[论文解读] What Is the Fractional Laplacian?

Anna Lischke, Guofei Pang+9 more|arXiv (Cornell University)|Jan 29, 2018

Advanced Mathematical Modeling in Engineering参考文献 104被引用 66

一句话总结

本文在有界区域比较多种分数拉普拉斯算子定义（Riesz、谱、方向、视界基非本地等），分析它们的边界行为，并为非零边界条件提出新的数值方法与见解。

ABSTRACT

The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a joint theoretical and computational approach to examining their different characteristics by studying solutions of related fractional Poisson equations formulated on bounded domains. In this work, we provide new numerical methods as well as a self-contained discussion of state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

研究动机与目标

• 在有界域上对常见的分数拉普拉斯算子定义（Riesz、谱、方向、非本地/视界）进行综述和比较。
• 分析边界条件和随机过程解释，以理解物理含义及适用性。
• 在零边界和非零边界条件下，针对各定义开发和评估数值方法。
• 指出在特定异常扩散应用中何时每个定义更合适。

提出的方法

• 在 R^d 和有界域中对定义进行理论比较，涵盖边界条件含义和随机过程解释。
• 给出非零边界条件的新方法：用于 Riesz 拉普拉斯的径向基函数列点法，以及用于谱拉普拉斯的非同伦提升。
• 通过基准问题（泊松方程）在各定义下的数值实验，以评估精度和边界行为。
• 使用随机解法（Walk-on-Spheres）来阐明概率解释并求解 Riesz 问题。
• 讨论不齐次谱定义及相关表示（如逆拉普拉斯算子）之间的等价性。
• 评估在各定义下边界处的正则性和解的行为。

实验结果

研究问题

• RQ1 在零边界条件和非零边界条件下，不同分数拉普拉斯定义（Riesz、谱、方向、视界基非本地）在有界域上有何不同？
• RQ2 当使用每种定义时，解的边界行为和正则性属性如何出现？
• RQ3 哪些数值方法在有界域上最好地逼近每种定义，包括非零边界条件？
• RQ4 这些算子的随机解释如何解释它们的边界条件及物理意义？

主要发现

• 在有界域上，Riesz 与谱分数拉普拉斯算子会呈现不同的边界行为，存在边界层且某些 α 与域大小下内部响应非单调。
• 在 Riesz 定义下，解的边界层会随着 α 减小而变得更陡，与在零边界条件下更平滑的谱解不同。
• 对于零边界条件，Riesz 解通常高于谱解，这是逆算子正性保持性质的结果。
• 为非零边界条件开发了新的数值方法：Riesz 的径向基函数列点法，以及谱定义的非同伦提升。
• 建立了某些非齐次谱定义的等价性，使得通过逆拉普拉斯表示可以一致地表述和求解。
• 随机解释（停止/次级布朗运动）阐明了为何需要不同的外部/边界数据以适用于不同定义。