Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations

Shixu Meng, Daniel Appelo, Yingda Cheng

TL;DR

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations and proposes a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.

Abstract

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations. We are interested in computing low rank solutions to matrix equations, e.g. arising from spatial discretization of stiff partial differential equations (PDEs). The low rank GMRES method is a particular class of Krylov subspace method where the iteration is performed on the low rank factors of the solution. Such methods can exploit the low rank property of the solution to save on computational and storage cost. Of critical importance for the efficiency and applicability of the low rank GMRES method is the availability of an effective low rank preconditioner that operates directly on the low rank factors of the solution and that can limit the iteration count and the maximal Krylov rank. The preconditioner we propose here is based on the basis update and Galerkin (BUG) method, resulting from the dynamic low rank approximation. It is a nonlinear preconditioner for the low rank GMRES scheme that naturally operates on the low rank factors. Extensive numerical tests show that this new preconditioner is highly efficient in limiting iteration count and maximal Krylov rank. We show that the preconditioner performs well for general diffusion equations including highly challenging problems, e.g. high contrast, anisotropic equations. Further, it compares favorably with the state of the art exponential sum preconditioner. We also propose a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.

Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations

TL;DR

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations and proposes a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.

Abstract

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations. We are interested in computing low rank solutions to matrix equations, e.g. arising from spatial discretization of stiff partial differential equations (PDEs). The low rank GMRES method is a particular class of Krylov subspace method where the iteration is performed on the low rank factors of the solution. Such methods can exploit the low rank property of the solution to save on computational and storage cost. Of critical importance for the efficiency and applicability of the low rank GMRES method is the availability of an effective low rank preconditioner that operates directly on the low rank factors of the solution and that can limit the iteration count and the maximal Krylov rank. The preconditioner we propose here is based on the basis update and Galerkin (BUG) method, resulting from the dynamic low rank approximation. It is a nonlinear preconditioner for the low rank GMRES scheme that naturally operates on the low rank factors. Extensive numerical tests show that this new preconditioner is highly efficient in limiting iteration count and maximal Krylov rank. We show that the preconditioner performs well for general diffusion equations including highly challenging problems, e.g. high contrast, anisotropic equations. Further, it compares favorably with the state of the art exponential sum preconditioner. We also propose a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.

Paper Structure

This paper contains 15 sections, 4 theorems, 49 equations, 10 figures, 5 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Review: low rank GMRES
  3. GMRES
  4. Low rank GMRES
  5. Preconditioner
  6. Review: basis update and Galerkin (BUG) method
  7. Proposed methods: BUG preconditioner and variants
  8. BUG preconditioner and implicit time stepping
  9. Hybrid preconditioner
  10. Numerical results
  11. Parameter study for the BUG preconditioner
  12. Comparison of BUG, ES, and hybrid preconditioners
  13. Higher order schemes
  14. Conclusions and future work
  15. ES preconditioner

Key Result

Theorem 1

Suppose the matrix differential equation eqn:mode satisfies one-sided Lipschitz condition The implicit midpoint method with low rank GMRES scheme as described in Algorithm algo:imlmbugc applied to a linear diffusion type problem with the property $\|\mathcal{A}\|_2 \le \frac{C \Delta t}{h^2},$ and mesh size $\Delta t=O(h),$ tolerance $\epsilon=O(h^3), \epsilon_2=O(h^2)$ is stable if Line Here, th

Figures (10)

  • Figure 1: Example \ref{['Optimal']}. Solving diffusion equation with variable coefficients \ref{['coefficient ex__parameter_restart']} and manufactured solution \ref{['mms ex__parameter_restart']} with BUG preconditioner. Restart every $3$ iterations. Stopping criteria $\eta_{\mathcal{A},b}(x_k) \le \delta.$$\delta=\epsilon=h^3.$ For $h=h_x = h_y \in \{1.56(-2), 7.81(-3), 3.90(-3)\}$, this figure displays the history of solution error, iteration number, $\eta_{\mathcal{A},b}$, solution rank, and maximal Krylov rank.
  • Figure 2: Example \ref{['Optimal']}. Solving diffusion equation with variable coefficients \ref{['coefficient ex__parameter_restart']} and manufactured solution \ref{['mms ex__parameter_restart']} with BUG preconditioner (which restarts every $3$ iterations) and no preconditioner (which restarts every $3$ iterations "None-3" and every $25$ iterations "None-25"). Stopping criteria $\eta_{\mathcal{A},b}(x_k) \le \delta.$$\delta=\epsilon=h^3.$ For $h=h_x = h_y \in \{ 7.81(-3)\}$, this figure displays the history of solution error, iteration number, and maximal Krylov rank.
  • Figure 3: Example \ref{['Example restart']}. Solving diffusion equation with variable coefficients \ref{['coefficient ex__parameter_restart']} and manufactured solution \ref{['mms ex__parameter_restart']}. Restart every $25$ iterations. Stopping criteria $\eta_{\mathcal{A},b}(x_k) \le \delta.$$\delta=\epsilon=h^3.$ For $h=h_x = h_y \in \{ 1.56(-2), 7.81(-3), 3.90(-3)\}$, this figure displays the history of solution error, iteration number, $\eta_{\mathcal{A},b}$, solution rank, and maximal Krylov rank.
  • Figure 4: Example \ref{['exa:stop']}. Solving diffusion equation with variable coefficients \ref{['coefficient ex__parameter_restart']} and manufactured solution \ref{['mms ex__parameter_restart']}. Restart every $3$ iterations using BUG preconditioner. Stopping criteria $\eta_b \le \delta$. $\delta=\epsilon=h^3.$ For $h= h_x = h_y =7.81(-3)$, we report the history of solution error, $\eta_{\mathcal{A},b}$, and iteration number.
  • Figure 5: Example \ref{['Example compare_es_le_']}. Solving diffusion equation with constant coefficients \ref{['coefficient ex_comparision_es<']} and manufactured solution \ref{['mms ex_comparision_es<']}. Preconditioner: ES, BUG, and hybrid. Rounding tolerance $\epsilon = h^3$. For $h=h_x = h_y =7.81(-3)$, this figure displays the history of solution error, iteration number, $\eta_{\mathcal{A},b}$, solution rank, and maximal Krylov rank. \newlabelfigure:ex_comparision_es<0
  • ...and 5 more figures

Theorems & Definitions (14)

  • Theorem 1: Stability
  • Proof 1
  • Theorem 2: Convergence
  • Proof 2
  • Example 5.1: Optimal parameter
  • Example 5.2: Varying the restart parameter
  • Example 5.3: Varying the stopping criteria
  • Example 5.4: Varying the rounding tolerance
  • Example 5.5
  • Example 5.6: High contrast variable coefficient example
  • ...and 4 more