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Generalized Optimal AMG Convergence Theory for Nonsymmetric and Indefinite Problems

Ahsan Ali, James Brannick, Karsten Kahl, Oliver A. Krzysik, Jacob B. Schroder, Ben S. Southworth

TL;DR

This work generalizes the two-grid optimal AMG convergence theory to nonsymmetric and indefinite problems by exploiting a matrix-induced orthogonality structure between left and right generalized eigenvectors of the pencil $(A,M)$. It shows that the asymptotic two-grid convergence rate can be characterized by the spectral radius $ ho(E_{TG})= ext{max}_{ ext{excluding chosen modes}}|1-\lambda|$, and that optimal intergrid operators $P$ and $R$ align with the subspace spanned by the smallest generalized eigenvalues of the pair $(A,M)$. The theory naturally extends to symmetric indefinite (saddle-point) problems through appropriate eigenvector layouts and $ ilde{M}$-based analysis, and is validated numerically on nonsymmetric advection-diffusion, the Hermitian-indefinite Wilson Dirac system, and saddle-point Darcy discretizations. By tying transfer operators directly to the relaxation via generalized eigenvectors, the framework provides a robust predictive tool for designing and assessing nonsymmetric AMG methods and suggests practical pathways for solver development in challenging applications. Future work includes building scalable AMG solvers for nonsymmetric and indefinite saddle-point systems that explicitly target the derived spectral-radius criteria.

Abstract

Algebraic multigrid (AMG) is known to be an effective solver for many sparse symmetric positive definite (SPD) linear systems. For SPD systems, the convergence theory of AMG is well-understood in terms of the $A$-norm, but in a nonsymmetric setting, such an energy norm is non-existent. For this reason, convergence of AMG for nonsymmetric systems of equations remains an open area of research. A particular aspect missing from theory of nonsymmetric and indefinite AMG is the incorporation of general relaxation schemes. In the SPD setting, the classical form of optimal AMG interpolation provides a useful insight in determining the best possible two-grid convergence rate of a method based on an arbitrary symmetrized relaxation scheme. In this work, we discuss a generalization of the optimal AMG convergence theory targeting nonsymmetric problems, using a certain matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem relating the system matrix and relaxation operator. We show that using this generalization of the optimal convergence theory, one can obtain a measure of the spectral radius of the two grid error transfer operator that is mathematically equivalent to the derivation in the SPD setting for optimal interpolation, which instead uses norms. In addition, this generalization of the optimal AMG convergence theory can be further extended for symmetric indefinite problems, such as those arising from saddle point systems so that one can obtain a precise convergence rate of the resulting two-grid method based on optimal interpolation. We provide supporting numerical examples of the convergence theory for nonsymmetric advection-diffusion problems, two-dimensional Dirac equation motivated by $γ_5$-symmetry, and the mixed Darcy flow problem corresponding to a saddle point system.

Generalized Optimal AMG Convergence Theory for Nonsymmetric and Indefinite Problems

TL;DR

This work generalizes the two-grid optimal AMG convergence theory to nonsymmetric and indefinite problems by exploiting a matrix-induced orthogonality structure between left and right generalized eigenvectors of the pencil . It shows that the asymptotic two-grid convergence rate can be characterized by the spectral radius , and that optimal intergrid operators and align with the subspace spanned by the smallest generalized eigenvalues of the pair . The theory naturally extends to symmetric indefinite (saddle-point) problems through appropriate eigenvector layouts and -based analysis, and is validated numerically on nonsymmetric advection-diffusion, the Hermitian-indefinite Wilson Dirac system, and saddle-point Darcy discretizations. By tying transfer operators directly to the relaxation via generalized eigenvectors, the framework provides a robust predictive tool for designing and assessing nonsymmetric AMG methods and suggests practical pathways for solver development in challenging applications. Future work includes building scalable AMG solvers for nonsymmetric and indefinite saddle-point systems that explicitly target the derived spectral-radius criteria.

Abstract

Algebraic multigrid (AMG) is known to be an effective solver for many sparse symmetric positive definite (SPD) linear systems. For SPD systems, the convergence theory of AMG is well-understood in terms of the -norm, but in a nonsymmetric setting, such an energy norm is non-existent. For this reason, convergence of AMG for nonsymmetric systems of equations remains an open area of research. A particular aspect missing from theory of nonsymmetric and indefinite AMG is the incorporation of general relaxation schemes. In the SPD setting, the classical form of optimal AMG interpolation provides a useful insight in determining the best possible two-grid convergence rate of a method based on an arbitrary symmetrized relaxation scheme. In this work, we discuss a generalization of the optimal AMG convergence theory targeting nonsymmetric problems, using a certain matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem relating the system matrix and relaxation operator. We show that using this generalization of the optimal convergence theory, one can obtain a measure of the spectral radius of the two grid error transfer operator that is mathematically equivalent to the derivation in the SPD setting for optimal interpolation, which instead uses norms. In addition, this generalization of the optimal AMG convergence theory can be further extended for symmetric indefinite problems, such as those arising from saddle point systems so that one can obtain a precise convergence rate of the resulting two-grid method based on optimal interpolation. We provide supporting numerical examples of the convergence theory for nonsymmetric advection-diffusion problems, two-dimensional Dirac equation motivated by -symmetry, and the mixed Darcy flow problem corresponding to a saddle point system.
Paper Structure (11 sections, 7 theorems, 62 equations, 6 figures, 2 tables)

This paper contains 11 sections, 7 theorems, 62 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Relaxations in matrix forms
  3. Algebraic two-grid HPD theory review
  4. Induced orthogonality of generalized eigenvectors
  5. Optimal AMG theory for non-HPD problems
  6. Numerical examples
  7. 2D constant direction advection-diffusion
  8. Sparse random matrix
  9. 2D Wilson Dirac System
  10. 2D mixed Darcy problem
  11. Conclusion

Key Result

Lemma 3.1

\newlabellem:optimal0 Given $n_{c}$ and $\widetilde{M}$, let $\mathbf{y}_{i}$ solve the generalized EVP where $i=1,2,\dots,n$; $P:\mathbb{C}^{n_{c}}\to\mathbb{C}^{n}$ is full rank, $\mu_{1}\leq \mu_{2} \leq \cdots \leq\mu_{n_{c}}\leq \cdots \leq \mu_{n}$ denote the eigenvalues and $\mathbf{y}_{i}$ denote the corresponding $\widetilde{M}$-orthonormal eigenvectors, i.e., $\left(\mathbf{y}_{i}, \m

Figures (6)

  • Figure 1: Advection-diffusion problems \ref{['eqn:advdiff1']},\ref{['eqn:advdiff2']} - sparsity pattern and verification of the generalized optimal convergence theory using Kaczmarz relaxation where we compare the theoretical prediction with the computed reduction factors.
  • Figure 2: Verification of the generalized optimal convergence theory using Jacobi and Gauss-Seidel method for the advection-diffusion problem \ref{['eqn:advdiff1']},\ref{['eqn:advdiff2']} with diffusion coefficient $\alpha=0.1$.
  • Figure 3: Verification of the generalized optimal convergence theory using Kaczmarz relaxation method for randomly generated sparse nonsymmetric operator $A$.
  • Figure 4: For the Wilson discretization of the Dirac equation problem \ref{['eq:dirac']} - sparsity pattern and numerical verification of the generalized optimal convergence theory.
  • Figure 5: For the 2D mixed Darcy problem \ref{['eqn:darcy1']}, \ref{['eqn:darcy2']} - Initial mesh, sparsity pattern and numerical verification of the generalized optimal convergence theory.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Lemma 3.1
  • Lemma 4.1
  • Proof 1
  • Corollary 4.2
  • Theorem 5.1
  • Proof 2
  • Corollary 5.2: Optimality of transfer operators
  • Proof 3
  • Corollary 5.3: Real-valued Hermitian operators
  • Proof 4
  • ...and 5 more