Hierarchical matrix approximations for space-fractional diffusion equations

Space fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N, full representation of a fractional diffusion matrix would require O(N²) memory storage requirement, with a similar estimate for matrix–vector products. In this work, we present H² matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O(N) asymptotically. Matrix–vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudo-transient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix–vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 10⁵ – 10⁶. In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H² approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H² matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.