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Sharpened PCG Iteration Bound for High-Contrast Heterogeneous Scalar Elliptic PDEs

Philip Soliman, Filipe Cumaru, Alexander Heinlein

TL;DR

The paper addresses PCG convergence for high-contrast heterogeneous scalar elliptic PDEs arising from finite element discretization, where eigenvalue clustering makes the classical $\kappa$-based bound overly pessimistic. It introduces a sharpened multi-cluster PCG iteration bound using edge eigenvalues of spectral clusters and demonstrates its computation from early Ritz values within a 2-level overlapping Schwarz preconditioner, comparing AMS, GDSW, and RGDSW coarse spaces. The key contributions include a two-cluster bound $m_2(\kappa,\kappa_1,\kappa_2)$ extended to $s$ clusters as $m_s(p_1,\ldots,p_s)$, a spectral-gap-based partitioning strategy with a Lambert $W$ threshold, and practical estimation of the bound from Ritz values, enabling differentiation of coarse-space performance. The results show the bound provides sharper iteration predictions, guides the choice of coarse space, and offers a feasible early estimator, with potential impact on preconditioner design for complex, high-contrast PDEs.

Abstract

A new iteration bound for the preconditioned conjugate gradient (PCG) method is presented that more accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too pessimistic. By using the edge eigenvalues of each cluster in the spectral distribution, the bound is shown to be orders of magnitude sharper than the classical bound for certain examples. Its effectiveness is demonstrated on a high-contrast elliptic PDE preconditioned with a two-level overlapping Schwarz preconditioner, where the performance of different (algebraic) coarse spaces is successfully distinguished. A key contribution of this work is the observation that, for certain high-contrast problems, simpler coarse spaces can be made competitive in terms of PCG convergence. Conversely, more complex preconditioners are not always required. Finally, it is shown that the bound can be estimated effectively from Ritz values computed during early PCG iterations.

Sharpened PCG Iteration Bound for High-Contrast Heterogeneous Scalar Elliptic PDEs

TL;DR

The paper addresses PCG convergence for high-contrast heterogeneous scalar elliptic PDEs arising from finite element discretization, where eigenvalue clustering makes the classical -based bound overly pessimistic. It introduces a sharpened multi-cluster PCG iteration bound using edge eigenvalues of spectral clusters and demonstrates its computation from early Ritz values within a 2-level overlapping Schwarz preconditioner, comparing AMS, GDSW, and RGDSW coarse spaces. The key contributions include a two-cluster bound extended to clusters as , a spectral-gap-based partitioning strategy with a Lambert threshold, and practical estimation of the bound from Ritz values, enabling differentiation of coarse-space performance. The results show the bound provides sharper iteration predictions, guides the choice of coarse space, and offers a feasible early estimator, with potential impact on preconditioner design for complex, high-contrast PDEs.

Abstract

A new iteration bound for the preconditioned conjugate gradient (PCG) method is presented that more accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too pessimistic. By using the edge eigenvalues of each cluster in the spectral distribution, the bound is shown to be orders of magnitude sharper than the classical bound for certain examples. Its effectiveness is demonstrated on a high-contrast elliptic PDE preconditioned with a two-level overlapping Schwarz preconditioner, where the performance of different (algebraic) coarse spaces is successfully distinguished. A key contribution of this work is the observation that, for certain high-contrast problems, simpler coarse spaces can be made competitive in terms of PCG convergence. Conversely, more complex preconditioners are not always required. Finally, it is shown that the bound can be estimated effectively from Ritz values computed during early PCG iterations.

Paper Structure

This paper contains 4 sections, 1 theorem, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multi-cluster PCG iteration bound
  3. Numerical experiments
  4. Conclusion

Key Result

theorem 1

Let $k_i, \ i = 0,\dots,s$ be the partition indices for a spectrum with $s$ clusters $\sigma_s$, $\hat{C}^{(i)}_{p_i}(\lambda)$ be as in eq:generalized_scaled_chebyshev_polynomial and let the $i^{\textrm{th}}$ cluster be denoted as $I_i = [\lambda_{k_{i-1}+1},\lambda_{k_i}]$. Then, for any $i > 1$ a

Figures (3)

  • Figure 1: Example of a periodic high-contrast coefficient distribution with multiple inclusions along subdomain interfaces for a subdomain size $H$ = 1/4 and local problem of size $H/h = 16$. Non-overlapping subdomain boundaries $\partial\Omega_i$ and grid elements are indicated with thick and thin lines, respectively. Light-blue: $\mathcal{C}=1$; red: $\mathcal{C}=10^8$.
  • Figure 2: Example of an eigenvalue spectrum consisting of multiple clusters.
  • Figure 3: Actual PCG iterations $m$ versus the classical ($m_1$) and multi-cluster ($m_s$) bounds computed at convergence for varying subdomain sizes $H$.

Theorems & Definitions (2)

  • theorem 1
  • proof