Hypergraph partitioning and reordering for parallel sparse triangular solves and tensor decomposition

Several scientific and real-world problems require computations with sparse ma-trices, or more generally, sparse tensors which are multi-dimensional arrays. For sparse matrix computations, parallelization of sparse triangular systems intro-duces significant challenges because of the sequential nature of the computations involved. One approach to parallelize sparse triangular systems is to use sparse triangular SPIKE (stSPIKE) algorithm, which was originally proposed for shared memory architectures. stSPIKE decouples the problem into independent smaller systems and requires the solution of a much smaller reduced sparse triangular sys-tem. We extend and implement stSPIKE for distributed-memory architectures. Then we propose distributed-memory parallel Gauss-Seidel (dmpGS) and ILU (dmpILU) algorithms by means of stSPIKE. Furthermore, we propose novel hy-pergraph partitioning models and in-block reordering methods for minimizing the size and nonzero count of the reduced systems that arise in dmpGS and dmpILU. For sparse tensor computations, tensor decomposition is widely used in the anal-ysis of multi-dimensional data. The canonical polyadic decomposition (CPD) is one of the most popular tensor decomposition methods, which is commonly computed by the CPD-ALS algorithm. Due to high computational and mem-ory demands of CPD-ALS, it is inevitable to use a distributed-memory-parallel algorithm for efficiency. The medium-grain CPD-ALS algorithm, which adopts multi-dimensional cartesian tensor partitioning, is one of the most successful dis-tributed CPD-ALS algorithms for sparse tensors. We propose a novel hypergraph partitioning model, CartHP, whose partitioning objective correctly encapsulates the minimization of total communication volume of multi-dimensional cartesian tensor partitioning. Extensive experiments on real-world sparse matrices and tensors validate the parallel scalability of the proposed algorithms as well as the effectiveness of the proposed hypergraph partitioning and reordering models.