Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis

TL;DR

This paper presents a sinc-basis spectral method for efficiently approximating the fractional Laplacian and solving related PDEs with applications in imaging and physics, achieving fast computation and good convergence.

Contribution

It introduces a novel sinc-basis spectral scheme for fractional Laplacian approximation with $ ext{O}(N ext{log} N)$ complexity, applicable in 2D and 3D, and demonstrates its effectiveness on various problems.

Findings

Achieves $ ext{O}(N ext{log} N)$ complexity for operator evaluation.

Recovers FEM convergence rates on benchmark problems.

Successfully applies to fractional Allen-Cahn and image denoising tasks.

Abstract

Fueled by many applications in random processes, imaging science, geophysics, etc., fractional Laplacians have recently received significant attention. The key driving force behind the success of this operator is its ability to capture non-local effects while enforcing less smoothness on functions. In this paper, we introduce a spectral method to approximate this operator employing a sinc basis. Using our scheme, the evaluation of the operator and its application onto a vector has complexity of $\mathcal O(N\log(N))$ where $N$ is the number of unknowns. Thus, using iterative methods such as CG, we provide an efficient strategy to solve fractional partial differential equations with exterior Dirichlet conditions on arbitrary Lipschitz domains. Our implementation works in both $2d$ and $3d$. We also recover the FEM rates of convergence on benchmark problems. We further illustrate the efficiency of our approach by applying it to fractional Allen-Cahn and image denoising problems.