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Algebraic Multigrid with Overlapping Schwarz Smoothers and Local Spectral Coarse Grids for Least Squares Problems

Ben S. Southworth, Hussam Al Daas, Golo A. Wimmer, Ed Threlfall

TL;DR

The paper tackles robust solvers for large SPD least-squares systems $A=G^TG$, proposing a fully algebraic multilevel framework that fuses aggregation-based AMG with overlapping Schwarz DD smoothers and locally constructed spectral coarse spaces. By exploiting a SPSD splitting $A=\sum_i \mathcal{R}_i^T\widetilde{A}_i\mathcal{R}_i$ and solving local generalized eigenproblems, it builds sparse, effective coarse spaces and transfers that work across multiple levels without global eigenproblem solves. The resulting LS–AMG–DD method demonstrates AMG-like operator complexity while achieving convergence that is robust to extreme anisotropy, including challenging fusion problems where classical AMG struggles. The approach is fully algebraic, scalable, and flexible in coarsening strategy, making it a practical black-box solver for a broad class of least-squares systems in scientific computing.

Abstract

This paper develops a new algebraic multigrid (AMG) method for sparse least-squares systems of the form $A=G^TG$ motivated by challenging applications in scientific computing where classical AMG methods fail. First we review and relate the use of local spectral problems in distinct fields of literature on AMG, domain decomposition (DD), and multiscale finite elements. We then propose a new approach blending aggregation-based coarsening, overlapping Schwarz smoothers, and locally constructed spectral coarse spaces. By exploiting the factorized structure of $A$, we construct an inexpensive symmetric positive semidefinite splitting that yields local generalized eigenproblems whose solutions define sparse, nonoverlapping coarse basis functions. This enables a fully algebraic and naturally recursive multilevel hierarchy that can either coarsen slowly to achieve AMG-like operator complexities, or coarsen aggressively-with correspondingly larger local spectral problems-to ensure robustness on problems that cannot be solved by existing AMG methods. The method requires no geometric information, avoids global eigenvalue solves, and maintains efficient parallelizable setup through localized operations. Numerical experiments demonstrate that the proposed least-squares AMG-DD method achieves convergence rates independent of anisotropy on rotated diffusion problems and remains scalable with problem size, while for small amounts of anisotropy we obtain convergence and operator complexities comparable with classical AMG methods. Most notably, for extremely anisotropic heat conduction operators arising in magnetic confinement fusion, where AMG and smoothed aggregation fail to reduce the residual even marginally, our method provides robust and efficient convergence across many orders of magnitude in anisotropy strength.

Algebraic Multigrid with Overlapping Schwarz Smoothers and Local Spectral Coarse Grids for Least Squares Problems

TL;DR

The paper tackles robust solvers for large SPD least-squares systems , proposing a fully algebraic multilevel framework that fuses aggregation-based AMG with overlapping Schwarz DD smoothers and locally constructed spectral coarse spaces. By exploiting a SPSD splitting and solving local generalized eigenproblems, it builds sparse, effective coarse spaces and transfers that work across multiple levels without global eigenproblem solves. The resulting LS–AMG–DD method demonstrates AMG-like operator complexity while achieving convergence that is robust to extreme anisotropy, including challenging fusion problems where classical AMG struggles. The approach is fully algebraic, scalable, and flexible in coarsening strategy, making it a practical black-box solver for a broad class of least-squares systems in scientific computing.

Abstract

This paper develops a new algebraic multigrid (AMG) method for sparse least-squares systems of the form motivated by challenging applications in scientific computing where classical AMG methods fail. First we review and relate the use of local spectral problems in distinct fields of literature on AMG, domain decomposition (DD), and multiscale finite elements. We then propose a new approach blending aggregation-based coarsening, overlapping Schwarz smoothers, and locally constructed spectral coarse spaces. By exploiting the factorized structure of , we construct an inexpensive symmetric positive semidefinite splitting that yields local generalized eigenproblems whose solutions define sparse, nonoverlapping coarse basis functions. This enables a fully algebraic and naturally recursive multilevel hierarchy that can either coarsen slowly to achieve AMG-like operator complexities, or coarsen aggressively-with correspondingly larger local spectral problems-to ensure robustness on problems that cannot be solved by existing AMG methods. The method requires no geometric information, avoids global eigenvalue solves, and maintains efficient parallelizable setup through localized operations. Numerical experiments demonstrate that the proposed least-squares AMG-DD method achieves convergence rates independent of anisotropy on rotated diffusion problems and remains scalable with problem size, while for small amounts of anisotropy we obtain convergence and operator complexities comparable with classical AMG methods. Most notably, for extremely anisotropic heat conduction operators arising in magnetic confinement fusion, where AMG and smoothed aggregation fail to reduce the residual even marginally, our method provides robust and efficient convergence across many orders of magnitude in anisotropy strength.
Paper Structure (22 sections, 1 theorem, 52 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 1 theorem, 52 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Algebraic multigrid
  4. Algebraic domain decomposition smoothers
  5. Spectral coarse grids
  6. Element based AMG
  7. Generalized multiscale finite elements
  8. Domain decomposition and GenEO methods
  9. Algebraic spectral coarse spaces
  10. Spectral DD-AMG
  11. A multilevel method for $A=G^TG$
  12. SPSD splitting and two-level method
  13. DD local eigenvalue problem
  14. Multilevel method
  15. Algorithmic discussion
  16. ...and 7 more sections

Key Result

Lemma 1

Define the multiplicity of node $j\in\{1,\ldots,m\}$ corresponding to rows of $G$ as $M(j) = |\{k\ : \ j \in \mathrm{nz}_k\}|$, that is, $M(j)$ is the number of overlapping aggregates that the $j$th row of $G$ is nonzero in. Define $W =$diag$(1/M(j)) \in \mathbb{R}^{m\times m}$ to average contributi Then $\widetilde{A}_i$ is SPSD for all $i\in[1,n_c]$ and

Figures (5)

  • Figure 1: Solver performance as a function of anisotropy coefficient $\epsilon$ for fixed $N=250,000$ and $\theta = \pi/6$. For reference, the average convergence factor at $\epsilon = 10^{-7}$ for $LS_{2,3,4}$, AMG, and SA are (respectively) $\rho = 0.5, 0.86, 0.94$.
  • Figure 2: Solver performance as a function of total DOFs $N$ for fixed $\epsilon=10^{-5}$ and $\theta = \pi/6$. For reference, the average convergence factor at $\epsilon = 10^{-5}$ for $LS_{2,3,4}$, AMG, and SA are (respectively) $\rho = 0.51, 0.9, 0.97$.
  • Figure 3: Solver performance as a function of angle $\theta$ for fixed total DOFs $250,000$ and $\epsilon=10^{-5}$.
  • Figure 4: Initial temperature field for text case \ref{['Nimrod_IC']}, with fixed magnetic field lines shown in white.
  • Figure 5: Solver performance and operator complexity as a function of total DOFs $N\approx 25$K for fixed mesh refinement $\Delta x=0.017$ and varying $\kappa_\| \in[10^2, 10^8]$ (left), and fixing $\kappa_\| = 10^6$ and considering three levels of mesh refinement, $\Delta x\in\{0.008,0.017,0.034\}$. For reference, the average convergence factors range from 0.64 at $\kappa_\| = 10^2$ to $0.78$ at $\kappa_\| = 10^5$, for fixed $\kappa_\perp = 1$.

Theorems & Definitions (3)

  • Lemma 1
  • proof
  • Remark 1