A distributed-memory hierarchical solver for general sparse linear systems

TL;DR

This paper introduces a parallel hierarchical solver for large sparse linear systems on distributed-memory machines, which is faster and more memory-efficient than traditional methods by leveraging low-rank structures and local communication.

Contribution

The paper proposes a novel parallel hierarchical solver that exploits low-rank structures for efficiency and can serve as a direct solver or preconditioner in distributed-memory environments.

Findings

Faster solution times compared to sparse direct solvers.

Reduced memory usage due to low-rank approximations.

Scalability demonstrated through numerical experiments.

Abstract

We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits the low-rank structure of fill-in blocks. Depending on the accuracy of low-rank approximations, the hierarchical solver can be used either as a direct solver or as a preconditioner. The parallel algorithm is based on data decomposition and requires only local communication for updating boundary data on every processor. Moreover, the computation-to-communication ratio of the parallel algorithm is approximately the volume-to-surface-area ratio of the subdomain owned by every processor. We present various numerical results to demonstrate the versatility and scalability of the parallel algorithm.