On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver

We consider the modeling of (polarized) seismic wave propagation on a rectangular domain via the discretization and solution of the inhomogeneous Helmholtz equation in 3D, by exploiting a parallel multifrontal sparse direct solver equipped with Hierarchically Semi-Separable (HSS) structure to reduce the computational complexity and storage. In particular, we are concerned with solving this equation on a large domain, for a large number of different forcing terms in the context of seismic problems in general, and modeling in particular. We resort to a parsimonious mixed grid finite differences scheme for discretizing the Helmholtz operator and Perfect Matched Layer boundaries, resulting in a non-Hermitian matrix. We make use of a nested dissection based domain decomposition, and introduce an approximate direct solver by developing a parallel HSS matrix compression, factorization, and solution approach. We cast our massive parallelization in the framework of the multifrontal method. The assembly tree is partitioned into local trees and a global tree. The local trees are eliminated independently in each processor, while the global tree is eliminated through massive communication. The solver for the inhomogeneous equation is a parallel hybrid between multifrontal and HSS structure. The computational complexity associated with the factorization is almost linear in the size, n say, of the matrix, viz. between O(n log n) and O(n4/3 log n), while the storage is almost linear as well, between O(n) and O(n log n). We exploit the use of a regular (Cartesian) mesh common in many seismic applications.