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Integrating Additive Multigrid with Multipreconditioned Conjugate Gradient Method

Hardik Kothari, Maria Giuseppina Chiara Nestola, Marco Favino, Rolf Krause

TL;DR

The paper tackles slow convergence of additive multigrid methods in large-scale diffusion problems by coupling additive MG with multipreconditioned conjugate gradient (MPCG). It leverages corrections from multiple MG levels as distinct preconditioners and selects coefficients via energy-norm minimization to form robust, $A$-conjugate search directions, implemented through a truncated CG with memory. Numerical experiments show the additive MG–MPCG method often outperforms additive MG–PCG and remains competitive with multiplicative MG–PCG, particularly when the multilevel hierarchy is not optimal. The approach offers a scalable, parallelizable preconditioning strategy for anisotropic diffusion and fractured media, with potential extensions to matrix-free and unfitted FEM contexts.

Abstract

Due to its optimal complexity, the multigrid (MG) method is one of the most popular approaches for solving large-scale linear systems arising from the discretization of partial differential equations. However, the parallel implementation of standard MG methods, which are inherently multiplicative, suffers from increasing communication complexity. In such cases, the additive variants of MG methods provide a good alternative due to their inherently parallel nature, although they exhibit slower convergence. This work combines the additive multigrid method with the multipreconditioned conjugate gradient (MPCG) method. In the proposed approach, the MPCG method employs the corrections from the different levels of the MG hierarchy as separate preconditioned search directions. In this approach, the MPCG method updates the current iterate by using the linear combination of the preconditioned search directions, where the optimal coefficients for the linear combination are computed by exploiting the energy norm minimization of the CG method. The idea behind our approach is to combine the $A$-conjugacy of the search directions of the MPCG method and the quasi $H_1$-orthogonality of the corrections from the MG hierarchy. In the numerical section, we study the performance of the proposed method compared to the standard additive and multiplicative MG methods used as preconditioners for the CG method.

Integrating Additive Multigrid with Multipreconditioned Conjugate Gradient Method

TL;DR

The paper tackles slow convergence of additive multigrid methods in large-scale diffusion problems by coupling additive MG with multipreconditioned conjugate gradient (MPCG). It leverages corrections from multiple MG levels as distinct preconditioners and selects coefficients via energy-norm minimization to form robust, -conjugate search directions, implemented through a truncated CG with memory. Numerical experiments show the additive MG–MPCG method often outperforms additive MG–PCG and remains competitive with multiplicative MG–PCG, particularly when the multilevel hierarchy is not optimal. The approach offers a scalable, parallelizable preconditioning strategy for anisotropic diffusion and fractured media, with potential extensions to matrix-free and unfitted FEM contexts.

Abstract

Due to its optimal complexity, the multigrid (MG) method is one of the most popular approaches for solving large-scale linear systems arising from the discretization of partial differential equations. However, the parallel implementation of standard MG methods, which are inherently multiplicative, suffers from increasing communication complexity. In such cases, the additive variants of MG methods provide a good alternative due to their inherently parallel nature, although they exhibit slower convergence. This work combines the additive multigrid method with the multipreconditioned conjugate gradient (MPCG) method. In the proposed approach, the MPCG method employs the corrections from the different levels of the MG hierarchy as separate preconditioned search directions. In this approach, the MPCG method updates the current iterate by using the linear combination of the preconditioned search directions, where the optimal coefficients for the linear combination are computed by exploiting the energy norm minimization of the CG method. The idea behind our approach is to combine the -conjugacy of the search directions of the MPCG method and the quasi -orthogonality of the corrections from the MG hierarchy. In the numerical section, we study the performance of the proposed method compared to the standard additive and multiplicative MG methods used as preconditioners for the CG method.
Paper Structure (8 sections, 6 equations, 6 figures, 1 algorithm)

This paper contains 8 sections, 6 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Problem definition
  2. Solution method
  3. Additive multigrid method
  4. Multipreconditioned Conjugate Gradient method (MPCG)
  5. Numerical Experiments
  6. Comparison with the other solution methods
  7. Comparing components of $\boldsymbol{\alpha}$
  8. Conclusion

Figures (6)

  • Figure 1: Fracture network and pressure distribution for two cases (Example 2)
  • Figure 2: Iterations $v/s$ corrections for two different test cases of Example$~1$
  • Figure 3: Iterations $v/s$ corrections for two different test cases of Example$~2$
  • Figure 4: Comparing the number of iterations required for convergence for various values of the diffusion coefficients $k_{xx}$ and fracture permeability $k_f$.
  • Figure 5: Example 1: Heatmap of components of $\alpha_\ell$ associated with different level of the multilevel hierarchy
  • ...and 1 more figures