On the ADI method for the Sylvester Equation and the optimal-$\mathcal{H}_2$ points

TL;DR

This paper demonstrates the equivalence of ADI and rational Krylov methods for Sylvester equations when using pseudo H2-optimal shifts, which produce near-optimal low-rank solutions.

Contribution

It introduces pseudo H2-optimal shifts, establishing their equivalence in ADI and Krylov methods and showing their near-optimality for Lyapunov equations.

Findings

Pseudo H2-optimal shifts make ADI and Krylov methods equivalent.

These shifts produce nearly optimal low-rank solutions.

Empirical results confirm the effectiveness of the proposed shifts.

Abstract

The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equation. In this paper we show that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods. We will call these shifts pseudo H2-optimal shifts. These shifts are also optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to the rational Krylov projection subspace. Via several examples, we show that the pseudo H2-optimal shifts consistently yield nearly optimal low rank approximations to the solutions of the Lyapunov equations.