A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization

TL;DR

This paper introduces a distributed-memory library for efficient computations with dense Hierarchically Semi-Separable matrices using randomized algorithms, enabling fast solutions and matrix-vector products in large-scale applications.

Contribution

It presents a novel distributed-memory implementation of HSS matrix algorithms with adaptive randomized sampling, improving scalability and efficiency for large dense structured matrices.

Findings

Demonstrates efficiency on large problems using up to 8,000 cores.

Provides parallel algorithms for HSS compression, matrix-vector multiplication, and factorization.

Part of the STRUMPACK software for structured matrix computations.

Abstract

We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices appear in many applications, e.g., finite element methods, boundary element methods, etc. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The compression algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. This work is part of a more global effort, the STRUMPACK (STRUctured Matrices PACKage) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver.