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

Abstract

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 \textttBLACS and \textttScaLAPACK 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...