A scalable framework for solving fractional diffusion equations

The study of fractional order differential operators (involving non-integer derivative terms) is receiving renewed attention in many scientific fields from photonics to speech modeling. While numerous scalable codes exist for solving integer-order partial differential equations (PDEs), the same is not true for fractional order PDEs. Therefore, there is a need for highly scalable numerical methods and codes for solving fractional order PDEs on complex geometries. The key challenge is that most approaches for fractional PDEs have at least quadratic complexity in both storage and compute, and are challenging to scale. We present a scalable framework for solving fractional diffusion equations using the method of eigen-function expansion. This includes a scalable parallel algorithm to efficiently compute the full set of eigenvalues and eigenvectors for a discretized Laplace eigenvalue problem and apply them to construct approximate solutions to fractional order model problems. We demonstrate the efficacy of our methods by performing strong and weak scalability tests using complex geometries on TACC's Frontera compute cluster. We also show that our approach compares favorably against existing dense and sparse solvers. In our largest solve, we estimated half a million eigenpairs using 28,672 cores.