What Is the Fractional Laplacian? (1801.09767v3)

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.

• The paper rigorously compares definitions of the fractional Laplacian and introduces innovative numerical methods for bounded domains.
• It addresses complex boundary condition handling using harmonic and nonharmonic lifting to optimize the solution of fractional Poisson problems.
• It establishes connections with stochastic processes, enhancing understanding of anomalous diffusion and informing more robust computational models.

Insights into the Fractional Laplacian: A Comparative Review

The paper "What Is the Fractional Laplacian? A Comparative Review with New Results" provides a comprehensive and technical examination of various definitions of the fractional Laplacian, particularly in bounded domains, and offers numerical methods for discretizing these operators. The authors contend with both uniform and nonuniform boundary conditions, evaluating their implications on computational methodologies and solution properties.

The fractional Laplacian, denoted as $(-\Delta)^{\alpha/2}$ for fractional orders $\alpha \in (0,2)$, is instrumental in modeling anomalous diffusion processes across disciplines. The fractional Laplacian in $\mathbb{R}^d$ can be equivalently described using spectral, integral, and stochastic frameworks. However, these definitions diverge significantly in bounded domains due to their boundary condition treatments. The Riesz definition, requiring a condition over the entire exterior domain, contrasts with the spectral definition, which needs specifications only on the domain boundary.

Key Contributions and Findings

1. Definition Comparisons: The authors provide a rigorous comparison of various definitions of the fractional Laplacian, carefully considering the mathematical subtleties involved in translating these definitions to bounded domains. This includes the Riesz (integral) and spectral definitions, each illuminating unique aspects of boundary behavior, regularity, and physical interpretation.
2. Numerical Methods: The paper showcases distinct numerical techniques for solving fractional Poisson problems, predominantly using the Riesz and spectral definitions. It introduces an adaptive finite element method (AFEM) and a novel radial basis function (RBF) collocation approach, in addition to Monte Carlo-based walk-on-spheres (WOS) methodology.
3. Boundary Condition Handling: A critical aspect discussed is the handling of nonzero boundary conditions using harmonic and nonharmonic lifting methods, especially in the spectral framework. These techniques facilitate the conversion of nonzero boundary problems to their homogeneous counterparts, thus exploiting well-established numerical techniques.
4. Stochastic Connections: The link between fractional Laplacians and stochastic processes, particularly $\alpha$-stable Lévy flights, is explored. This not only aids in comprehending the operator's impact but also provides a foundation for stochastic numerics, which are essential for G-wave and Monte Carlo simulations.
5. Regularity and Well-posedness: The paper carefully navigates through the existing literature on the regularity and well-posedness of fractional Poisson problems, providing insights into how fractional operators affect solution smoothness and boundary value problem formulation. It underscores the nuanced differences between achieving regularity gains using spectral versus Riesz operators.

Practical Implications and Future Work

The distinctions between the various definitions mark critical decision points for practitioners aiming to model physical phenomena accurately. For example, the spectral definition might be more suitable for problems requiring local boundary conditions and smooth solutions, while the Riesz definition could better model scenarios with nonlocal interactions.

The paper suggests future exploration into the less-trod areas of the fractional Laplacian with Neumann or Robin boundary conditions, as well as the development of robust preconditioners for improving the computational efficiency of the proposed numerical methodologies like the RBF collocation technique.

Additionally, the extension of these methodologies to non-isotropic and directional fractional Laplacians offers a robust avenue for tailoring these mathematical tools to a broader suite of applications beyond the isotropic cases studied.

In summary, this paper provides an essential comparative analysis and methodological advancements for fractional Laplacians, setting a precedent for both theoretical and practical engagement with fractional calculus models in bounded domains. The implications of choosing specific definitions on the boundaries present a significant consideration for future research and application.