使用单调重排进行位移插值
Displacement interpolation using monotone rearrangement
AI总结：
本文提出了一种基于最优传输映射的位移插值方法，用于减少超几何偏微分方程中高维现象的维度，同时扩展了Radon变换的 intertwining性质以实现多维插值。
摘要：
当近似依赖于参数的函数时，许多实际例子表明线性插值或参数的线性近似效果不佳。这在超几何偏微分方程（PDEs）的响应中尤为明显，其中线性、低维基函数难以构造。我们提出使用位移插值，即在相邻参数处的函数之间的最优传输映射上进行插值，以实现超几何现象的有效维度降低。我们进一步提出一个多维扩展，通过利用Radon变换的 intertwining性质。此扩展是对经典Lax-Philips翻译表示法的推广。
英文摘要：
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].
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