Lagrangian Reduced Order Modeling Using Finite Time Lyapunov Exponents

Round 1

Reviewer 1 Report

This manuscript describes the construction of two new Lagrangian reduced order models (ROMs) - alpha ROM and lambda ROM - for solving the QGE mathematical model. The paper argues that the Lagrangian ROMs are more accurate than the Eulerian ROMs generally employed. The paper is well written and the scientific arguments are well presented. However, a few comments need to be addressed. Therefore the article is recommended for publication but after minor revisions.

The manuscript currently claims that both Lagrangian ROMs lead to orders of magnitude improvement in accuracy. However, based on the results only the alpha-ROM exhibits such an improvement that too when lower number of basis functions (r) are considered. Lambda-ROM does not show orders of magnitude improvement in accuracy. This must be corrected in the paper (especially in lines 471-474, in the abstract and other places in the paper).

Since the ROM methods are being compared with the FOM method, it is important that the colorbar scales used in figures 1, 4 and 5 must be the same at least for r=30 cases. Especially the statement in lines 386-387 regarding the qualitative metrics cannot be verified until the scales are the same. 

The authors mention in the conclusion that higher alpha always yielded most accurate results, this is not always true from the tabulated results provided in the paper. This is somewhat true for the Eulerian metrics (Table 2) but there too you observe alpha=1000 yielding better results than alpha=10000 even at lower r values. It is not clear that choosing a higher alpha always yields better results. Tables 3 indicate that as the dimension increases, lower alphas lead to better accuracy in Eulerian metrics for predictive regime. The Lagrangian metrics show this trend in both reconstructive and predictive regimes. As r increases, the general trend suggests that lower alphas lead to better accuracy. Mentioning this observation and a discussion on this would be helpful to the readers and is suggested to be included.

As the entire discussion in the paper mostly centers around alpha-ROM and it clearly demonstrates better results in almost all cases, I am wondering if lambda-ROM needs to be included in the paper?

From Figures 4 and 5 (looking at r=30 cases) it seems that in the predictive regime E-ROM and lambda-ROM perform better than in the reconstructive regime. However, alpha-ROM seems to deomonstrate the opposite effect. What might be causing this?

Editorial Correction: In line 188, the equation reference must be to (19) instead of (17).

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

I enjoyed reading this manuscript. It develops an interesting new idea in the construction of ROMs with clear exposition and well-designed experiments. I have only minor comments and suggestions for revisions, listed below. 1. There are two places in the construction of a ROM where inner products play a role: (i) constructing the basis and (ii) computing the evolution equations [equation (11)]. The entire paper, but particularly the introduction, doesn't spend much time explaining how the inner product is used in the basis construction. I know how it works, but other readers might benefit from a little more detail on how the new inner product changes the POD analysis that generates the basis. 1a. Also, you define a new inner product (19) but you only use it to construct the basis, not to define the evolution equations [cf. equation (11)]. It would be interesting to explore that. 2. In section 2.2 there's a mix of Finite Element analysis with PDEs that's not always clear. For example, if \varphi_j is defined in the FEM space and equation (9) is solved using FEMs then there's going to be an error, i.e. \phi_j will not solve equation (9) *exactly*. This is really a minor issue, but you might clarify if you think it's worthwhile. 3. The sentence on lines 139-142 suggests that using a different inner product re-orders the modes in the POD computation, but that's not really correct. It changes the whole structure of the eigenmodes, not just their order. Perhaps this could be clarified. 4. Equation (18) is simply not right - presumably a typo. The Lagrangian norm would be [||\omega||^2 + \alpha ||\lambda||^2]^{1/2}. 5. It's not clear to me that the largest FTLE is always positive "by definition", as claimed on line 182. The FTLE is the natural log of something positive, but "by definition" that allows the FTLE to be negative. This needs clarification. 6. On line 212 you say "We do not necessarily add new information..." I disagree. A standard ROM doesn't use the FTLE, whereas your new ROMs do; so it seems like you're adding information. 7. Line 248 "the the" 8. In section 3, I'm not sure if you ever list the time window T that you use to compute the FTLEs - maybe I missed it. 9. On line 259 "a spectral method" is not a lot of information. Is it based on Chebyshev polynomials or Fourier modes? Does it use collocation or Galerkin projection? Maybe one more sentence here would help. 10. Doing tests on the same time window as the training data and then on a longer time window is kind of redundant here. In both cases you're computing the same thing: the time mean. There's not really any reason to suspect that the results would be any different on a longer time window. A more relevant test would have been to change the parameters Ro and/or Re. I'm not saying this needs to be done, but it would have made a better paper.

Author Response

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Author Response File: Author Response.pdf