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 , in contrast to the existing finite element methods (FEM) whose implementations are currently limited to 2d . 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.
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.
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, in contrast to the existing finite element methods (FEM) whose implementations are currently limited to 2d