Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity
Round 1
Reviewer 1 Report
I find the paper quite inaccessible.
Comments for author File: 
Comments.pdf
Author Response
Answers to reviewer 1
Our answers to your comments and the corresponding modifications in the article are in red.
This paper is not written for the purpose of being understandable by others than the few people who already know what this is about. After working on this review for quite some time, I do understand the idea of figure 1, but I still don’t know how this is obtained.
è The Figure does not aim to explain how the tensor train decomposition is performed. It solely explained how a given tensor train is evaluated. We have changed the figure and modified the caption for better understanding.
I didn’t understand what χ is until I got to the example. The readability would improve if you would show the example first and connect the symbols to those in the actual example. For example, you specify a set which is specific to each . Later on it turns out, that this is connected to the number of components in the χth QOI.
è Section 2 has been modified at line 73 to 82.
The definitions of and of j* and j** are quite puzzling. My current understanding is that is one QOI component at one time instant for one parameter setting. Then how can it be thathas columns?
è The paragraph explaining matricization has been rewritten with additional details. Your understanding of is correct, it denotes a single element of the tensor . Since is the first matricization of the tensor , it has indeed rows and columns (precision added in the new version of the manuscript).
I think that at “The Frobenius norm …” a new paragraph should start. I has nothing to do with definition of . Anyways, I don’t think it is used anywhere in the rest of the paper.
è A paragraph break has been added as rightfully requested (line 89). The Frobenius norm of a tensor is used in Proposition 1, equation (16).
In 87-108: Is there a connection between and A or is A just any matrix? Or is it may be defined in (7)?
è denotes an arbitrary matrix. But in the context of the present work, especially Algorithm 1, will always be some matrix as defined in equation (7) for the special case k=1, and equation (13) for other values of k. A remark has been added in the paragraph from line 94 to 102 in order to clarify that point.
line 112: “the column sampling is restricted to indices iq+1, . . . id-1 and is replicated for all values of id in .” Does that mean, that for each QOI a different sampling is performed, even though one integration of your DAE yields values for all QOI’s?
è The previous wording was confusing to say the least. It has been changed (line 116), hopefully for the better.
Appendix A: The final objective is to be able to estimate parameters when experimental data are available. The experiments that you describe in relation to the model you use to describe the results of the experiments, will give you only one observable quantity: σ11.
è The final objective of our proposed approach is not "to be able to estimate parameters when experimental data are available" but rather to be able to construct a surrogate model enabling to predict the quantity of interest of complex models. In order to illustrate the relevance of our approach, we have chosen this application for the following reasons 1) We believe that using a well-known mechanical model (with existing analytical solution in our case) can help the reader not to concentrate on details unlinked to our approach 2) This model has many parameters and feature highly nonlinear behaviors which makes it difficult to construct an accurate surrogate model of.
Dear authors, from my questions it may be clear, that I didn’t understand much from your paper. I think that should worry you.
Author Response File: Author Response.pdf
Reviewer 2 Report
The work proposes an improved version of the tensor train approximation for parametric estimation considering a nonlinear elasto-viscoplastic constitutive law as a case study.
The manuscript is very well written, complete and rigorous and is of interest for the audience of MDPI. Accordingly, I recommend the manuscript for publication listing below minor amendments.
In the introduction, when motivating the importance of the study, Authors should discuss the wide literature and audience dealing with biological tissues and biomechanics in particular. Here, recent contributions have tried to set a state-of-the-art in constitutive modeling and parameters identifications from both a theoretical and computational point of view (see e.g. Gilchrist et al. 2017 https://doi.org/10.1007/s11012-017-0646-9, Saccomandi & Vergori 2018, Marino 2018 https://doi.org/10.1007/s10237-018-1009-8 and DOI: 10.1016/B978-0-12-801238-3.99926-4).
Again in the introduction, I would suggest the Authors discuss in terms of “nonlinearity” other than “complexity”.
Is the tensor-train related to the mechanical multiplicative decomposition (see Spencer)? Since the Authors take a well-established mechanical model it could be interesting for the readers setting such a link.
Please, reshape Figure 1. It is not understandable in its present form.
Please, in the discussion section, summarise the most important contribution of the work.
What happens if the truncation tolerance is lowered?
What happens if parameters are randomly selected from a different distribution other than “uniform law”?
“Effectivity” needs to be better introduced.
In Tab. A1 it is not clear the physical dimension s^-n
What about if a different applied strain is considered?
When entering \linenumbers with the equation is necessary to set \linenomath before and after in order to avoid numbering breaking.
Author Response
Answers to reviewer 2
Our answers to your comments and the corresponding modifications in the article are in blue.
The work proposes an improved version of the tensor train approximation for parametric estimation considering a nonlinear elasto-viscoplastic constitutive law as a case study.
The manuscript is very well written, complete and rigorous and is of interest for the audience of MDPI. Accordingly, I recommend the manuscript for publication listing below minor amendments.
In the introduction, when motivating the importance of the study, Authors should discuss the wide literature and audience dealing with biological tissues and biomechanics in particular. Here, recent contributions have tried to set a state-of-the-art in constitutive modeling and parameters identifications from both a theoretical and computational point of view (see e.g. Gilchrist et al. 2017 https://doi.org/10.1007/s11012-017-0646-9, Saccomandi & Vergori 2018, Marino 2018 https://doi.org/10.1007/s10237-018-1009-8 and DOI: 10.1016/B978-0-12-801238-3.99926-4).
è We understand the suggestion you make about discussing the wide literature in the field of biological tissues and biomechanics. However, those are fields that we are not acquainted with and we believe that it might mislead the reader on the core purpose of this article. Finally, the concern of surrogate modeling is actually addressed in many fields and our intent is not to try to be exhaustive about this matter but rather introduce one working method in the field of mechanical science.
Again in the introduction, I would suggest the Authors discuss in terms of “nonlinearity” other than “complexity”.
è We have modified the introduction as suggested.
Is the tensor-train related to the mechanical multiplicative decomposition (see Spencer)? Since the Authors take a well-established mechanical model it could be interesting for the readers setting such a link.
è The multiplicative decomposition in mechanics aims at splitting the deformation gradient into an elastic part and a plastic part. This is not the purpose of the tensor-train decomposition. No tensor carriage can be considered as either solely elastic or solely plastic.
Please, reshape Figure 1. It is not understandable in its present form.
è We changed the figure and modified the caption for better understanding.
Please, in the discussion section, summarise the most important contribution of the work.
è We added a summary of the most important contribution at the beginning of the section.
What happens if the truncation tolerance is lowered?
è As explained in line 217, the lower the tolerance the smaller the approximation errors.
What happens if parameters are randomly selected from a different distribution other than “uniform law”?
è The random selection of entries during the training phase remains an open issue. It deserves more research.
“Effectivity” needs to be better introduced.
è We modified (above line 230) the way the notion of effectivity is introduced.
In Tab. A1 it is not clear the physical dimension s^-n
è We added a remark in the corresponding caption.
What about if a different applied strain is considered?
è For the sake of simplicity, the loading condition was not treated as an input parameter in the article. It is not an intrinsic limitation of the proposed methodology. On the other hand, we must acknowledge that the ability of this method to consider a very large number of parameters (20 or more) was not considered in the present work.
When entering \linenumbers with the equation is necessary to set \linenomath before and after in order to avoid numbering breaking.
è We thank you for the tip. The line numbering has been fixed accordingly.
è We thank you for your valuable comments and the time dedicated to reviewing the article.
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper presents a fruitful approach to overcoming the curse of dimensiality in solving tensor equations, namely, TT approximation, and its applicaiton to simulation of solids. The main ideas of the paper are clearly outlined and careful mathematical descriptions are provided.
Nevertheless, I can address some comments.
In the line 3, I would recommend to place the parenthesis "for every output..." in the end of the sentence to improve the style.
In the line 198, the empty square leaves an impression of an error, but not the mathematical symbol, so please introduce another notation.
Due to high abstraction in the mathematical notation, it would be difficult for some readers to reproduce the presented results, so the link to the source code or any appendix with actual code implementation of the proposed algorithm would amplify the readers' interest to the paper.
In spite of these comments, the theoretical part is correct and detailed and the experimental section provides demonstrative data. I enjoyed reviewing this paper and can recommend it to be published after a minor revision.
Author Response
Answers to reviewer 3
Our answers to your comments and the corresponding modifications in the article are in green.
The paper presents a fruitful approach to overcoming the curse of dimensiality in solving tensor equations, namely, TT approximation, and its applicaiton to simulation of solids. The main ideas of the paper are clearly outlined and careful mathematical descriptions are provided.
Nevertheless, I can address some comments.
In the line 3, I would recommend to place the parenthesis "for every output..." in the end of the sentence to improve the style.
è We have modified the introduction as rightfully suggested.
In the line 198, the empty square leaves an impression of an error, but not the mathematical symbol, so please introduce another notation.
è The square symbol was indeed confusing. We changed it to the letter Z (line 201).
Due to high abstraction in the mathematical notation, it would be difficult for some readers to reproduce the presented results, so the link to the source code or any appendix with actual code implementation of the proposed algorithm would amplify the readers' interest to the paper.
è Unfortunately, for copyright reasons, we are not able to share the actual code implementation of the proposed algorithm. We are currently looking for solutions about that matter.
In spite of these comments, the theoretical part is correct and detailed and the experimental section provides demonstrative data. I enjoyed reviewing this paper and can recommend it to be published
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
I read the manuscript and find it fit for publication. I only have one small comment.
line 80: For outputs p you define tensor A1, where as in line 216 p is apparently A4.
