论文标题
通过SINC-BASIS的积分分数拉普拉斯和分数PDE的近似
Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis
论文作者
论文摘要
在随机过程中的许多应用，成像科学，地球物理学等中的推动下，分数laplacians最近受到了极大的关注。该操作员成功背后的关键推动力是它能够捕获非本地效应的能力，同时对功能的平滑度降低。在本文中，我们引入了一种光谱方法，以近似使用SINC基础的操作员。使用我们的方案，对矢量的运算符及其应用程序的评估的复杂性为$ \ MATHCAL O（n \ log（n））$，其中$ n $是未知数的数量。因此，使用诸如CG之类的迭代方法，我们提供了一种有效的策略，可以在任意Lipschitz域上使用外部迪里奇条件来求解分数偏微分方程。我们的实施价格均为$ 2D $和$ 3D $。我们还恢复了基准问题上的FEM收敛速度。我们通过将其应用于分数Allen-Cahn和图像降解问题来进一步说明我们的方法的效率。
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.
