systems in a completely data-driven, essentially equation-free pipeline. Discrete- or continuous-time numerical algorithms (integrators, nonlinear equation solvers, optimization algorithms are themselves dynamical systems. In this paper, we use this insight to leverage the Koopman operator framework in the data-driven study of such algorithms and discuss benefits for analysis and acceleration of numerical computation. For algorithms acting on high-dimensional spaces by quickly contracting them toward low-dimensional manifolds, we demonstrate how basis functions adapted to the data help to construct efficient reduced representations of the operator. Our illustrative examples include the gradient descent and Nesterov optimization algorithms as well as the Newton--Raphson algorithm.

中文翻译：

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关于算法的考夫曼算子

SIAM应用动力系统杂志，第19卷，第2期，第860-885页，2020年1月。
用于数值算法研究的系统数学框架将允许进行比较，促进共轭论证，并能够发现改进的，加速的，数据驱动的算法。在过去的一个世纪中，Koopman算子为动力学系统的研究提供了数学框架，该框架促进了共轭论证并且可以提供有效的简化描述。最近，算子的数值逼近使得能够在完全由数据驱动的，基本上无方程的管道中分析大量确定性和随机动力系统。离散或连续时间数值算法（积分器，非线性方程求解器，优化算法本身就是动力学系统。在本文中，我们利用这种见识来利用Koopman运算符框架进行此类算法的数据驱动研究，并讨论分析和加速数值计算的好处。对于通过将高维空间快速收缩为低维流形而作用于高维空间的算法，我们演示了适用于数据的基函数如何帮助构造算子的有效简化表示。我们的示例包括梯度下降和Nesterov优化算法以及Newton-Raphson算法。我们演示了适应数据的基础函数如何帮助构造有效的操作员简化表示形式。我们的示例包括梯度下降和Nesterov优化算法以及Newton-Raphson算法。我们演示了适应数据的基础函数如何帮助构造有效的操作员简化表示形式。我们的示例包括梯度下降和Nesterov优化算法以及Newton-Raphson算法。