Quadrature Integration Techniques for Random Hyperbolic PDE Problems
Round 1
Reviewer 1 Report
1. The forward and inverse Laplace transforms (4) and (5) are classical.
What new features are included by stating that these transforms are random?
2. Boundary conditions should be defined for equation (11).
3. It is well-known that numerical computation of the inverse Laplace transform is a challenging task, since the inverse Laplace transform is ill-posed problem. Thus all algorithms should include some regularization property. This point is not explained and analyzed in the paper.
4. Please explain what non-standard techniques are proposed to define the MC
algorithm. It seems that a classical version of this method is used straightforwardly.
5. EF-GL algorithm is not defined at all in the text.
6. The presented 1D deterministic examples are too simple toy problems
in order to estimate a real value of the presented algorithms.
7. Please explain the results in Table 2: CPU time depends on the number of nodes N_{GL} in a very strange form: a) compare N=5 and N=10 for GL algorithm, b) the complexity of CPU time for EF-GL depends exponentially on N.
8. It should be explained how a mesh with $N_x = 10, N_t = 5$ was selected. I recommend to present some basic convergence analysis results for the selected
finite difference scheme. At least, experimental convergence rates should
be given. It is important to compare CPU times of MQ, TI and GL algorithms
with CPU times for the finite difference scheme (explicit and implicit cases).
Author Response
We answer to the referee’s comments. We use italic font for the comments and queries by the referee and Times New Roman for our reply.
- The forward and inverse Laplace transforms (4) and (5) are classical. What new features are included by stating that these transforms are random?
The new features are related to the fact that Laplace transform is not applied to a function but to a stochastic process and although we have used it before in order to make the manuscript readable and understandable, some minimal information is needed.
- Boundary conditions should be defined for equation (11).
Transformed boundary conditions and corresponding explanation have been included.
- It is well-known that numerical computation of the inverse Laplace transform is a challenging task, since the inverse Laplace transform is ill-posed problem. Thus all algorithms should include some regularization property. This point is not explained and analyzed in the paper.
The reviewer is right. Apart from the fact that we use result of [9] transforming the inverse Laplace transform into a Fourier integral, we analyse numerical results to select the most appropriate, see Table 1. One of the methods considered in the manuscript is Talbot inverse algorithm that is found to have a regularization property, and, as expected, performs better.
- Please explain what non-standard techniques are proposed to define the MC algorithm. It seems that a classical version of this method is used straightforwardly.
For us MC is just a tool not an innovative technique and always the results are checked. Even more, we use MC as less as possible and only to be able to compute the expectation and the variance of the approximate stochastic process solution, as it has been stated in the introduction.
- EF-GL algorithm is not defined at all in the text.
The reviewer is right, we have now presented the EF-GL method in subsection 3.3 with corresponding reference.
- The presented 1D deterministic examples are too simple toy problems in order to estimate a real value of the presented algorithms.
If reviewer pays attention to all examples of section 5, s(he) would check that there are 4 examples with increasing difficulty. What the reviewer calls “toy examples” are included because in such cases the exact solution is known and numerical results can be checked. Please, check that examples 2-4 are not “toy”.
- Please explain the results in Table 2: CPU time depends on the number of nodes N_{GL} in a very strange form: a) compare N=5 and N=10 for GL algorithm, b) the complexity of CPU time for EF-GL depends exponentially on N.
Firstly, the results of Table 2 have been corrected. We have also included comments explaining the results and also why the EF-GL is discarded in the following experiments.
- It should be explained how a mesh with $N_x = 10, N_t = 5$ was selected. I recommend to present some basic convergence analysis results for the selected finite difference scheme. At least, experimental convergence rates should be given. It is important to compare CPU times of MQ, TI and GL algorithms with CPU times for the finite difference scheme (explicit and implicit cases).
In order to visualize the shape of the numerical solution by the Algorithm 1 we choose a mesh with Nx =10 nodes of the spatial domain as well as five values of the time choosing Nt =5. In the manuscript, FDM in refined grid are used to get the reference solution, the step sizes are chosen to guarantee the stability of the scheme. The study of convergency of the FDM is not the objective of the paper, since it is focused on the numerical integration techniques. In this respect, the numerical convergence of the MQ and GL are studied, see section 5.1. Moreover, we extend the numerical convergence study of the MQ with respect to the chosen step size, results have been reported in new table 3. The CPU time of all considered methods, including the FDM for the PDE, has been reported.
Reviewer 2 Report
The work “Quadrature Integration Techniques for Random Hyperbolic PDE Problems” deals with the numerical solution of the random hyperbolic partial differential equations. The manuscript is written in a professional language and is well structured. The text contains the results of a computational experiment, illustrated with graphs and tables. The authors consider the application of classical numerical methods, including the midpoint rule, the Gauss-Laguerre quadrature, and the Talbot inverse method. The conclusions are well-founded and convincing enough.
However, there are the following notes and comments:
- The content of the abstract is quite general and does not reflect the results obtained by the authors.
- The manuscript (subsection 5.2) does not consider the case of noisy data and does not discuss the stability of algorithms to perturbations of the initial data (for example, u(0,t)=g_{\delta}(t), where \delta>0, ||g_0-g_{\delta}||_C<=\delta), which is especially important for solving practical problems (see, for example, [1]).
- The effectiveness of the Talbot inverse method for the deterministic case is commented in comparison with the midpoint rule (see, lines 139, 140). The study [2] compares the Talbot algorithm with other methods. It might make sense to explain in more detail the benefits of the Talbot inverse method.
[1] A.N. Tikhonov and A.Y. Arsenin (1977) Solutions of Ill Posed Problems. John Wiley Publications.
[2] Colin L. Defreitas & Steve J. Kane (2020) The Noise Handling Properties of the Talbot Algorithm for Numerically Inverting the Laplace Transform. https://arxiv.org/pdf/1703.02857.pdf
Author Response
We answer to the referee’s comments. We use italic font for the comments and queries of the referee and Times New Roman for our reply.
The content of the abstract is quite general and does not reflect the results obtained by the authors.
The abstract has been re-written in more detail.
The manuscript (subsection 5.2) does not consider the case of noisy data and does not discuss the stability of algorithms to perturbations of the initial data (for example, u(0,t)=g_{\delta}(t), where \delta>0, ||g_0-g_{\delta}||_C<=\delta), which is especially important for solving practical problems (see, for example, [1]).
The main objective of the manuscript is the study of numerical integration methods for random problems (check subsections 5.3 and 5.4), where the stochastic perturbations in parameters take place. Subsection 5.2 deals with the deterministic problem only as a particular case of more general random model.
The effectiveness of the Talbot inverse method for the deterministic case is commented in comparison with the midpoint rule (see, lines 139, 140). The study [2] compares the Talbot algorithm with other methods. It might make sense to explain in more detail the benefits of the Talbot inverse method.
We appreciate the reviewer’s recommendation. The study by Defreitas and Kane (2020) has enriched our knowledge about the Talbot algorithm. The new subsection 3.4 has been extended by the discussion about the method and its advantages. The reference Defreitas and Kane (2020) has been included.
Round 2
Reviewer 1 Report
- The applied MC method can be equally well used with many other numerical methods developed to solve hyperbolic problems and I see no specific results targeted for the Laplace transform method.
- The presented results can be considered as one more approach how to solve the formulated problem. The new technique is not compared with the known state of the art methods used to solve this class of problems.
- In order to promote this technique as a robust method for solving real world applications, one or two applied examples should be presented with some links in literature indicating that these problems are important and no efficient methods exist till now.
Author Response
In our opinion, the role of reviewer 1 is not professional in the opinion of the revised version, and has no relation with the first opinion and with our answers and our revised version. The second opinion by reviewer 1 has no concrete reference with the revised version. The reviewer role must be professional.
