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Improving Smoothed Aggregation AMG Robustness on Stretched Mesh Applications

Chris Siefert, Raymond Tuminaro, Daniel Sunderland

TL;DR

This work analyzes robustness of Smoothed Aggregation AMG on stretched meshes and identifies four sub-steps that govern coarsening: SOC matrix choice, scaling, classification, and dropping with diagonal lumping. It introduces distance Laplacian SOC, non-symmetric scaling, gap-based (cut-drop) classification, and a distributed lumping scheme to preserve row sums and diagonal positivity, demonstrating improved convergence on FE Poisson problems across 2D, 3D, and application-mesh scenarios. The results show that distance-Laplacian SOC with non-symmetric scaling and Gap or Val classification, combined with distributed lumping, yields the most robust AMG convergence for stretched meshes, with reasonable operator complexities. The authors discuss limitations to diffusion-dominated problems and outline future work toward heterogeneous and anisotropic materials, mass-matrix based SOC, and broader PDE applicability.

Abstract

Strength-of-connection algorithms play a key role in algebraic multigrid (AMG). Specifically, they determine which matrix nonzeros are classified as weak and so ignored when coarsening matrix graphs and defining interpolation sparsity patterns. The general goal is to encourage coarsening only in directions where error can be smoothed and to avoid coarsening across sharp problem variations. Unfortunately, developing robust and inexpensive strength-of-connection schemes is challenging. The classification of matrix nonzeros involves four aspects: (a) choosing a strength-of-connection matrix, (b) scaling its values, (c) choosing a criterion to classify scaled values as strong or weak, and (d) dropping weak entries which includes adjusting matrix values to account for dropped terms. Typically, smoothed aggregation AMG uses the linear system being solved as a strength-of-connection matrix. It scales values symmetrically using square-roots of the matrix diagonal. It classifies based on whether scaled values are above or below a threshold. Finally, it adjusts matrix values by modifying the diagonal so that the sum of entries within each row of the dropped matrix matches that of the original. While these procedures can work well, we illustrate failure cases that motivate alternatives. The first alternative uses a distance Laplacian strength-of-connection matrix. The second centers on non-symmetric scaling. We then investigate alternative classification criteria based on identifying gaps in the values of the scaled entries. Finally, an alternative lumping procedure is proposed where row sums are preserved by modifying all retained matrix entries (as opposed to just diagonal entries). A series of numerical results illustrates trade-offs demonstrating in some cases notably more robust convergence on matrices coming from linear finite elements on stretched meshes.

Improving Smoothed Aggregation AMG Robustness on Stretched Mesh Applications

TL;DR

This work analyzes robustness of Smoothed Aggregation AMG on stretched meshes and identifies four sub-steps that govern coarsening: SOC matrix choice, scaling, classification, and dropping with diagonal lumping. It introduces distance Laplacian SOC, non-symmetric scaling, gap-based (cut-drop) classification, and a distributed lumping scheme to preserve row sums and diagonal positivity, demonstrating improved convergence on FE Poisson problems across 2D, 3D, and application-mesh scenarios. The results show that distance-Laplacian SOC with non-symmetric scaling and Gap or Val classification, combined with distributed lumping, yields the most robust AMG convergence for stretched meshes, with reasonable operator complexities. The authors discuss limitations to diffusion-dominated problems and outline future work toward heterogeneous and anisotropic materials, mass-matrix based SOC, and broader PDE applicability.

Abstract

Strength-of-connection algorithms play a key role in algebraic multigrid (AMG). Specifically, they determine which matrix nonzeros are classified as weak and so ignored when coarsening matrix graphs and defining interpolation sparsity patterns. The general goal is to encourage coarsening only in directions where error can be smoothed and to avoid coarsening across sharp problem variations. Unfortunately, developing robust and inexpensive strength-of-connection schemes is challenging. The classification of matrix nonzeros involves four aspects: (a) choosing a strength-of-connection matrix, (b) scaling its values, (c) choosing a criterion to classify scaled values as strong or weak, and (d) dropping weak entries which includes adjusting matrix values to account for dropped terms. Typically, smoothed aggregation AMG uses the linear system being solved as a strength-of-connection matrix. It scales values symmetrically using square-roots of the matrix diagonal. It classifies based on whether scaled values are above or below a threshold. Finally, it adjusts matrix values by modifying the diagonal so that the sum of entries within each row of the dropped matrix matches that of the original. While these procedures can work well, we illustrate failure cases that motivate alternatives. The first alternative uses a distance Laplacian strength-of-connection matrix. The second centers on non-symmetric scaling. We then investigate alternative classification criteria based on identifying gaps in the values of the scaled entries. Finally, an alternative lumping procedure is proposed where row sums are preserved by modifying all retained matrix entries (as opposed to just diagonal entries). A series of numerical results illustrates trade-offs demonstrating in some cases notably more robust convergence on matrices coming from linear finite elements on stretched meshes.
Paper Structure (22 sections, 2 theorems, 26 equations, 15 figures, 5 tables, 2 algorithms)

This paper contains 22 sections, 2 theorems, 26 equations, 15 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Smoothed Aggregation and its Shortcomings
  3. Structured Geometric Multigrid Semi-Coarsening
  4. Smoothed Aggregation
  5. Smoothed Aggregation Enhancements
  6. SOC matrix
  7. SOC Scaling and Nonsymmetry
  8. Classification Based on Gaps
  9. Lumping the dropped matrix
  10. Computational Examples
  11. 2D Tensor-Product Stretched Poisson
  12. 3D Tensor-Product Stretched Poisson
  13. 3D Radial Trisection Stretched Poisson
  14. 2D Boundary Layer Meshes with Circular Cavities
  15. Meshes from Applications
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let $A$ be an $n \times n$ matrix with sparsity pattern ${\@fontswitch{}{\mathcal{}} S}$, having positive diagonal entries and and non-negative row sums. Consider a matrix $\tilde{A}$ whose sparsity pattern $\tilde{\@fontswitch{}{\mathcal{}} S} \in {\@fontswitch{}{\mathcal{}} S}$ whose nonzero entri where That is, $e_i$ is the sum of nonzero entries in $A$ excluded from the $i^{th}$ row of $\tild

Figures (15)

  • Figure 1: Stencil for $z$-stretch example. Identical stencil values are highlighted with identical boxes or ovals (dotted rectangles, solid rectangles, dotted ovals, solid ovals)
  • Figure 1: Mesh with 2 nearby vertices relative to distances between other adjacent vertices.
  • Figure 1: 2x de-refinement of 2D tensor-product mesh with stretch factors of $\gamma_1 = 4.5459$ and $\gamma_2 = 1.2877$.
  • Figure 1: Traditional SA criterion values for 2D stretched stencil.
  • Figure 2: Traditional SA criterion values for Figure \ref{['fig:bleez']} stencil.
  • ...and 10 more figures

Theorems & Definitions (7)

  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Corollary 3.3
  • Proof 2
  • Remark 3.4
  • Remark 3.5