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What Is the Fractional Laplacian?

Anna Lischke, Guofei Pang, Mamikon Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai, Mark M. Meerschaert, Mark Ainsworth, George Em Karniadakis

arXiv:1801.09767v3math.NA

TL;DR

The paper addresses the lack of consensus over which fractional Laplacian definition is appropriate on bounded domains, where boundary conditions make otherwise equivalent characterizations mathematically distinct. It compares several definitions through theory, stochastic interpretations, numerical methods, and bounded-domain Poisson benchmarks, finding distinct boundary regularity and process behavior while proving equivalence among considered inhomogeneous formulations. The results help practitioners relate operator and numerical-method choices to their applications.

  • Problem

    Fractional Laplacian definitions that are equivalent in R^d become mathematically distinct on bounded domains through different boundary conditions, with no consensus on the most appropriate choice for applications.

  • Method

    The paper combines theoretical analysis, stochastic-process interpretations, numerical-method development, and one- and two-dimensional benchmark comparisons across several fractional Laplacian definitions.

  • Results

    The study finds contrasting boundary behavior between Riesz and spectral solutions and analytically proves equivalence among the considered methods for inhomogeneous benchmark problems.

  • Takeaways & Limitations

    Operator choice in bounded-domain anomalous-transport models should account for the associated boundary behavior and stochastic interpretation.

  • Takeaways & Limitations

    The paper does not compute Riesz fractional Neumann problems because their nonlocal boundary condition lacks consensus and suitable numerical approaches were unavailable.

Abstract

from arXiv · show

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.

1. Introduction

The paper compares fractional Laplacian definitions on bounded domains, where equivalent whole-space characterizations produce distinct operators through their boundary conditions. It combines theoretical analysis, stochastic interpretations, numerical methods, and benchmark computations to clarify these differences and guide model selection.

  • Definitions and boundary conditions: Equivalent fractional Laplacian characterizations in R^d become distinct operators on bounded domains because their associated boundary conditions differ.The paper focuses on the Riesz, spectral, directional, and horizon-based nonlocal definitions.
  • Approach: The study combines theoretical comparisons, stochastic interpretations, and quantitative benchmark computations using state-of-the-art and new numerical methods.The benchmarks include one- and two-dimensional fractional Poisson problems and emphasize nonzero boundary conditions.
  • Definitions and boundary conditions: The Riesz definition requires prescribing u(x) throughout the exterior of the domain, whereas the spectral definition requires only a local boundary condition.These formulations correspond to different treatments of α-stable Lévy processes at the boundary.
  • Motivating examples: Changing the computational-domain size can switch solution maxima from decreasing to non-monotonic and then increasing behavior as the fractional order varies.The scaling relation uL(x) = L^αu1(x/L) produces different transition behavior for the Riesz and spectral operators.
  • Motivating examples: Riesz and spectral solutions differ most sharply near boundaries: Riesz solutions develop boundary layers or singularities as α decreases, while spectral solutions remain smoother for smooth sources.For f = 1, the difference concentrates in the interior when 1 < α < 2 and forms a sharpening boundary layer when α < 1.

Section Overview

The paper reviews multiple characterizations of the fractional Laplacian in R^d and explains how these formulations motivate distinct operators and boundary treatments in bounded domains. It combines theoretical derivations, stochastic connections, and computational perspectives to compare these definitions.

  • The paper reviews the Riesz, spectral, directional, and regional representations, including their derivations, regularity properties, and connections to Lévy processes.
  • On R^d, the fractional Laplacian can be defined through spectral calculus and represented as a Fourier multiplier with symbol |ξ|^α.The spectral-theorem construction applies positive powers of −∆, while the Fourier transform diagonalizes the operator.
  • The real-space Riesz representation uses a singular integral whose principal value is regularized by the vanishing difference u(x) − u(y).This formulation is developed for 0 < α < 2 after the inverse-Fourier representation for positive α fails.
  • The article explains that equivalent characterizations on R^d can diverge in bounded domains, where the chosen operator and boundary conditions must be matched.For the spectral definition with zero Dirichlet data, the spectral-theorem and elliptic-extension perspectives remain applicable; bounded-domain analogues depend on the fractional Laplacian considered.

2.2. Fractional Laplacians on Bounded Domains

On bounded domains, equivalent whole-space formulas yield distinct fractional Laplacians with different boundary conditions, stochastic interpretations, and solution regularity. The section compares these distinctions, including Riesz and spectral formulations, their lifting constructions, and regularity limits.

  • Definitions: Equivalent whole-space characterizations become distinct fractional Laplacians on bounded domains because their equivalences break down after restriction.The section surveys definitions formulated by restriction and definitions introduced directly on bounded domains.
  • Boundary conditions: The spectral and regional definitions admit local boundary conditions, whereas the Riesz and directional definitions require exterior conditions on R^d \ Ω.For Riesz and directional operators, values outside the domain are part of the boundary-value formulation.
  • Stochastic interpretations: The Riesz operator with Dirichlet data generates stopped α-stable Lévy motion, whose discontinuous paths exit Ω by jumps into the exterior.This contrasts with spectral operators, which generate subordinated stopped or reflected Brownian motion depending on the boundary condition.
  • Well-posedness and regularity: For bounded Lipschitz Ω, the fractional Poisson problem is well-posed when f ∈ H^-α/2(Ω), with solution u ∈ H^α/2(Ω).This result follows using the Lax–Milgram Lemma.
  • Well-posedness and regularity: For the Riesz operator, increasing source regularity may improve interior but not global regularity once α/2 + s ≥ 1/2.The section also reports strict boundary Hölder regularity and cases where the α-gain in Sobolev regularity is sharp.
  • Well-posedness and regularity: For f ∈ L2(Ω), spectral solutions satisfy u ∈ H^α(Ω) for all α ∈ (0,2), while Riesz solutions have this property only when α < 1.For smoother sources, spectral regularity can reach H^{s+α}(Ω) under boundary compatibility, whereas Riesz regularity is capped by α/2 + 1/2 − ε.
  • Inhomogeneous spectral operators: Inhomogeneous spectral operators reduce to the homogeneous spectral operator applied to a lifted function, and formulations from two recent works are equivalent.With suitable boundary-data regularity, the fractional harmonic lifting is related to a standard harmonic problem.

2.7. Summary

The section compares fractional Laplacian definitions in bounded domains, emphasizing their distinct operators, domains, boundary treatments, and stochastic interpretations.

  • The Riesz and spectral fractional Laplacians differ stochastically: Riesz processes leave the domain’s closure, whereas spectral processes remain confined to it.
  • The regional fractional Laplacian is defined on functions over Ω rather than over all of R^d.
  • The regional definition differs from the Riesz definition even when the function vanishes outside Ω.
  • The regional fractional Poisson problem has been studied through the Feynman–Kac formula, with Neumann and Robin boundary conditions also discussed.
  • Probabilistic studies of stable-type processes in bounded domains also address related fractional Laplacians and spectral-Laplacian eigenvalue estimates.

Section Overview

The paper compares numerical approaches for several fractional Laplacian definitions, emphasizing benchmark solutions, boundary regularity, convergence, and computational limitations. It introduces new collocation methods while identifying unresolved theoretical and preconditioning problems.

  • Numerical methods: The study discretizes Riesz, spectral, directional, and horizon-based nonlocal fractional Laplacians using several state-of-the-art numerical methods.Riesz discretizations use AFEM and WOS; spectral discretizations use SEM and related approaches; the directional method uses new RBF collocation with a vector Grünwald scheme; the horizon-based operator uses finite volumes.
  • Numerical methods: The paper introduces an RBF collocation method for the directional representation and reports the first numerical results using the vector Grünwald scheme.The method focuses on the uniform directional measure, yielding the Riesz fractional Laplacian, and can extend to nonisotropic directional operators.
  • Computational limitations: Theoretical support is incomplete for convergence of truncated directional operators in inhomogeneous problems and for error bounds connecting truncated and full boundary-value problems.The numerical example makes truncation error close to machine precision with sufficiently large K2, but theoretical justification remains open.
  • Convergence: The modified vector Grünwald scheme achieves first-order convergence, with error convergence O(h) as the step size h decreases.The observed slope is 1, consistent with the theoretical convergence rate cited for the scheme.
  • Convergence: For zero exterior data, the RBF method converges, but accuracy deteriorates for stronger boundary gradients and at larger point counts because the collocation matrix becomes increasingly ill-conditioned.Errors are larger near the boundary where collocation points are sparser, while increasing condition numbers reduce the convergence rate.
  • Computational limitations: The RBF method has quadratic assembly complexity, and effective preconditioners remain an open problem for iterative solution of its collocation systems.Direct solution can exhibit cubic complexity as the number of collocation points increases.

Section Overview

The benchmark section compares Riesz and spectral fractional Poisson solutions across domains, forcings, and fractional orders using several numerical methods. The solutions differ most visibly near boundaries and corners, while the Riesz computations face conditioning and convergence limitations.

  • Benchmark setup: Four benchmark problems compare Riesz and spectral solutions with zero Dirichlet conditions across square, disk, and L-shaped domains.The study uses multiple numerical methods for the Riesz definition and spectral element discretization for the spectral definition.
  • Square domain: On the square domain with f = 1, the spectral solution lies below the Riesz solution for α = 0.5 and 1.5.For α = 0.5, apparent failure to enforce the Riesz zero boundary condition is an AFEM artifact; the true solution has zero trace.
  • Square domain: For f = sin(πx) sin(πy), boundary-condition oscillations in the Riesz solution at α = 0.5 are less pronounced but remain visible in uRiesz − uspectral.One- and two-dimensional profiles are qualitatively similar across the four square-domain cases.
  • Boundary behavior: Riesz boundary layers sharpen as α decreases, whereas spectral solutions remain smooth near the boundary for smooth forcing.The Riesz boundary layer for the sine forcing is particularly notable because of the singularity in the Riesz definition.
  • L-shaped domain: In the L-shaped domain, difference plots show a relatively minor inside-corner spike for α = 0.5 that is absent or much weaker for α = 1.5.The spike intensifies in the α = 0.5 cases, while the Riesz solution remains above the spectral solution when f ≥ 0.

Section Overview

The section extends the computational study to nonzero boundary conditions and compares formulations and methods for inhomogeneous fractional Laplacians. It establishes numerical equivalence for several spectral formulations and for directional and Riesz solutions, while identifying important computational and formulation boundaries.

  • Inhomogeneous problems: Nonzero boundary conditions can substantially increase computational cost and may require modifying the fractional Laplacian definition, especially for the spectral operator.The paper develops computationally feasible adaptations for these inhomogeneous problems.
  • Limitations: The Riesz fractional Neumann problem is not computed because its nonlocal Neumann condition lacks consensus and suitable numerical approaches are unavailable.Spectral Neumann conditions are more straightforward because they require only local boundary conditions.
  • Spectral formulations: The APR harmonic lifting, heat semigroup, and nonharmonic lifting formulations are numerically equivalent descriptions of the inhomogeneous spectral fractional Laplacian.The paper compares these formulations through numerical experiments and presents the nonharmonic lifting approach for discretization.
  • Lifting methods: For one-dimensional tests with f = x or f = −x and nonzero Dirichlet data, harmonic and nonharmonic lifting solutions differ by approximately machine precision.The tests use lifting functions v = x and v = x^3 across different fractional orders α.
  • Computational formulations: The heat semigroup formulation requires time integration of a discretized heat equation, whereas harmonic lifting applies a homogeneous spectral discretization after lifting the boundary data.The comparison evaluates computational cost using spectral element discretizations.
  • Operator comparison: For inhomogeneous tests with α = 1.5, directional and Riesz solutions agree up to numerical error, while the spectral solution has greater magnitude.Boundary data are prescribed locally for the spectral definition and on the exterior for the Riesz and directional definitions.
  • RBF computation: The RBF solution converges when the truncation parameter reaches K2 = 6000 because the exterior condition g(x) = exp(−|x|^2) decays rapidly.This identifies a concrete truncation requirement for the reported collocation computation.

6. Summary and Discussion

The work combines theoretical and computational analysis to compare fractional Laplacians and related Poisson problems on bounded domains. It reports contrasting boundary behavior, equivalent inhomogeneous formulations, and a scope limitation in the numerical-method sample.

  • The study examines spectral, horizon-based nonlocal, and several Riesz fractional Laplacian formulations through theory and refined numerical comparisons.The work also uses different numerical methods to compare their characteristics and solutions.
  • All considered methods produced equivalent solutions for inhomogeneous benchmark problems, and this equivalence was proved analytically for the first time.The methods included recently proposed approximations and nonharmonic lifting to a homogeneous reformulation.
  • The numerical-method sample is not representative of the field’s full breadth because the work prioritizes fundamental questions over a comprehensive survey.Finite-difference approaches are specifically identified as receiving limited discussion.
  • Riesz solutions show singular behavior near boundaries, contrasting sharply with the smooth boundary behavior of spectral solutions.
  • The authors present the work as a starting point for researchers modeling anomalous transport, while noting that fractional-Laplacian research remains incomplete.

Appendix A Sobolev Spaces and the Trace Theorem

Appendix A assembles commonly used fractional Sobolev spaces and introduces the trace framework used in the paper. It restricts the stated trace theorem to bounded, simply connected Lipschitz domains and 1/2 < s < 3/2.

  • The appendix assembles commonly used Sobolev spaces for fractional-Laplacian analysis and follows the exposition and notation of a cited reference.
  • For 0 < s < 1, the appendix defines a fractional Sobolev space together with its seminorm and norm.
  • The appendix defines a trace operator for smooth functions and states a trace theorem on bounded, simply connected Lipschitz domains.
  • The stated trace-theorem range is 1/2 < s < 3/2, which the article considers sufficient for its discussion of traces.
  • A separate fractional Sobolev space is required when s = 1/2.

The associated norm is defined

The appendix defines the Sobolev space used to characterize spectral fractional-Laplacian regularity and distinguishes it from related fractional Sobolev spaces and their dual.

  • The space Hs(Ω) is used to characterize regularity properties of the spectral fractional Laplacian.
  • Its construction uses eigenpairs of the integer Laplacian −∆ on Ω with zero Dirichlet boundary conditions.
  • The appendix records relationships among the fractional Sobolev spaces introduced in Definitions A.1, A.3, A.4, and A.5.
  • For s ≥ 0, H−s(Ω) is defined as the dual space of Hs(Ω).

Appendix B Grids

Appendix B provides the meshes used in the two-dimensional numerical comparisons. The comparison figures organize meshes and collocation points across domains and fractional-Laplacian definitions.

  • The appendix includes the meshes used for the two-dimensional numerical comparisons in Section 4.
  • Square: On the square, the figure compares meshes and collocation points for directional, spectral, and Riesz fractional Poisson equations.
  • Disk: On the disk, the figure compares meshes and collocation points for directional, spectral, and Riesz fractional Poisson equations.
  • L-shaped domain: On the L-shaped domain, the figure compares meshes and collocation points for directional, spectral, and Riesz fractional Poisson equations.

Appendix C Additional Disk Comparisons

The disk comparisons examine Riesz and spectral solutions across different forcing, fractional orders, and numerical methods. For α = 1.5, three methods compute the Riesz solution while SEM computes the spectral solution.

  • For f = sin(πr2) and α = 0.5, the disk comparison shows spectral and Riesz solutions alongside their difference.
  • For α = 1.5, Figure 34 compares the Riesz and spectral solutions and their difference on the disk.
  • For α = 1.5, the disk comparison uses RBF collocation, AFEM, and WOS for the Riesz solution, and SEM for the spectral solution.

Appendix D Additional L-shape Comparisons

The L-shaped-domain comparison tests Riesz and spectral solutions for α = 1.5 using several numerical methods and an exterior-corner view. The spectral solution is the only one with a significant difference.

  • For f(x) = 1, g(x) = 0, and α = 1.5, the comparison uses three Riesz methods and one spectral method.
  • The spectral solution is the only solution with a significant difference; the other solutions are equivalent up to numerical error.
  • Figure 37 repeats the L-shaped-domain comparison from Figure 36 with a view facing the outside corner.
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