Approximation Of Integral Fractional Laplacian And Fractional Pdes Via Sinc-Basis

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 nonlocal effects while enforcing less smoothness on functions. In this article, 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 O(N log(N)), where N is the number of unknowns. Thus, using iterative methods such as conjugate gradient, we provide an efficient strategy to solve fractional PDEs with exterior Dirichlet conditions on arbitrary Lipschitz domains. Our implementation works in both two and three dimensions. We also recover the FEM rates of convergence on benchmark problems. For fractional exponent s = 1/4, our current three-dimensional implementation can solve the Dirichlet problem with 5.10(6) unknowns in under 2 hours on a standard office workstation. We further illustrate the efficiency of our approach by applying it to fractional Allen-Cahn and image denoising problems.