SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2897-A2922, January 2021.
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 $\mathcal O(N\log(N))$, 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 $5\cdot 10^6$ 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.

中文翻译：

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通过 sinc-Basis 逼近积分分数拉普拉斯算子和分数偏微分方程

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2897-A2922 页，2021 年 1 月。
在随机过程、成像科学、地球物理学等领域的许多应用的推动下，分数拉普拉斯算子最近受到了极大的关注。该算子成功背后的关键驱动力是它能够捕捉非局部效应，同时对函数执行较低的平滑度。在本文中，我们介绍了一种使用 sinc 基近似该算子的谱方法。使用我们的方案，算子的评估及其在向量上的应用具有 $\mathcal O(N\log(N))$ 的复杂度，其中 $N$ 是未知数的数量。因此，使用诸如共轭梯度之类的迭代方法，我们提供了一种有效的策略来解决任意 Lipschitz 域上具有外部 Dirichlet 条件的分数阶偏微分方程。我们的实现在两个和三个维度上都有效。我们还恢复了基准问题的 FEM 收敛率。对于分数指数 $s=1/4$，我们当前的三维实现可以在标准办公室工作站上在 2 小时内解决具有 $5\cdot 10^6$ 未知数的狄利克雷问题。我们通过将其应用于分数 Allen-Cahn 和图像去噪问题来进一步说明我们方法的效率。