Continuous and Discrete Data Assimilation with Noisy Observations for the Rayleigh-Bénard Convection: A Computational Study

The paper is primarily computational and reports empirical scalings for Λ versus noise, space, and time resolutions, but it contains an internal inconsistency on the temporal scaling for CDA (body text says Λ varies linearly with S, whereas the conclusion asserts quadratically; compare the discussion around Fig. 6d with the conclusion) . It also leaves rigorous DDA theory open, noting future work for error estimates . The model provides a thoughtful analytic outline: it matches the paper’s noise-quadratic scaling for both CDA and DDA and the spatial laws (CDA ~ R^2, DDA ~ R^3) consistent with the paper’s numerics , and it explains the faster CDA transient but lower DDA plateau in line with the paper’s observations . However, parts of the model’s analysis (notably sharp DDA lower bounds and the precise S^2/S^3 temporal exponents) are presented as likely open as of 2022-11-05 and rely on assumptions (mixed H^{-1} bounds, semigroup smoothing) not fully justified in the provided sources. Hence, both sides are incomplete: the paper lacks consistent or rigorous proofs for time-scaling, and the model identifies open points while giving only a partial theoretical justification.