Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis
Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis
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DOI：
10.1137/20m1374122
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发表时间：
2020-10
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通讯作者：
Harbir Antil;P. Dondl;Ludwig Striet
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作者：
Harbir Antil;P. Dondl;Ludwig Striet
分数拉普拉斯因其在随机过程、成像科学、地球物理等领域的广泛应用而受到广泛关注。该操作符成功背后的关键驱动力是其捕捉非局部效应的能力，同时对函数实施较不平滑的处理。在这篇文章中，我们引入了一种谱方法来逼近这个算子，它使用了SINC基。使用我们的方案，算子的计算及其在向量上的应用的复杂性为$\数学O(N\log(N))$，其中$N$是未知数的个数。因此，利用CG等迭代方法，我们提供了一种求解任意Lipschitz域上具有外Dirichlet条件的分数阶偏微分方程解的有效策略。我们的实现同时适用于$2d$和$3d$，而现有的有限元方法目前的实现仅限于$2d$。我们还恢复了基准问题的有限元收敛速度。通过将其应用于分数Allen-Cahn问题和图像去噪问题，进一步说明了该方法的有效性。
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.