Efficient Parallel Algorithms for Hierarchically Semiseparable Matrices

Recently, hierarchically semiseparable (HSS) matrices have been used in the development of fast direct sparse solvers. Key applications of HSS algorithms, coupled with multifrontal solvers, appear in solving certain large-scale computational inverse problems. Here, we develop massively parallel HSS algorithms appearing in these solution methods, namely, parallel HSS construction using the rank revealing QR (RRQR) method, parallel HSS ULV factorization, and parallel HSS solution. HSS representations have a nice binary tree structure called HSS tree. HSS operations can be conducted following the traversal of this tree, and communications are generally limited to between siblings and between parents and children. Thus, HSS algorithms are often highly scalable. BLACS [1] and ScaLapack [3] are used as our porTable libraries. We construct contexts (sub-communicators) on each node of the HSS tree and exploit the governing 2D-block-cyclic data distribution scheme widely used in ScaLapack. Computational examples confirm the weak scaling, strong scaling and accuracy of our implementation.