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Coarse grid corrections in Krylov subspace evaluations of the matrix exponential

Mike A. Botchev

TL;DR

This work introduces a coarse grid correction (CGC) strategy to accelerate matrix-exponential and $\varphi$-function actions in iterative solvers by decomposing the source into a smooth coarse-grid component and a nonsmooth fine-grid remainder. The two-grid and multigrid formulations provide provable error bounds based on exponential residuals and grid compatibility, while practical error estimators enable on-the-fly assessment. Numerical experiments on 1D and 3D heat equations show that CGC can substantially reduce computational work, especially when paired with Krylov subspace methods, though gains with Chebyshev methods are more modest. The approach offers a scalable and parallelizable path for efficient exponential-time integration in large-scale PDE discretizations, with potential extensions to nonsymmetric problems and adaptive multigrid cycles.

Abstract

A coarse grid correction (CGC) approach is proposed to enhance the efficiency of the matrix exponential and $\varphi$ matrix function evaluations. The approach is intended for iterative methods computing the matrix-vector products with these functions. It is based on splitting the vector by which the matrix function is multiplied into a smooth part and a remaining part. The smooth part is then handled on a coarser grid, whereas the computations on the original grid are carried out with a relaxed stopping criterion tolerance. Estimates on the error are derived for the two-grid and multigrid variants of the proposed CGC algorithm. Numerical experiments demonstrate the efficiency of the algorithm, when employed in combination with Krylov subspace and Chebyshev polynomial expansion methods.

Coarse grid corrections in Krylov subspace evaluations of the matrix exponential

TL;DR

This work introduces a coarse grid correction (CGC) strategy to accelerate matrix-exponential and -function actions in iterative solvers by decomposing the source into a smooth coarse-grid component and a nonsmooth fine-grid remainder. The two-grid and multigrid formulations provide provable error bounds based on exponential residuals and grid compatibility, while practical error estimators enable on-the-fly assessment. Numerical experiments on 1D and 3D heat equations show that CGC can substantially reduce computational work, especially when paired with Krylov subspace methods, though gains with Chebyshev methods are more modest. The approach offers a scalable and parallelizable path for efficient exponential-time integration in large-scale PDE discretizations, with potential extensions to nonsymmetric problems and adaptive multigrid cycles.

Abstract

A coarse grid correction (CGC) approach is proposed to enhance the efficiency of the matrix exponential and matrix function evaluations. The approach is intended for iterative methods computing the matrix-vector products with these functions. It is based on splitting the vector by which the matrix function is multiplied into a smooth part and a remaining part. The smooth part is then handled on a coarser grid, whereas the computations on the original grid are carried out with a relaxed stopping criterion tolerance. Estimates on the error are derived for the two-grid and multigrid variants of the proposed CGC algorithm. Numerical experiments demonstrate the efficiency of the algorithm, when employed in combination with Krylov subspace and Chebyshev polynomial expansion methods.

Paper Structure

This paper contains 14 sections, 3 theorems, 47 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Basic multigrid concepts
  3. Problem setting and baseline methods
  4. Krylov subspace methods
  5. Chebyshev polynomial expansion
  6. Coarse grid corrections
  7. Coarse grid corrections, a two-grid version
  8. CGC algorithm, a multigrid version
  9. Towards a full multigrid cycle
  10. Evaluation of the $\varphi_k$ matrix functions
  11. Numerical experiments
  12. 1D heat equation
  13. 3D heat equation
  14. Conclusions

Key Result

Lemma 1

BGH13 Let $A\in\mathbb{R}^{N\times N}$ satisfy omega and let $y_{\mathrm{ex}}(t)$ be the exact solution of ivp. If $r_k(t)$ is the residual of an approximate solution $y_k(t)\approx y_{\mathrm{ex}}(t)$ then the error $\epsilon_k(t)=y_{\mathrm{ex}}(t)-y_k(t)$ can bounded in norm for any $t\geqslant 0

Figures (4)

  • Figure 1: The smoothing properties of conventional iterative schemes for solving linear systems $A x = b$. $A$ is a 1D discretized Laplacian with homogeneous Dirichlet boundary conditions in the space domain $[0,1]$. Standard second order three-point finite differences are used on a uniform grid of $N=100$ nodes. Zero initial guess vector $x_0=0$ is taken, so that $r_0=b$. Top left: entries of the right-hand side vector $b$ being a sum of two normalized eigenvectors of $A$, corresponding to the smallest in magnitude and 51st smallest in magnitude eigenvalues. Top right and bottom left: entries of the residual vector at iterations 1 and 2. Bottom right: restriction $Q^T$ followed by prolongation $Q$ as a smoother, with a uniform coarse grid of $n=50$ nodes.
  • Figure 1: A two-grid version of our CGC (coarse grid correction) algorithm
  • Figure 2: A two-grid method iteration for solving linear system $Ax=b$
  • Figure 2: A multigrid version of the CGC algorithm

Theorems & Definitions (5)

  • Lemma 1
  • Proposition 1
  • Proof 1
  • Proposition 2
  • Proof 2