DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems
Round 1
Reviewer 1 Report
In this paper, authors have proposed a data-informed active subspace (DIAS) regularization framework that aims to improve fidelity of Tikhonov inverse solutions. Overall paper is well written but it requires some minor improvements that are mentioned below.
1) Designate a separate section for related work and novelties of the proposed method.
2) After related work, please make a comparison table that should mention the strengths and weaknesses of the previous method as well as proposed method.
3) On page number 10, definition of the reconstruction operator is missing.
Author Response
Dear Reviewer,
We are thankful for your thorough reviews with critical comments and suggestions that help improve the manuscript substantially.
Yours sincerely,
Hai Nguyen, Jonathan Wittmer, and Tan Bui-Thanh
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In this work, the authors combined two pieces of their previous work, the active subspace method and the data informed regularization idea, to develop the DIAS framework. I am not sure whether it is the manuscript or some technical reasons, the most important theoretical part, Theorem 1, in incomplete. A large empty space presents under Eq. (16) and the definition of reconstruction operator is missing.
Other than this issue. Mathematical illustration of some definitions needs to be improved. For example, all the norms of matrices, vectors need to be explicitly defined. Also, the definition of noise percentage needs to be explained. For benchmark problems, model setup and explaination need to be provided for at least one problem, instead of referring to literatures.
The essential idea of either DI or AS is truncated SVD, so the innovation of this work is limited. Further, the inverse problem is less challenging. It is linear, convex, low-noise and limited to L2 regularization. By truncated SVD of the forward operator, the problem can be easily solved. The authors may want to test their tools on more challenging problems to convince reviewers.
Author Response
Dear Reviewer,
We are thankful for your thorough reviews with critical comments and suggestions that help improve the manuscript substantially. You also can see the changes based on other reviewers in the attached file.
Yours sincerely,
Hai Nguyen, Jonathan Wittmer, and Tan Bui-Thanh
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper illustrates a procedure to improve the ability of the Tikhonov regularization to manage ill-posedness in inverse problems.
The topic is very relevant and the proposed approach appears very interesting, but in the reviewer opinion, to make the paper suitable for publication, the Authors should deepen the critical discussion about the validation of the proposed approach, which is based on the results obtained on some benchmark problems extracted from the Regularization Tools v. 4.0 for Matlab 7.3 (§ 5.2), supplemented by another relevant case, X-ray tomography. More precisely, the following remarks should be addressed:
- Since benchmark problems provided in Regularization Tools for Matlab are generally considered too simplistic and not fully representative of modern problems, their relevance should be better discussed and justified, underlining how the main findings of the study apply or can be extended to more advanced problems;
- In the examples, both for benchmark problems and X-ray tomography, the Authors “corrupt” the observational data with a 1% additive white noise, but discussion about the sensitivity of the results on white noise percentage is lacking: in the reviewer opinion, even considering just one or two paradigmatic cases, a sensitivity analysis could be very helpful, for example considering also white noise percentages of 0.5% or 2%.
Author Response
Dear Reviewer,
We are thankful for your thorough reviews with critical comments and suggestions that help improve the manuscript substantially.
Yours sincerely,
Hai Nguyen, Jonathan Wittmer, and Tan Bui-Thanh
Author Response File: Author Response.pdf
Reviewer 4 Report
In this paper, the authors considered the data-informed active subspace (DIAS) regularization strategy. Various numerical results for linear inverse problems are presented. The main results are interesting. Therefore, I recommend it accepted if the authors can make some revisions as follows.
1) Please check the Theorem 1 and supplement the missing information.
Author Response
Dear Reviewer,
We are thankful for your thorough reviews with critical comments and suggestions that help improve the manuscript substantially.
Yours sincerely,
Hai Nguyen, Jonathan Wittmer, and Tan Bui-Thanh
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
The authors successfully and sufficiently addressed all reviewer's concerns. The quality is obviously improved so it can be accepted as it for publication.
Reviewer 3 Report
The Authors addressed the comments, improving the quality of the manuscript.
