Efficiency Investigation of Langevin Monte Carlo Ray Tracing

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors provide a detailed analysis of the computational costs associated with Langevin Monte Carlo ray tracing. Notably, the rate of convergence and the impact of the preconditioning matrix are thoroughly discussed, with the preconditioning matrix contributing most significantly to the improvement in convergence. In my opinion, the presented results are both novel and interesting. However, before further consideration for publication in Mathematics, I have a few suggestions that could enhance the overall quality of the manuscript.

1. The labels of Figure should be enlarge. They currently look very small.

2. The derivation of the determinant that appears right above Eq. (20)  should be shown.

3. More details on the Langevin equation would be welcome.

 

Author Response

Our response is in attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Review for the manuscript

Efficiency Investigation of Langevin Monte Carlo Ray Tracing 

 ([Mathematics] Manuscript ID: mathematics-3274476).

 

 

Content. Inthis paper, the authors studied the Langevin Monte Carlo ray tracing, generates samples using time series of a system of the Langevin dynamics. The main conclusions is that the computationally expensive drift term can be dropped because it does not improve convergence. Another important conclusion is that the preconditioning matrix makes the greatest contribution to improvement of convergence. At the same time, calculation of this matrix is not so expensive because it does not require calculating the gradient of the potential. The results of our study allow to significantly speed up the method.

 

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Comment. The questions addressed in this manuscript are of interest to various mathematics. The manuscript is well written, which makes it comfortable to read and check. The questions addressed in this manuscript are of interest to various mathematical communities.   

 

Recommendation. Due to the relevance of the subject, the manuscript needs minor revision. The manuscript cannot be accepted for publication as it stands now. I would like to ask the authors to address the questions and comments below and to improve their presentation.

 

 

1. I would like to see the programming/codes of the simulation. As nowadays the research is

supported by associated programmes/software codes for reproducibility. This will have a wider

appeal to audience and practitioners.

 

2. Please also provide details on the machine specification and programming language/software

used. Please let us know which software was used to create the figures in the article, and provide

the name of the software

 

 

 

Author Response

Our response is in attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper discusses an efficient method  for lighting simulation with applications in computer graphics.   Numerical experiments are conducted to evaluate the performance of the algorithm and to gain computational insights  that can be  useful to indicate possible improvements.  The paper does not present any  novel findings  both  at problem setting and theoretical results.  Its main contribution relies upon the experimental analysis of the considered algorithm.  I think it can be interestingly enough to motivate its publication.

However, I would ask the authors to  revise the paper by addressing the following issues: 

1) The construction of the preconditioning matrix T  should be made clear. Also its relationship with the matrix Q should be discussed more explicitly. 

2) The computational complexity of the construction of the preconditioning matrix T should be  analyzed. 

 

Author Response

Our response is in attached file.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

 

In my opinion this paper is very interesting and deserve to be published in mathematics. 

However, i just have a question :  The method used is interesting but how can we get around the complexity of the calculation ? 

 

 

Author Response

Our response is in attached file.

Author Response File: Author Response.pdf