Effective error estimation for model reduction with inhomogeneous initial conditions

The paper’s Theorem 2 derives the augmented error system, partitions its observability Gramian, and uses the standard energy identity to obtain ||y_{x0} - ŷ_{x0}||_L2^2 = x0^T Q x0 + 2 x0^T Q̊ W^T x0 + x0^T W Q̂ W^T x0 (their eq. (12)), then inserts an arbitrary low-rank factor U with Q ≈ U^T U to get the exact identity ||y_{x0} - ŷ_{x0}||_L2 = sqrt(Δ_{x0}^2 + x0^T (Q - U^T U) x0) and the spectral-norm bound (Theorem 2) under A, Â Hurwitz. This is exactly the model’s approach: same augmented system, same Gramian block equations (11a–c), same energy argument, same decomposition with U^T U, same norm bound, and the same exactness when U is a Cholesky factor. Therefore both are correct and follow substantially the same proof path, differing only in minor notation and level of detail. Key steps are explicitly stated in the paper’s Sections 4.1–4.2 and Theorem 2 (see the construction of the error system and Gramian blocks and eq. (12) in 4.2, and the theorem statement and its proof) , with the underlying observability energy identity recalled in Section 2 (eq. (4)) .