Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsAs far as I could check there are no mistakes, results are interesting and meet high standards. Authors in Theorem 3 can perhaps use the following known results:
$$S_n(p)~=~1^p+2^p+3^p+\cdots+n^p$$
If $p$ is not an integer, then for $S_n(p)$ only bounds are available, for example,
$$n^{p+1}~<~(p+1)S_n(p)~<~(n+1)^{p+1}-1,~~~0<p<1$$
$$n^{p+1}~<~(p+1)S_n(p)~<~(n+1)^pn,~~~p\geq 1$$
$$(n+1)^{p+1}-n^{p+1}~<~(p+1)[S_n(p)-S_{n-1}(p)]~<~n^{p+1}-(n-1)^{p+1},~~~-1<p<0$$
$$(p+1)[S_n(p)-1]~<~n^{p+1}-1<(p+1)S_{n-1}(p),~~~p<-1.$$
Given in
R.P. Agarwal, Difference Equations and Inequalities: Second Edition, Revised and Expended, Marcel Dekker, New York, 2000.
S.L. Guo and Feng Qi, Recursion Formulae for $\sum_{m=1}^n m^k,$, Journal for Analysis and its Applications, 18(1999), 1123-1130.
J.C. Kuang, Applied Inequalities, 2nd ed. (in Chinese), Changsha: Hunan Education Press 1993.
On page 5, what is k (do we have some estimates).
Author Response
We thank the anonymous Reviewer for the comments and suggestions, which helped us to improve the presentation quality of our work.
Below we give detailed answers to the raised questions.
General Comment: As far as I could check there are no mistakes, results are interesting and meet high standards
Answer: We thank the Reviewer for the appreciation of our results.
Query: Authors in Theorem 3 can perhaps use the following known results:
$$S_n(p)~=~1^p+2^p+3^p+\cdots+n^p$$
If $p$ is not an integer, then for $S_n(p)$ only bounds are available, for example,
$$n^{p+1}~<~(p+1)S_n(p)~<~(n+1)^{p+1}-1,~~~0<p<1$$
$$n^{p+1}~<~(p+1)S_n(p)~<~(n+1)^pn,~~~p\geq 1$$
$$(n+1)^{p+1}-n^{p+1}~<~(p+1)[S_n(p)-S_{n-1}(p)]~<~n^{p+1}-(n-1)^{p+1},~~~-1<p<0$$
$$(p+1)[S_n(p)-1]~<~n^{p+1}-1<(p+1)S_{n-1}(p),~~~p<-1.$$
Given in
R.P. Agarwal, Difference Equations and Inequalities: Second Edition, Revised and Expended, Marcel Dekker, New York, 2000.
S.L. Guo and Feng Qi, Recursion Formulae for $\sum_{m=1}^n m^k,$, Journal for Analysis and its Applications, 18(1999), 1123-1130.
J.C. Kuang, Applied Inequalities, 2nd ed. (in Chinese), Changsha: Hunan Education Press 1993.
Answer: We preferred to maintain our computations in order to be self-contained and because we do not need precise constants but only asymptotic relationships. However, after the end of the proof, we added two lines for explaining that there exist more sophisticate estimates by referring to the indicated paper and to the indicated books.
Query: On page 5, what is k (do we have some estimates).
Answer: The Reviewer is right. We added an estimate of k which is indeeed a constant indipendent of n.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors present a fractional Helmholtz equation approximated by ad-hoc centered differences with variable wave number µ(x, y), in the specific case where the complex-valued function µ(x, y) has a pole of order γ. Eigenvalue distribution and clustering results have been derived. The numerical results presented in the paper corroborate well the presented analysis.
A small comment to the authors: in the introduction section of the paper they could better highlight what has been already done in the field and what is their contribution to the field of knowledge.
Author Response
We thank the anonymous Reviewer for the comments and suggestions, which helped us to improve the presentation quality of our work.
Below we give detailed answers to the raised questions.
General Comment: The authors present a fractional Helmholtz equation approximated by ad-hoc centered differences with variable wave number µ(x, y), in the specific case where the complex-valued function µ(x, y) has a pole of order γ. Eigenvalue distribution and clustering results have been derived. The numerical results presented in the paper corroborate well the presented analysis.
Answer: We thank the Reviewer for the positive evaluation of our work.
Query: A small comment to the authors: in the introduction section of the paper they could better highlight what has been already done in the field and what is their contribution to the field of knowledge.
Answer: We followed the suggestion by adding a short discussion on the matter at the end of the introduction section, with an ad hoc comparison with the previous relevant literature.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe manuscript is well done and can be accepted almost as it is. I appreciate
that it has a theoretical and a numerical part. The bibliography is well
formatted without missing fields, however more than half of the references
being self-citations is too much: please reduce self-citations to the bare
necessary minimum. The equations are written professionally; only the text in
between needs minor attentions, e.g.
"We consider Helmholtz equation" -> "We consider the Helmholtz equation"
"described by the equations below" -> "described by the equations"
"the case of a bounded μ(x,y) being studied in [1,18]." -> "the case of a bounded μ(x,y) has been studied by Adriani et al. [1] and Serra-Capizzano [18]." (when references are endnotes, they cannot be a part of the sentence)
"For approximating (1)" -> "To approximate Eq. (1)"
An equation, e.g. the unnumbered equation before Eq. (18), cannot form a sentence of their own. All equations should be numbered, like all pages, sections, theorems, figures, tables and references.
"has the following diagonalization form" -> "has the diagonalization form"
In tables, numbers should be right-aligned and vertical lines should be avoided unless necessary to structure large and complex tables.
This list is not comprehensive. Please revisit carefully your text
extrapolating from the above examples.
Comments on the Quality of English LanguageMinor edits required; a few examples are provided in the main report.
Author Response
We thank the anonymous Reviewer for the comments and suggestions, which helped us to improve the presentation quality of our work.
Below we give detailed answers to the raised questions.
General Comment: The manuscript is well done and can be accepted almost as it is. I appreciate that it has a theoretical and a numerical part. The bibliography is well formatted without missing fields, however more than half of the references being self-citations is too much: please reduce self-citations to the bare necessary minimum. The equations are written professionally; only the text in between needs minor attentions
Answer: We thank the Reviewer for both the technical and scientific appreciation of our results.
Query: The bibliography is well formatted without missing fields, however more than half of the references being self-citations is too much: please reduce self-citations to the bare necessary minimum.
Answer: We appreciate that the Reviewer recognizes the professional style of the references. Regarding self-citations we have eliminated two of them, but we were unable to cut more self-citations. Indeed we emphasize that the topic treated in the present manuscript represents a special situation, since the GLT theory, also in the non-Hermitian setting, started with the work of one of the authors and this necessarily leads to citing the work of the second author.
Query: The equations are written professionally; only the text in between needs minor attentions, e.g.
"We consider Helmholtz equation" -> "We consider the Helmholtz equation"
"described by the equations below" -> "described by the equations"
"For approximating (1)" -> "To approximate Eq. (1)"
"has the following diagonalization form" -> "has the diagonalization form"
This list is not comprehensive. Please revisit carefully your text extrapolating from the above examples.
Answer: We have corrected all the inaccuracies indicated by the Reviewer and we have read the manuscript carefully and other minor things have been modified
Query: "the case of a bounded μ(x,y) being studied in [1,18]." -> "the case of a bounded μ(x,y) has been studied by Adriani et al. [1] and Serra-Capizzano [18]." (when references are endnotes, they cannot be a part of the sentence)
Answer: We followed the indication, but with an obvious change since paper [18] (the numbering refers to the original manuscript) is authored by Li et al.
Query: An equation, e.g. the unnumbered equation before Eq. (18), cannot form a sentence of their own. All equations should be numbered, like all pages, sections, theorems, figures, tables and references.
In tables, numbers should be right-aligned and vertical lines should be avoided unless necessary to structure large and complex tables.
Answer: These are the only items that we did not follow since the Journal has already changed the style of our manuscript before submission. Hence we leave this matter to the professional editing work of the Journal, even if we really appreciated the careful and pertinent suggestions of the Reviewer.
