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A multigrid reduction framework for domains with symmetries

Àdel Alsalti-Baldellou, Carlo Janna, Xavier Álvarez-Farré, F. Xavier Trias

TL;DR

The paper addresses the computational bottleneck of Poisson solvers arising from divergence constraints in CFD and similar contexts. It introduces AMGR, a symmetry-aware multigrid-reduction framework that aggressively coarsens the top level and uses a compute-intensive SpMM on the smoothers while preserving AMG convergence. It also develops low-rank corrections for factorable and non-factorable preconditioners (LRCFSAI(k), LRCAMG(k)) and Schur-based AMG, and shows AMGR's superior performance on industrial CFD problems. The results demonstrate up to about 70% speedups and good scalability, with applicability to asymmetric boundary conditions and recurring geometries.

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.

A multigrid reduction framework for domains with symmetries

TL;DR

The paper addresses the computational bottleneck of Poisson solvers arising from divergence constraints in CFD and similar contexts. It introduces AMGR, a symmetry-aware multigrid-reduction framework that aggressively coarsens the top level and uses a compute-intensive SpMM on the smoothers while preserving AMG convergence. It also develops low-rank corrections for factorable and non-factorable preconditioners (LRCFSAI(k), LRCAMG(k)) and Schur-based AMG, and shows AMGR's superior performance on industrial CFD problems. The results demonstrate up to about 70% speedups and good scalability, with applicability to asymmetric boundary conditions and recurring geometries.

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.
Paper Structure (10 sections, 2 theorems, 56 equations, 5 figures, 9 tables, 3 algorithms)

This paper contains 10 sections, 2 theorems, 56 equations, 5 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Symmetry-aware spatial discretisation
  3. Low-rank corrections for factorable preconditioners
  4. Low-rank corrections for non-factorable preconditioners
  5. Inner-interface spatial discretisation
  6. Schur complement-based AMG
  7. Multigrid reduction
  8. Practical implementation
  9. Experimental results
  10. Conclusions

Key Result

Theorem 2.1

\newlabelthm:correction0 Given the two SPD matrices $A$ and $B$, let $L$ be the lower Cholesky factor of $B$, i.e., $B = LL^T$. Then, given $Y \coloneqq ( \mathbb{I} - L^{-1} A L^{-T} )$, the following holds: where $Y=Y^T$ and $Y=V \Sigma V^T$ is the eigendecomposition of $Y$.

Figures (5)

  • Figure 1: Single-symmetry 1D mesh with a random mirrored ordering.
  • Figure 1: 1D meshes with a random inner-interface ordering.
  • Figure 1: Adequate partitioning of a mesh with 2 reflection symmetries.
  • Figure 1: Industrial test cases used in the numerical experiments. Top: DrivAer car model and finned-tube heat exchanger. Bottom: $6 \times 4$ wind farm.
  • Figure 2: AMG and AMGR scaling on the unit cube. Left: weak scaling with a workload of $384^3$ unknowns per node. Right: strong scaling on a $256^3$ mesh.

Theorems & Definitions (4)

  • Theorem 2.1
  • Proof 1
  • Theorem 2.2
  • Proof 2