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

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.

💡 Research Summary

The paper investigates the relationship between two widely used iterative techniques for solving large‑scale Sylvester equations, especially the Lyapunov case: the Alternating Direction Implicit (ADI) method and Rational Krylov Projection Methods (RKPM). Both methods rely heavily on the choice of complex shifts, yet a rigorous connection between them has been missing. The authors introduce “pseudo H₂‑optimal shifts,” a set of shifts that satisfy the interpolation conditions of H₂‑optimal model reduction (the transfer‑function values match at the mirror images of the reduced poles) but omit the derivative matching condition.

Lemma 1 shows that when ADI is run with shifts αᵢ = −σᵢ and βᵢ = −σᵢ, the low‑rank factors generated at each iteration lie in the rational Krylov subspaces K_r(A,b,σ) and K_r(B⁎,c,¯σ). This extends earlier results for Lyapunov equations to the general Sylvester case.

The central result (Theorem 2) proves that, if the orthonormal bases Q_r and U_r span the rational Krylov spaces built with the same shift set σ, then the solution of the projected Sylvester equation  Q_r⁎ A Q_r · X̃_r + X̃_r · U_r⁎ B U_r + Q_r⁎ b c⁎ U_r = 0 lifted back as X_r = Q_r X̃_r U_r⁎ coincides exactly with the rank‑r ADI approximation obtained after r steps with the same shifts. The equivalence holds precisely when either the eigenvalues of Q_r⁎ A Q_r or those of U_r⁎ B U_r equal the negatives of the shift set (λ(Q_r⁎ A Q_r) = −σ). The proof relies on Galerkin projection interpolation properties, establishing that the rational functions generated by ADI are reproduced by the Krylov bases.

For the Lyapunov equation (B = A⁎, Y = bb⁎) the authors further prove (Theorem 3) that the residual R = A X_r + X_r A⁎ + bb⁎ is orthogonal to the Krylov subspace (Q_r⁎ R = 0) exactly when the same eigenvalue‑shift condition holds. This orthogonality implies that the residual lives entirely outside the subspace spanned by the basis, which is a hallmark of optimal projection.

Numerical experiments on several large‑scale systems (heat diffusion, circuit simulation, structural dynamics) demonstrate that pseudo H₂‑optimal shifts lead to ADI and RKPM approximations with virtually identical error curves and convergence rates. Compared with traditional heuristic shift selections (e.g., Zolotarev points, evenly spaced shifts), the pseudo H₂ shifts achieve near‑optimal low‑rank approximations with far fewer iterations, especially for Lyapunov problems where the residual orthogonality is observed. The paper also notes that these shifts can be computed efficiently via an IRKA‑like iteration.

In summary, the authors establish a rigorous equivalence between ADI and rational Krylov projection for Sylvester equations when pseudo H₂‑optimal shifts are employed, and they show that these shifts confer optimality properties (residual orthogonality) for Lyapunov equations. This provides a theoretically grounded, practically effective strategy for shift selection in large‑scale model reduction and control applications.