Space-Fractional Diffusion with Variable Order and Diffusivity: Discretization and Direct Solution Strategies

Abstract: 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.