  A systematic mathematical framework for the study of numerical algorithms
would allow comparisons, facilitate conjugacy arguments, as well as enable the
discovery of improved, accelerated, data-driven algorithms. Over the course of
the last 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 towards 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.
