A Gentle Introduction to Interpolation on the Grassmann Manifold

Abstract. This paper offers a self-contained exposition of the fundamental mathematical and computational tools for interpolation on the Grassmann manifold, including detailed derivations of geodesics and explicit formulations of the exponential and logarithmic maps. The presentation emphasizes intuition and draws continuous parallels with the Euclidean setting. This pedagogical approach facilitates the understanding of linear, piecewise linear, and high-order interpolation algorithms, as well as their extension to more general manifolds. Two numerical examples are finally used to illustrate the potential of these algorithms: one in the context of parametric model order reduction, and another drawn from stationary iterative methods for linear systems.