p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Round 1

Reviewer 1 Report

show the method efficiency.

In total, the paper methodology is interesting and can be employed for different engineering problems.

The reviewer has the following comments to improve the paper quality.

 

1) Abstract, on 2nd paragraph: the authors mentioned that the multilevel methods can speed up the method. The sentence is ambiguous.

The main advantage of multilevel techniques compared to (single-level) MC is a significant computational effort reduction (see, e.g., [8]) by distributing the complexity among the levels. However, the convergence rate is still O(N^{-1/2}) by using random points. Please rewrite this sentence.

 

2) Page 6: $M$ denotes the degree of freedom please mention it. Also, please mention the refinement strategy (i.e., uniform) of Fig. 1 and Fig. 2.

 

3) Algorithm 1: the reviewer did not understand how the number of samples is chosen. Is it according to Eq (10)? If so, please mention it.

 

Normally an optimization approach to minimize the total cost should be taken into account in MLMC methods. How it is included in the algorithm?

Why at each level, the number of samples multiplied by 1.2?

 

4) Theorem 1: It is for MLMC or MLQMC?

The convergence rates are different.

 

The theorem is Gile's Complexity Theorem which should be cited by authors.

Usually to the mesh refinement is shown with (h_\ell)^\alpha in MLMC papers.

 

5) The authors should cite the application of MLMC and MLQMC in different real-world problems. Here, a list of the necessary prior paper (which must be cited) is provided:

 

a) Current fluctuation in FinFETs:

Optimal multilevel randomized quasi-Monte-Carlo method for the stochastic drift-diffusion-Poisson system, CMAME, 2018.

b) Current variation in silicon nanowire biosensors:

Three-dimensional optimal multi-level Monte-Carlo approximation of the stochastic drift-diffusion-Poisson system in nanoscale devices. J. Comput. Electron. 2018.

c) Thermal fluctuation in the phase separation process

A multilevel Monte Carlo finite element method for the stochastic Cahn-Hilliard-Cook equation, Computational Mechanics, 2019.

d) Noise and variation in double-gate MOSFETs

https://arxiv.org/pdf/1904.05851.pdf

 

6) minor: please replace Gauß with Gauss.

Author Response

Dear referee,

Please see the attachment.

Sincerely,

Philippe Blondeel

Author Response File: Author Response.pdf

Reviewer 2 Report

In this manuscript, the authors have introduced a p-refined multilevel quasi-Monte Carlo (p-MLQMC) method for uncertainty quantification (UQ) which involves the finite element simulation of the underlying physical process. After the seminal work by Giles [8], multilevel methods have been investigated quite intensively in the UQ context and recognized as one of the standard choices for this kind of applications. The novelty of this manuscript is to use p-refinement (instead of h-refinement) in the finite element simulation, through which the authors observed a huge reduction of the variances for finer correction terms in the multilevel methods and succeeded in saving total runtimes required to achieve a prescribed root-mean-squared error.

However, when I went through the manuscript, I came up with some doubts about applicability of their proposed methodology to more general settings. Therefore, I cannot recommend the publication from the journal in the current form. I want the authors to revise the manuscript thoroughly by incorporating the following major/minor comments.

 

Major comments:

1) As Algorithm 2 requires the maximum level $L$ of multilevel corrections as an input, the proposed method is clearly not extensible for higher levels, which is an immense disadvantage over the h-refinement. That is, if the prescribed error is not achieved by the prescribed maximum level $L$, one needs to re-run Algorithm 2 with $L+1$ and re-simulate all realizations. How do you address this issue? What if you start from large $L$ but only use smaller corrections terms in numerical experiments? The current Section 4 is definitely not enough to verify the effectiveness of the proposed method. The authors must conduct experiments for smaller prescribed errors.

2) Also, Algorithm 2 looks useful only for two-dimensional finite element discretization. How do you extend this algorithm to three-dimensional cases, which are most important in practical use?

3) Why do you stick to using (shifted) rank-1 lattice rules? It seems to me that using something like (digitally shifted/scrambled) Sobol' sequences is much more natural. Can you justify your choice? Also, how did you choose generating vectors in your experiments?

 

Minor comments:

1) page 3, lines 6 and 7 from the bottom: Replace $:=$ by $=$.

2) page 5, line 17: $x^(r,n)$ should be in $[0,1)^s$ not in $(0,1]$.

3) page 5, line 18: Should (0,1] be [0,1)?

4) page 5, line 29: Why do you choose the multiplication constant as 1.2? Choosing 2 looks more reasonable.

5) page 6, line 1: Is $T\in \mathbb{R}$ correct?

6) page 7, Algorithm 1: When and how do you choose $R_{\ell}$?

7) page 7, line 9 from the bottom: Regarding the sentence ``Choosing the actual ...'', why do you need inclusion property? Do not multilevel methods work without such property?

8) page 7, line 5 from the bottom: Regarding the sentence ``This will lead to ...'', if multilevel methods also work without inclusion property, numerical experiments should be conducted also for ``square points'' case to justify this statement.

9) page 10, line 12 from the bottom: Please check the correctness of the equality. Do you need expectation for the right-hand side?

10) page 18, Fig. 17: Please include the results for p-QMC and h-QMC.

 

Author Response

Dear referee,

Please see the attachment.

Sincerely,

Philippe Blondeel

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors considered the reviewer's comments and improved the manuscript's quality. I recommend acceptance for the paper.

Reviewer 2 Report

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

Dear referee,

Please see the attachment.

Sincerely,

Philippe Blondeel

Author Response File: Author Response.pdf