Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
Round 1
Reviewer 1 Report
I think the paper is interesting and it deserves to be published. The proofs are detailed and mostly clear. There are, however, several ways to improve this work, and so I would ask the authors to review their manuscript before publication. My specific comments are the following:
line 16: for historical reasons, it would be nice if the authors include some references of the classical works by Tikhonov, Lavrent'ev, etc.
lines 23-26: the authors speak about elliptic equations and then about hyperbolic equations. It is not clear in line 25 ("The question of the existence...") to which equations they refer therein.
line 33: "This problem concerns ill-posed problems". It's a bit weird. What problem do they refer to?
line 38: "image of the operator"... which operator? The authors could write down a model equation so that they can refer precisely to the terms appearing in it.
line 39: there is an obvious English mistake.
line 50: the authors speak about a function U and a set G that have not appeared before. See comment on line 38 above...
lines 51-55: the authors explain what they do in their manuscript. Their explanation is very clear, although for a moment I thought that $\partial U(x,f_\delta) / \partial x_j$ did not need to be a derivative of $U(x,f_\delta)$ (I thought the authors may be speaking about a "formal solution" in the sense of Gromov). The proof of Theorem 1 (line 139) shows that $\partial U(x,f_\delta) / \partial x_j$ really is a derivative of $U(x,f_\delta)$, and therefore I would recommend a slight rewriting of line 51.
line 70: what is n(x,y)?
lines 75, 77, 84: there are some obvious English mistakes.
equations (2.3), (2.4): what is $H_0^{(1)}$ ? what is B(u,p)?
line 120: where does the parameter sigma come from in g_\sigma? I think it's important to explain this.
line 123: Theorem 1 is called Theorem 3.1 in line 151. There is also an English mistake .
line 152: what is $\psi(x_1)$ ?
lines 156 and 157: the authors speak about a curve S and about a surface S
line 186: I don't think I understand this line. $f_\delta(y)$ is constructed or assumed to exist? It seems to me that it didn't appear before. I think this question is answered on lines 199-202.
Author Response
Response to Reviewer 1
We would like to thank you for the timely and insightful comments and suggestions. Wehave made improvements to the manuscript and feel that these have increased the qualityof our manuscript. The changes are detailed below.
line 16: for historical reasons, it would be nice if the authors include some references of the classical works by Tikhonov, Lavrent'ev, etc.
Author’s response: In Line 16 some reference are cited.
lines 23-26: the authors speak about elliptic equations and then about hyperbolic equations. It is not clear in line 25 ("The question of the existence...") to which equations they refer therein.
Author’s response: Lines 23-26 are revised and corrected.
line 33: "This problem concerns ill-posed problems". It's a bit weird. What problem do they refer to?
Author’s response: We changed it to "The Cauchy problem for matrix factorizations of the Helmholtz equation is among ill-posed and unstable problems."
line 38: "image of the operator"... which operator? The authors could write down a model equation so that they can refer precisely to the terms appearing in it.
Author’s response: It is done.
line 39: there is an obvious English mistake.
Author’s response: We checked whole paper and changed many cases.
line 50: the authors speak about a function U and a set G that have not appeared before. See comment on line 38 above...
Author’s response: Here we are briefly talking about the meaning of the work.
lines 51-55: the authors explain what they do in their manuscript. Their explanation is very clear, although for a moment I thought that $\partial U(x,f_\delta) / \partial x_j$ did not need to be a derivative of $U(x,f_\delta)$ (I thought the authors may be speaking about a "formal solution" in the sense of Gromov). The proof of Theorem 1 (line 139) shows that $\partial U(x,f_\delta) / \partial x_j$ really is a derivative of $U(x,f_\delta)$, and therefore I would recommend a slight rewriting of line 51.
Author’s response: Everything is written here clearly.
line 70: what is n(x,y)?
Author’s response: Here n(x,y) is removed, i.e. this is a technical error. Corrections introduced)
lines 75, 77, 84: there are some obvious English mistakes.
Author’s response: They are corrected.
equations (2.3), (2.4): what is $H_0^{(1)}$ ?
Author’s response:$H_0^{(1)}$ is the fundamental solution of the Helmholtz equation in $ R^{2}$.
what is B(u,p)?
Author’s response: B(u,p) is the limited function.
line 120: where does the parameter sigma come from in g_\sigma? I think it's important to explain this.
Author’s response: The function $\Phi $ depends on \sigma. see (7) - (8).
line 123: Theorem 1 is called Theorem 3.1 in line 151.There is also an English mistake.
Author’s response: It is done.
line 152: what is $\psi(x_1)$ ?
Author’s response: $\psi(x_1)$ shows the vector- function.
lines 156 and 157: the authors speak about a curve S and about a surface S.
Author’s response: The word surface S is replaced by the curve S.
line 186: I don't think I understand this line. $f_\delta(y)$ is constructed or assumed to exist? It seems to me that it didn't appear before. I think this question is answered on lines 199-202.
Author’s response: Regarding $f_\delta(y)$ , this is the given approximate value of the function U(y) on S.
In the article, the numbering of the formulas has been changed. for example (1.1) on (1), (2.1) on (2) and, ...,
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Nevertheless, some drawbacks are still not be resolved:
(1) The authors have to compare their results with the other literature and explain them in detail.
(2) The authors have to interpret the mentioned literature in detail and add more related literature in Introduction.
(3) The content of this manuscript was not enough to be published in this high quality journal.
(4) The reviewer interests in the error analysis of the proposed algorithm in the manuscript.
(5) The originality of this manuscript was not enough.
(6) The reviewer concerns about the stability of the proposed approach when the input data are polluted by random noise for different issues. Nevertheless, the results do not demonstrate in the manuscript.
(7) The authors should interpret how to employ their method to the real applications in the manuscript.
Author Response
Thank you very much for your reviewing. We revised the paper and answer your concerns. Please check the attachment for reply and the latest version of paper.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
This version is much better than the first one, but I still have some comments. Please make the following changes:
- Line 23. I think the remark "There are some classes of the well-posed equations which are called the hyperbolic equations." is not really necessary. Consider deleting it.
- Line 53: what is G? you only explain what it is on line 103, so consider a wording of the type "$x \in G$, in which $G$ is a subset of R^2 to be specified momentarily", j=1,2, etcetera
- Line 83: I think you mean "In [42], the continuation problem of the Helmholtz equation was ..."
- 4. Line 149: What is \psi(x_1) ? You said (to me, not in the paper) that it's a vector function, but then on line 153 \psi indicates a curve, so please be explicit.
Author Response
Response to Reviewer 1(Round 2)
We would like to thank you for the timely and insightful comments and suggestions. We have made improvements to the manuscript and feel that these have increased the quality of our manuscript. The changes are detailed below.
This version is much better than the first one, but I still have some comments. Please make the following changes:
- Line 23. I think the remark "There are some classes of the well-posed equations which arecalled the hyperbolic equations." is not really necessary. Consider deleting it.
Author’s response:The inscription "There are some classes of the well-posed equations which arecalled the hyperbolic equations." was removed from the article at the request of the reviewer.
- Line 53: what is G? you only explain what it is on line 103, so consider a wording of the type "$x \in G$, in which $G$ is a subset of R^2 to be specified momentarily", j=1,2, etcetera
Author's response: Here, so that there is no misunderstanding in the article. We have translated this part at the end of the article, in section 4. Conclusions. Please see lines 189-198.
- Line 83: I think you mean "In [42], the continuation problem of the Helmholtz equation was ..."
Author’s response:It is done. We changed “continuity” to “continuation”.
- Line 149: What is \psi(x_1) ? You said (to me, not in the paper) that it's a vector function, but then on line 153 \psi indicates a curve, so please be explicit.
Author’s response: \ psi (x_1) is a curve. In Line 139, we have described about.
We will be thankful if you accept our revisions and recommend the paper for publication.
Author Response File: Author Response.pdf
Reviewer 2 Report
The reviewer does not agree the replies of authors.
Author Response
Response to Reviewer 2
We would like to thank you for the timely and insightful comments and suggestions. We have made improvements to the manuscript and feel that these have increased the quality of our manuscript. The changes are detailed below. (1) The authors have to compare their results with the other literature and explain them in detail. Author’s response: We agree with you! But as you know in the papers based on pure mathematics we do not add the numerical considerations. Thus we have focused only on theories and lemmas. We will be so thankful if you accept our study in the present form. (2) The authors have to interpret the mentioned literature in detail and add more related literature in Introduction. Author’s response: Thank you for your comment. It is done. (3) The content of this manuscript was not enough to be published in this high quality journal. Author’s response: Thank you for your comment. Some modifications are done. (4) The reviewer interests in the error analysis of the proposed algorithm in the manuscript. Author’s response: Working on the stability and error analysis is the topic of our next paper. We will submit it to one of journals as soon as possible. (5) The originality of this manuscript was not enough. Author’s response: Some modifications are done! (6) The reviewer concerns about the stability of the proposed approach when the input data are polluted by random noise for different issues. Nevertheless, the results do not demonstrate in the manuscript.
Author’s response: We agree with you, but the regularization method can help
us for this concern. As you know, for solving the problem we do not apply the
input data. So we think we will not have problem about it.
(7) The authors should interpret how to employ their method to the real
applications in the manuscript.
Author’s response: About numerical illustrations, we described before. This
paper is based on pure mathematics and we do not want to focus on the
numerical considerations.
Author Response File: Author Response.pdf
