Rapidly convergent series expansions for a class of resolvents

TL;DR

This paper introduces rapidly converging series expansions for a class of resolvent operators, enabling efficient computation especially when the spectrum is within a known interval, with convergence rates comparable to conjugate gradient methods.

Contribution

The authors develop a new series expansion method for resolvents involving non-orthogonal projections, extending the abstract theory of composites and improving convergence properties.

Findings

Series converges in the entire complex plane excluding a known cut.

Convergence rate matches that of conjugate gradient for real z.

Method leverages subspace substitution with non-orthogonal projections.

Abstract

Following advances in the abstract theory of composites, we develop rapidly converging series expansions about $z=\infty$ for the resolvent ${\bf R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ where ${\bf Q}$ is an orthogonal projection and ${\bf P}$ is such that ${\bf P}{\bf P}^\dagger$ is an orthogonal projection. It is assumed that the spectrum of ${\bf P}^\dagger{\bf Q}{\bf P}$ lies within the interval $[z^-,z^+]$ for some known $z^+\leq 1$ and $z^-\geq 0$ and that the actions of the projections ${\bf Q}$ and ${\bf P}{\bf P}^\dagger$ are easy to compute. The series converges in the entire $z$-plane excluding the cut $[z^-,z^+]$. It is obtained using subspace substitution, where the desired resolvent is tied to a resolvent in a larger space and ${\bf Q}$ gets replaced by a projection $\underline{{\bf Q}}$ that is no longer orthogonal. When $z$ is real the rate of convergence of the new method matches that of the conjugate gradient method.