A Direct Elliptic Solver Based on Hierarchically Low-rank Schur Complements

TL;DR

This paper introduces a parallel fast direct solver for rank-compressible block tridiagonal systems, combining cyclic reduction and hierarchical matrix operations to achieve efficient complexity and memory usage, outperforming some existing methods.

Contribution

The paper presents a novel parallel direct solver leveraging hierarchical low-rank Schur complements, improving efficiency and applicability over existing hierarchical matrix solvers.

Findings

Achieves $O(N \, \log^2 N)$ complexity and $O(N \log N)$ memory footprint.

Comparable performance to $\\mathcal{H}$-LU and algebraic multigrid methods.

Can solve problems where algebraic multigrid fails to converge.

Abstract

A parallel fast direct solver for rank-compressible block tridiagonal linear systems is presented. Algorithmic synergies between Cyclic Reduction and Hierarchical matrix arithmetic operations result in a solver with $O(N \log^2 N)$ arithmetic complexity and $O(N \log N)$ memory footprint. We provide a baseline for performance and applicability by comparing with well known implementations of the $\mathcal{H}$-LU factorization and algebraic multigrid with a parallel implementation that leverages the concurrency features of the method. Numerical experiments reveal that this method is comparable with other fast direct solvers based on Hierarchical Matrices such as $\mathcal{H}$-LU and that it can tackle problems where algebraic multigrid fails to converge.