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The effect of approximate coarsest-level solves on the convergence of multigrid V-cycle methods

Petr Vacek, Erin Carson, Kirk M. Soodhalter

TL;DR

This work analyzes how inexact solves on the coarsest level of a multigrid V-cycle affect convergence for symmetric positive definite problems. By treating the inV-cycle as a perturbation of the exact exV-cycle, it derives propagation matrices and two accuracy assumptions (relative with a factor $\gamma$ and absolute with $\epsilon$) to bound both the convergence rate and the deviation from the exact-cycle solution. The authors formulate practical coarsest-level stopping criteria—one based on a relative residual tolerance and another on an absolute error bound—ensuring the inV-cycle converges to a chosen finest-level accuracy with nearly the same number of V-cycles as the exact coarsest-level solve. Numerical experiments on Poisson-type and jump-1024 problems validate the theory, show when the criteria yield the same iteration counts as exact solves, and discuss trade-offs in coarse-level work and precision. The results offer a principled framework for choosing coarsest-level tolerances and point to extensions to algebraic MG, non-symmetric problems, and different cycle schemes.

Abstract

The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsest-level problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm.

The effect of approximate coarsest-level solves on the convergence of multigrid V-cycle methods

TL;DR

This work analyzes how inexact solves on the coarsest level of a multigrid V-cycle affect convergence for symmetric positive definite problems. By treating the inV-cycle as a perturbation of the exact exV-cycle, it derives propagation matrices and two accuracy assumptions (relative with a factor and absolute with ) to bound both the convergence rate and the deviation from the exact-cycle solution. The authors formulate practical coarsest-level stopping criteria—one based on a relative residual tolerance and another on an absolute error bound—ensuring the inV-cycle converges to a chosen finest-level accuracy with nearly the same number of V-cycles as the exact coarsest-level solve. Numerical experiments on Poisson-type and jump-1024 problems validate the theory, show when the criteria yield the same iteration counts as exact solves, and discuss trade-offs in coarse-level work and precision. The results offer a principled framework for choosing coarsest-level tolerances and point to extensions to algebraic MG, non-symmetric problems, and different cycle schemes.

Abstract

The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsest-level problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm.
Paper Structure (17 sections, 2 theorems, 66 equations, 8 figures, 1 algorithm)

This paper contains 17 sections, 2 theorems, 66 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and motivating experiments
  3. Motivating experiments
  4. Convergence analysis of the inV-cycle method
  5. Relative coarsest-level accuracy
  6. Absolute coarsest-level accuracy
  7. Effects of the choice of the tolerance in relative residual stopping criterion
  8. Absolute coarsest-level stopping criteria
  9. Numerical experiments
  10. inV-cycle method satisfying the relative coarsest-level accuracy assumption
  11. Accuracy of the estimates for inV-cycle methods with a relative residual coarsest-level stopping criterion
  12. inV-cycle method satisfying the absolute coarsest-level accuracy assumption
  13. inV-cycle method with absolute coarsest-level stopping criteria
  14. Performance of inV-cycle methods with absolute coarsest-level stopping criteria
  15. Conclusions and open problems
  16. ...and 2 more sections

Key Result

Theorem 3.1

\newlabelthm:oneV-cycle0 Let $\mathbf{x}^{\mathrm{new}}_{\mathrm{ex}}$ be the approximation of $\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$ computed after one iteration of the exV-cycle method with error propagation matrix $\mathbf{E}$, $\| \mathbf{E}\|_{\mathbf{A}}<1$, starting with an approximation for some constant $\gamma>0$. Then the following estimate on the $\mathbf{A}$-norm of the relative d

Figures (8)

  • Figure 1: Comparison of inV-cycle methods with CG as the coarsest-level solver with various choices of relative residual tolerance $\tau$. The bright yellow and green color highlight variants that converge in the same number of V-cycles as the variant with MATLAB backslash operator on the coarsest-level. The bright yellow variants achieve this in the least total number of CG iterations on the coarsest-level.
  • Figure 1: Properties of inV-cycle methods with CG as the solver on the coarsest level, which is stopped when the assumption on the relative coarsest-level accuracy \ref{['eq:thm:gamma_def']} is satisfied with $\gamma=0.3$ (\ref{['line:ex1_gamma_0.3']}), $\gamma=10^{-3}$ (\ref{['line:ex1_gamma_1e-3']}), or $\gamma=10^{-4}$ (\ref{['line:ex1_gamma_1e-4']}). The dashed lines corresponds to the estimates $\| \mathbf{E} \|_{\mathbf{A}} + \gamma$. For comparison we also include results of the exV-cycle method (\ref{['line:ex1_backslash']}).
  • Figure 2: $\mathbf{A}$-norm of the error of the inV-cycle methods with CG as the solver on the coarsest level, which is stopped when the assumption on the relative coarsest-level accuracy \ref{['eq:thm:gamma_def']} is satisfied with $\gamma=0.3$ (\ref{['line:ex1_fix_gamma_0.3']}), $\gamma=10^{-3}$ (\ref{['line:ex1_fix_gamma_1e-3']}), or $\gamma=10^{-4}$ (\ref{['line:ex1_fix_gamma_1e-4']}). For comparison we also include the $\mathbf{A}$-norm of the error of the exV-cycle method (\ref{['line:ex1_fix_backslash']}). Every third point is marked.
  • Figure 3: Testing accuracy of the estimate discussed in \ref{['sec:effects_rel_res']}. We consider the V-cycle method with CG as the solver on the coarsest level. CG is stopped using the relative residual stopping criterion \ref{['eq:relative_res_stop_crit']} with $\tau = 10^{-4} \| \mathbf{T} \|^{-1} \| \mathbf{A}\|^{-\frac{1}{2}} \| \mathbf{A}^{-1}_0\|^{-\frac{1}{2}}$. The computation is done in standard MATLAB double precision (\ref{['line:ex3_relres']}) and in simulated quad precision using the Advapix toolbox (\ref{['line:ex3_relres_quad']}).
  • Figure 4: Properties of inV-cycle methods with CG as the solver on the coarsest level, which is stopped when the assumption on the absolute coarsest-level accuracy \ref{['eq:thm:eps_def']} (approximately) holds with $\epsilon = \theta( 1 - \| \mathbf{E} \|_{\mathbf{A}})$, where $\theta=10^{-4}$ (\ref{['line:ex2_delta_1e-4']}) or $\theta=10^{-11}$ (\ref{['line:ex2_delta_1e-11']}). For comparison we also include the $\mathbf{A}$-norm of the error of the exV-cycle method (\ref{['line:ex2_backslash']}).
  • ...and 3 more figures

Theorems & Definitions (2)

  • Theorem 3.1
  • Theorem 3.2