A multigrid reduction framework for domains with symmetries

TL;DR

This paper introduces AMGR, a multigrid reduction framework that leverages symmetries to accelerate Poisson equation solutions, achieving up to 70% speed-up without sacrificing convergence or scalability.

Contribution

It proposes a novel symmetry-aware multigrid reduction method that replaces standard operations with more compute-intensive ones for faster convergence.

Findings

Achieved up to 70% speed-up in CFD applications

Reduced memory footprint and setup costs

Maintained convergence and scalability

Abstract

Divergence constraints are present in the governing equations of numerous physical phenomena, and they usually lead to a Poisson equation whose solution represents a bottleneck in many simulation codes. Algebraic Multigrid (AMG) is arguably the most powerful preconditioner for Poisson's equation, and its effectiveness results from the complementary roles played by the smoother, responsible for damping high-frequency error components, and the coarse-grid correction, which in turn reduces low-frequency modes. This work presents several strategies to make AMG more compute-intensive by leveraging reflection, translational and rotational symmetries. AMGR, our final proposal, does not require boundary conditions to be symmetric, therefore applying to a broad range of academic and industrial configurations. It is based on a multigrid reduction framework that introduces an aggressive coarsening to the multigrid hierarchy, reducing the memory footprint, setup and application costs of the top-level smoother. While preserving AMG's excellent convergence, AMGR allows replacing the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments on industrial CFD applications demonstrated up to 70% speed-ups when solving Poisson's equation with AMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significant degradation.