We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these equations due both to the kernel singularity in the fractional integral operator and to the resulting dense discretized operators, which quickly become prohibitively expensive to handle because of their memory and arithmetic complexities. In this work, we present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. We also present a block low rank representation to handle the dense matrix representations, by exploiting the ability to approximate blocks of the resulting formally dense matrix by low rank factorizations. A Cholesky factorization solver operates directly on this representation using the low rank blocks as its atomic computational tiles, and achieves high performance on multicore hardware. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the first-order convergence rate allowed by the regularity of the solutions. They also show that considerable savings are obtained in storage ($O(N^{1.5})$) and computational cost ($O(N^2)$) compared to dense factorizations. This translates to orders-of-magnitude savings in memory and time on multi-dimensional problems, and shows that the proposed methods offer practical tools for tackling large nonlocal fractional diffusion simulations.
翻译:我们考虑的是具有不同空间差异和分数顺序的多维空间偏差扩散方程式。 在解决这些方程式时,由于分数整体操作器的内核奇特性以及由此产生的密集离散操作器,我们遇到了重大的计算挑战。由于其记忆和算术的复杂性,这些超异性迅速变得过于昂贵。在这项工作中,我们提出了一个单异性分解方案,通过一个单异性减色技术使单异差分解器正规化,适应于异差和分数顺序的空间变异性。这一正规化战略的制定很方便,是一种稀疏的矩阵校正,添加到密度操作操作操作器,适用于分数扩散方方形方程式的不同配方。我们还提出一个区块低级代表,通过低级因子分解法来接近所生成的超稠密矩阵的区块。Choloysky因数解解解解方案直接运行这个代表系统,使用低级级块作为原子计算图,并取得高额的硬件性。 数字结果显示, 奇异性矩阵处理方法是坚固、 大幅降低N- 级递解解的不相级缩缩缩缩化方法, 并显示常规存储法的递解的缩化方法, 并实现了Oalcalalalalalalalalalalalalalalalalalalalalalalalbildalxxxxxxxxx 。