Displacement interpolation using monotone rearrangement

TL;DR

This paper introduces displacement interpolation via optimal transport maps for better parameter-dependent function approximation, especially in hyperbolic PDEs, and extends it to multiple dimensions using Radon transform properties.

Contribution

It presents a novel displacement interpolation method based on optimal transport for hyperbolic PDEs and extends it to multiple dimensions with Radon transform intertwining.

Findings

Effective dimensionality reduction for hyperbolic phenomena

Improved approximation over linear methods

Multi-dimensional extension via Radon transform

Abstract

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].