Displacement interpolation using monotone rearrangement
For researchers in scientific computing and PDEs, this offers a new interpolation method to handle hyperbolic phenomena, but the results are theoretical without concrete numerical validation.
The paper proposes displacement interpolation via optimal transport maps to improve approximation of functions depending on a parameter, particularly for hyperbolic PDEs where linear interpolation fails. A multi-dimensional extension using the Radon transform is introduced, generalizing the Lax-Philips translational representation.
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].