A Parallel Geometric Multifrontal Solver Using Hierarchically Semiseparable Structure

We present a structured parallel geometry-based multifrontal sparse solver using hierarchically semiseparable (HSS) representations and exploiting the inherent low-rank structures. Parallel strategies for nested dissection ordering (taking low rankness into account), symbolic factorization, and structured numerical factorization are shown. In particular, we demonstrate how to manage two layers of tree parallelism to integrate parallel HSS operations within the parallel multifrontal sparse factorization. Such a structured multifrontal factorization algorithm can be shown to have asymptotically lower complexities in both operation counts and memory than the conventional factorization algorithms for certain partial differential equations. We present numerical results from the solution of the anisotropic Helmholtz equations for seismic imaging, and demonstrate that our new solver was able to solve 3D problems up to 600³ mesh size, with 216M degrees of freedom in the linear system. For this specific model problem, our solver is both faster and more memory efficient than a geometry-based multifrontal solver (which is further faster than general-purpose algebraic solvers such as MUMPS and SuperLU_DIST). For the 600³ mesh size, the structured factors from our solver need about 5.9 times less memory.