Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
Round 1
Reviewer 1 Report (Previous Reviewer 1)
The authors have solved a main objection and I recommend the pubication of the manuscript.
Reviewer 2 Report (New Reviewer)
I found the paper a very well organized nicely presented but too concise reading -- at least in view of the short peer reviewing deadlines and the time demand of checking computer experinces with the results.
At first sight (and checking some, but not all, detais) I can agree with the content of the paragraph on page 3 lines 94-103 of the Introduction.
Hence I do not require any change.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
This paper considers an interesting problem and the authors are known specialists in this area of mathematical research. The text is well written and well documented, the English is of good quality (I have found just one mistake, line 116: Let…satisfy…!). Another minor writing problem is that the authors use the same symbol with different meanings (for instance s may be space dimension or regularity order, etc.)
The hypotheses used in this paper are quite strong: the authors assume that (7) can be solved explicitly with respect to the control and, in Thm.1, the regularity conditions are very high (although f is assumed just of class C2). Another unclear point in Thm.1 is that they assume that the approximate solution exists, while this should follow via some argument.
Let me also mention that, in the examples, the cost functional is not of type (1). While this may not be a problem, some comments in this respect are necessary. In general, the authors introduce many conditions during the exposition that are rather implicit and it would be better to be expressed in terms of the data. This is a paper on approximation and it should have a constructive character.
The paper is very much based on the reference [12] of the authors, which it extends, at least partially. Results and arguments from [12] are used very frequently, everywhere in this article. However, [12] is just a preprint of the authors from February 2020 (two years ago). The readers have to check all these arguments from [12] by themselves, since no indication exists on the publication of [12] (at least whether it is submitted somewhere). Moreover, the reference [22] of one of the authors is again a preprint from 2015 with no hints on the publication process.
Let me also add that in January 2022, the authors have written another preprint on ArXiv, on a related subject. I think that the authors should clarify the situation of the old preprints first.
In conclusion, I recommend this work to be rejected.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper discusses novel symplectic multistep numerical integration techniques, evaluating the convergence and performance as an experimental part of the study. The manuscript is well-written and is of interest in the field. Please, find my comments below.
1. "which are all implicit for general Hamiltonian systems" - indeed, formally, they are implicit, but in the case of semi-implicit methods (e.g., CD semi-implicit \ semi-explicit methods), the implicitness is only one-dimensional and can be resolved analytically \ through a simple iterations method. This trick reduces the computational costs of the method almost down to the explicit case. Is this approach suitable for simplifying your methods?
2. Please consider generalized semi-explicit methods suitable for simulating non-Hamiltonian systems, including composition ODE solvers, in the literature review.
3. The experimental part is also to be improved. The traditional test problem for A-stable methods is the VDPL system with high values of parameter m. Despite the Reyleigh equation is a pretty classical problem, too, I recommend adding van der Pol oscillator to a list of the test problems.
3. From the famous work of R.M. Corless, it is known that the A-stable numerical method can suppress chaos in discrete models of chaotic systems. Thus, I recommend choosing a chaotic test problem or driving any of the existing ones into a chaotic regime to check this property.
4. Authors declare that the reference solution is obtained using RK4, which is not enough for studying stiff systems. I recommend using high-order A-alfa stable extrapolation methods or at least DOPRI8 solver for references solution. I understand, that the provided plots are targeted to compare the accuracy of the methods with the traditional precision vs. stepsize approach, but in stiff case, the "reference" plot may be distorted.
5. I recommend expanding the performance study by thorough comparison with competing techniques. First, Please, provide some comparison with single-step schemes. What about explicit multistep methods? Predictor-corrector methods, including semi-implicit cases? Second, the standard tool for studying the performance of integration methods is the performance plot. Please, provide "computation time vs. error" plots for proposed techniques instead or together with error vs. timestep plots.
6. While studying symplectic integrators, long-term simulation tests are required. The theoretically predicted properties of the method can be quite different in a real computer.
7. As far as I see, the coefficients of proposed methods require high-precision data representation. How may roundoff errors and digital noise affect the simulation while using the reported techniques?
8. Which variable stepsize techniques are applicable to your methods? How can the local truncation error be evaluated in these schemes? Is it possible to develop a symplectic step controller for your algorithms?
9. It is of interest to evaluate and compare the stability regions of the proposed methods with competing state-of-the-art techniques.
Minor issues:
Please, check Equation 75 for a possible typo.
Nevertheless, I like the reported study and believe it can be accepted after moderate revisions.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The revised version includes several minor improvements. The reference [13] plays a very important role in this manuscript, even in the proofs, and its status is unclear. I think that even for the journal Comput. Appl. Math. (where [13] waits for approval since 2020) is not appropriate (from the authors side) to publish already an extension/continuation.
I maintain the decision that this manuscript cannot be published before the situation of [13] is completely clarified.
Reviewer 2 Report
Thank you for providing a revised version of the manuscript and a fruitful discussion. It was of great interest to discuss the applicability of the semi-implicit methods. I highly appreciate adding the VDPL problem to the experimental study. It should be mentioned that in my previous review, I suggested considering semi-implicit Euler-type methods, not linearly implicit ones. There is a slight difference because Euler-Cromer integration can possibly be a nonlinear integration operator (this fact needs some theoretical proof, however).
I also recommend using traditional performance plots in further research because order plots do not allow comparing the performance of competing solvers.
Nevertheless, my overall impression is very high, and I will recommend the paper to be accepted in the present form.
