Sharp estimates for the convergence rate of Orthomin(k) for a class of linear systems

TL;DR

This paper establishes sharp upper bounds for the convergence rate of Orthomin(k) iterative method on specific linear systems involving unitary operators, showing it cannot outperform Orthomin(1) in general.

Contribution

It provides the first sharp estimates for Orthomin(k)'s convergence rate on a class of systems and demonstrates cases where higher k does not improve convergence.

Findings

Orthomin(k) convergence rate is at most ρ for systems with (I+ρU) x = b.

Examples show the convergence rate can be exactly ρ, confirming the estimate's sharpness.

In some cases, Orthomin(k) for k≥2 has the same rate as Orthomin(2), not faster than Orthomin(1).

Abstract

In this work we show that the convergence rate of Orthomin($k$) applied to systems of the form $(I+\rho U) x = b$, where $U$ is a unitary operator and $0<\rho<1$, is less than or equal to $\rho$. Moreover, we give examples of operators $U$ and $\rho>0$ for which the asymptotic convergence rate of Orthomin($k$) is exactly $\rho$, thus showing that the estimate is sharp. While the systems under scrutiny may not be of great interest in themselves, their existence shows that, in general, Orthomin($k$) does not converge faster than Orthomin(1). Furthermore, we give examples of systems for which Orthomin($k$) has the same asymptotic convergence rate as Orthomin(2) for $k\ge 2$, but smaller than that of Orthomin(1). The latter systems are related to the numerical solution of certain partial differential equations.