Accelerated Cyclic Reduction: A Distributed-Memory Fast Solver for Structured Linear Systems

TL;DR

Accelerated Cyclic Reduction (ACR) is a distributed-memory direct solver for 3D structured linear systems, combining cyclic reduction and hierarchical matrix techniques for efficient, scalable solutions especially suited for multiple right-hand sides.

Contribution

The paper introduces ACR, a novel distributed-memory solver that leverages hierarchical matrix arithmetic to efficiently solve rank-compressible block tridiagonal systems from elliptic PDE discretizations.

Findings

ACR has $O(k N ext{log} N ( ext{log} N + k^2))$ arithmetic complexity.

ACR demonstrates good scalability and robustness compared to multifrontal and multigrid methods.

ACR outperforms or matches existing methods in large-scale elliptic system solutions.

Abstract

We present Accelerated Cyclic Reduction (ACR), a distributed-memory fast direct solver for rank-compressible block tridiagonal linear systems arising from the discretization of elliptic operators, developed here for three dimensions. Algorithmic synergies between Cyclic Reduction and hierarchical matrix arithmetic operations result in a solver that has $O(k~N \log N~(\log N + k^2))$ arithmetic complexity and $O(k~N \log N)$ memory footprint, where $N$ is the number of degrees of freedom and $k$ is the rank of a typical off-diagonal block, and which exhibits substantial concurrency. We provide a baseline for performance and applicability by comparing with the multifrontal method where hierarchical semi-separable matrices are used for compressing the fronts, and with algebraic multigrid. Over a set of large-scale elliptic systems with features of nonsymmetry and indefiniteness, the robustness of the direct solvers extends beyond that of the multigrid solver, and relative to the multifrontal approach ACR has lower or comparable execution time and memory footprint. ACR exhibits good strong and weak scaling in a distributed context and, as with any direct solver, is advantageous for problems that require the solution of multiple right-hand sides.