Sensitivity-Driven Adaptive Surrogate Modeling for Simulation and Optimization of Dynamical Systems

The paper’s Theorem 5.7 bounds the approximate QoI error by combining the OCP adjoint Fréchet derivative representation (their Eq. (3.22)) with post‑refinement envelope bounds on g and its first partial derivatives (their Eqs. (5.3)–(5.4)), yielding the explicit time‑integral bound (their Eq. (5.18), followed by Theorem 5.7). The candidate solution repeats the same core steps: it starts from q̃g(gc)Δg, invokes the adjoint representation to separate terms with Δg, Δg_x, Δg_u, Δg_p, applies componentwise absolute‑value inequalities, and substitutes the envelope bounds, pulling out c = max_{|α|≤1,i} c_i^α. This reproduces exactly the claimed inequality with the same weights evaluated at (gc, xc, uc, pc, λc). The only difference is stylistic: the paper presents the bound via an equivalent linear program on the envelope‑bounded perturbations (then solves it analytically), whereas the model directly applies triangle/absolute‑value bounds. Under the paper’s stated assumptions for the adjoint sensitivity (Theorem 3.11) and the post‑refinement envelopes (5.3)–(5.4), both arguments are valid and yield the same bound .