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On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver
Author(s) -
Wang Shen,
de Hoop Maarten V.,
Xia Jianlin
Publication year - 2011
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1111/j.1365-2478.2011.00982.x
Subject(s) - solver , discretization , computer science , helmholtz equation , computational science , speedup , computational complexity theory , helmholtz free energy , context (archaeology) , factorization , domain decomposition methods , parallel computing , algorithm , mathematics , finite element method , mathematical analysis , physics , paleontology , biology , quantum mechanics , programming language , boundary value problem , thermodynamics
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 ( n 4/3 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.