Improved Polynomial Bounds and Acceleration of GMRES by Solving a min-max Problem on Rectangles, and by Deflating

TL;DR

The paper addresses GMRES convergence under left, right, and split preconditioning with a $W$-weighted inner product, focusing on positive definite $A$. It advances polynomial convergence analysis by recasting the GMRES bound as a min-max problem on the field of values and solving it on rectangles in the complex plane, using bounds derived from the Crouzeix-Palencia framework. It introduces two deflation strategies—based on high-frequency eigenvectors of $H N$ and of $M^{-1} N$—and demonstrates both analytic bounds and numerical acceleration of GMRES, showing substantial reductions in iteration counts. The findings guide practical choices of preconditioners, weights, and deflation spaces to speed up GMRES for large-scale, nonnormal problems while preserving stability and robustness.

Abstract

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.

Figures (12)

• Figure 1: $\Omega$ is defined by $\mu = 3$ and $\rho = 4$. Left: Disk enclosing $\Omega$ ($a = 17$ and $r = 4 \sqrt{17}$). Center: Disk-segment circumscribing $\Omega$. Right: Family of ellipses $E(c,d,b)$ circumscribing $\Omega$. The dot on the real axis is the center of all ellipses. The other dots are the foci of the ellipses.
• Figure 2: For various choices of $\Omega$ parametrized by $\mu$ and $\rho$, asymptotic convergence rate $| ({ a + \sqrt{a^2 - d^2}})/({c + \sqrt{c^2 - d^2}})|$ from \ref{['eq:ellasymptotic']} with respect to the distance from the ellipse to zero. For each $\Omega$ there is an optimal ellipse that gives the best (i.e., lowest) convergence rate.
• Figure 3: Bound for $K_k(\Omega)$ computed using \ref{['eq:boundellipse']} for different choices of the enclosing ellipse including the one that gives the optimal convergence rate. Top: $(\mu, \rho) = (2, 10)$ -- Bottom: $(\mu, \rho) = (2, 40)$. The vertical axis is not the same for the two plots.
• Figure 4: Contour plot (in the imaginary plane) of $|\hat{C}_k|$, the modulus of the polynomial from which the bound in \ref{['eq:boundellipse']} is obtained for polynomial orders $k = 1,\,2,\,3,\,4$. The red dot-dashed rectangle shows $\Omega$ (for $\mu =2$ and $\rho = 4$). The black dashed ellipse is the optimal ellipse from which $\hat{C}_k$ is defined. The resulting upper bound for $K_k(\Omega)$ is also reported. The real and imaginary axes have different scales.
• Figure 5: Comparison between all the bounds for $\mu = 2$ and $\rho = 4$. The bottom plot is a zoom on the first 10 iterations.

Theorems & Definitions (16)

• Theorem 2.1: Minimization property
• proof
• Remark 2.1
• Remark 2.2
• Theorem 2.2: Crouzeix-Palencia bound
• proof
• Theorem 2.3
• Definition 3.1
• Theorem 3.1
• Remark 3.1