Efficient Scalable Algorithms for Solving Dense Linear Systems with Hierarchically Semiseparable Structures

Hierarchically semiseparable (HSS) matrix techniques are emerging in constructing superfast direct solvers for both dense and sparse linear systems. Here, we develop a set of novel parallel algorithms for key HSS operations that are used for solving large linear systems. These are parallel rank-revealing QR factorization, HSS constructions with hierarchical compression, ULV HSS factorization, and HSS solutions. The HSS tree-based parallelism is fully exploited at the coarse level. The BLACS and ScaLAPACK libraries are used to facilitate the parallel dense kernel operations at the fine-grained level. We appply our new solvers for discretized Helmholtz equations for multifrequency seismic imaging and iteratively solve time-harmonic seismic inverse boundary value problems. In particular, we use the HSS algorithms to solve the dense Schur complement systems associated with the root separator of the separator tree obtained from nested dissection of the graph of discretized Helmholtz equations. We demonstrate that the new approach is much faster and uses much less memory than the LU factorization algorithm for both two-dimensional and three-dimensional problems, using up to 8912 processing cores. This is the first work in parallelizing HSS algorithms and conducting detailed performance analysis on a large parallel machine. This also lays a good foundation for developing scalable sparse structured factorization algorithms for general sparse linear systems.