  When approximating a function that depends on a parameter, one encounters
many practical examples where linear interpolation or linear approximation with
respect to the parameters prove ineffective. This is particularly true for
responses from hyperbolic partial differential equations (PDEs) where linear,
low-dimensional bases are difficult to construct. We propose the use of
displacement interpolation where the interpolation is done on the optimal
transport map between the functions at nearby parameters, to achieve an
effective dimensionality reduction of hyperbolic phenomena. We further propose
a multi-dimensional extension by using the intertwining property of the Radon
transform. This extension is a generalization of the classical translational
representation of Lax-Philips [Lax and Philips, Bull. Amer. Math. Soc. 70
(1964), pp.130--142].
