On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring
Round 1
Reviewer 1 Report
Comments for author File: 
Comments.pdf
Author Response
Please see the attached file letter-to-referee-1.pdf
Author Response File: Author Response.pdf
Reviewer 2 Report
The manuscript "On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring" studies the convergence of the additive Schwarz method for the solution of variational inequalities and equalities. Inspired by recent works in the literature, the author aims to identify the dependence of the convergence on the number of subdomain colouring. Some numerical results validate the theoretical findings.
This referee thinks that the manuscript is certainly interesting and of good mathematical quality.
Nevertheless, this referee also thinks that the manuscript is very badly written, and that the author has put very little effort to make his manuscript (which, again, is very interesting), pedagogical and comprehensible to a reasonably large audience.
Therefore, this referee recommends the editor to ask for a deep major revision.
Major comments:
- As a very general comment, this referee suggests the author to improve the readability, both by improving the English fluency (several sentences just read weird), and by making the manuscript accessible to a broader audience.
Below, there are a few (but not all) parts that the author is invited to improve.
- It is quite difficult to understand the content of the paper from the abstract. Colouring of subdomains, assumptions on convex sets (?), reflexive Banach spaces, additive Schwarz, damping.. all these words come before what it is actually the goal of the paper: solving variational inequalities/equalities.
- Similarly, the introduction is difficult to follow. The term convex set is repeated, but still it is not clear to what exactly it refers.
- End of page 2: Two assumptions are made on $F$, yet no comment at all is provided. What's their interpretation? How strict are these assumptions? Are they easily satisfied?
- (2) and (3) come from nowhere.
- Up to line 83, the reader has just seen assumptions, references to other works and claims without a single sentence to explain.
- Equation (10): s is fixed somehow. The reader cannot understand why this choice and not something else.
- Assumption 2: What does it mean this assumption?
- Page 6, a sequence of badly formatted inequalities is presented. Throughout the manuscript, this referee recommends to format the inequalities properly, and to add clear explanations how to move from one step to the other.
- This referee has honestly not checked Theorem 2. It is basically necessary to redo all the computations from scratch.
- In the first part of the numerical experiments, the convergence with respect to $\rho$ is analyzed. This referee is a bit lost. In Table 1, setting all relaxation parameters equal to 1/mc is the best choice (among those reported). This is claimed to be supported by Remark 3. However, in Figure 2, one can see that other values are better. No comments on this discrepancy are provided.
- Why the number of iterations increases as the overlapping parameter increases in left figure 3?
- From figure 5, this referee deduces that the subdomain decomposition can be such that actually two nonoverlapping subdomains that are separeted by another nonoverlapping subdomains between them, can actually become overlapping between them, once they are extended (see Figure 5, right, green and red subdomains, with the blue in the middle). Is this a reasonable domain decomposition?
- Regarding the numerical experiments, this referee does not have an explanation for the following observation: the author considers a global domain $\Omega$ of fixed size. Then he increases the number of subdomains, so each subdomain becomes smaller. It is well-known that Schwarz methods are not scalable in this setting (see Dolean, Nataf book).
Why the results seem to show the opposite? A clear discussion would be much appreciated.
Minor comments:
- In several places, the article "the" is superfluous, in others it is missing.
- A more standard notation for the set of real numbers if $\mathbb{R}$.
- Why not showing a couple of pictures of the decomposition of the 2D domain, with the subdomain colouring?
Author Response
Please see the attached file letter-to-referee-2.pdf
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
This referee is sufficiently satisfied by the responses of the author and therefore recommends the editor to accept the manuscript.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
This is an interesting manuscript with several results that are clearly of value
for the analysis of domain decomposition methods. The paper is generally well written, and I could find no errors in the proofs. However, before considering publication in this journal, the following points should be addressed:
- In the introduction the author claims that it follows from his theoretical
analysis that the convergence speed of the damped additive Schwarz method does not depend on the number of subdomains but only on the minimum number of colors the subdomains can be colored with s.t. subdomains with the same color are disjoint. In this context the author refers to [17-22] where, indeed, weak scalability of one-level Schwarz methods has been observed and proven. However, as for instance discussed in [Chaouqui, Ciaramella, Gander, Vanzan 2018], this behavior crucially depends on the geometric situation and on the boundary conditions active in each subdomain.
Weak scalability also does not seem to follow from the author's
results in Section 2.2 as the coercivity constant alpha_M will generally
decrease with the size of the domain Omega. It is also easily checked
numerically that for the Poisson problem in one spatial dimension and an
overlapping domain decomposition of equally sized subdomains with fixed
overlap, the convergence rate does depend on the chosen number of subdomains even when the size of the overlap stays the same.
- There is no discussion of the sharpness of the derived convergence rates. I
would suggest to at least compare the theoretically derived rates with empirical rates observed for some model problem.
- In Remark 2 it is stated that 'it follows from error estimations in the above
theorem that the best convergence rate of Algorithms 1 and 2 is obtained for
rho_min = rho/m_c = rho_1 = rho_m_c'. While this choice does give the best
rates in Theorem 1, without knowing anything about the optimality of these
rates it is unclear whether the same is true for the actually observed rates.
In particular, if the number of overlapping subdomains varies locally, it is
plausible to assume that locally adaptive dampening parameters might give better convergence speeds.
- The results from Section 3 should be put into context of existence results on the convergence of block Jacobi methods.
- In the introduction it is concluded that 'In the case of the equations in
finite element spaces [...] the additive Schwarz method converges only if the
subdomains can be colored with two colors'. However, I do not see, how this
conclusion is possible from the results. In particular, the existence of a
non-converging example for m_c=3 does not show that convergence never is
possible for minimal m_c > 2.
- For the interpretation of the results in Section 3 it would be helpful to give
some bounds on the value of \tilde{\lambda}, at least for the situation
described in Section 3.3.
As a suggestion: The paper already is relatively long. As Sections 2 and 3 are
mostly independent, it might make sense to split the results into two papers,
such that there is more room to discuss the individual results in more detail
and look at concrete examples.
Author Response
Please see the attachment
Author Response File: Author Response.pdf
Reviewer 2 Report
This is an article about the Schwarz method for nonlinear problems when the domain decomposition forms a "chain". The paper seems to contain two entirely different things. In Sections 1 and 2, I think that the author tried to combine [20] and [25] but I don't think they've succeeded. Section 3 is almost completely independent of Sections 1 and 2, and attempted to generalize [20] using standard techniques, but again I believe the author has not achieved the objective.
In equation (5), one sees that the author is finding the minimum of a convex functional over a fixed domain, whose function space is denoted K, and this function space does not grow with the number of subdomains. This scenario does not corresponding to "growing chains of subdomains" as in [20] (because the domain is indeed not "growing") and instead corresponds exactly to the case treated in [25] and the standard Schwarz literature.
If we consider strictly case of a fixed domain $\Omega$, I don't think there is original research value in the present paper. The nonlinear part of the paper (sections 1 and 2) is entirely subsumed by [25]. The linear theory (section 3) is also subsumed by the standard theory of Schwarz methods in standard textbooks.
In the scenario of a fixed domain $\Omega$, there are also errors. The number of subdomains will enter into the constants $C_0$ of Assumption 2, and the constant $\tilde{\lambda}$ of (64). The latter is because of the implied truncation in the definition of $\|\cdot\|_a$ compared to $|\cdot|_a$; the constant $C_0$ is also large because of implied truncation. As a result, the claims that the methods presented scale independently of the subdomains, made in Remark 2 and elsewhere, is false.
If the subdomain sizes were fixed (and hence $\Omega$ varies), as well as fixing the overlap $\delta>0$ (instead of minimal overlap), this would help control the truncation issue, although one would have to carefully examine what happens to the various other constants as a function of the varying domain $\Omega$.
Apart from the above main problems, I have a few lesser (but still major) issues. The fact that the domain is decomposed as a "chain" could have been stated succinctly as $\Omega_i \cap \Omega_j = \emptyset$ if $|i-j|>1$ but instead the authors chose to write this in an extremely obscure way in (9), (80), (81), (82), (83), (84), (85), (86). Equations (40), (41) and the surrounding discussion also seem needlessly cryptic.
The fact that the undamped Schwarz iteration diverges when there are 3 or more colors, is part of the standard literature and has no research value.
It was shown in [20] that, even in the case of actually "growing chains" of subdomains, the convergence factor depends on the overlap $\delta$, so one should not use the "minimal overlap" approach $\delta = h$, since then the convergence factor will deteriorate with refinement.
To support claims that convergence does not depend on number of subdomains or other parameters, I would have expected significant scaling experiments, but there is no numerical experiment at all.
Author Response
Please see the attachment
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper considers some additive Schwarz method, and some sound theoretical results are obtained. However, there is lack of numerical evidence to verify the theory. Is it possible to produce some numerical examples?
Author Response
Please see the attachment
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The main addition of the revised version of the manuscript is a numerical
experiment which is supposed to confirm the author's main result that the
convergence rate of the damped additive Schwarz method depends on the number of colors, but not on the number of subdomains. As clearly the number of iterations increases with the number of subdomains in Figure 2, the author has rephrased his claim and states that the convergence rate only has an upper limit which does not depend on the number of subdomains.
However, contrary to what the author says, the numerical results do not support this statement in any meaningful way. Over the entire range it can be observed that the number of iterations scales approximately linearly with the square root of the number of subdomains. This is expected as in the experiment the number of iterations required to let information flow from one subdomain to any other also scales with the square root of the number of subdomains. So no asymptotic behavior that might indicate an independence of the number of subdomains is visible.
This may be explained by the fact that the derived bounds are very pessimistic. Combined with the fact that increasing the number of subdomains while keeping a constant overlap is only possible up to a finite number of subdomains such that there is no true asymptotic behavior to reason about, I find it hard to see how the bounds would give any new insight. In particular, it obviously makes no sense to derive 'optimal' dampening parameters from them, and I would still expect that for domain decompositions with varying numbers of locally overlapping subdomains, locally adapted dampening parameters would give better convergence rates than choosing 1/m_c uniformly.
Overall, I cannot see any significant new insight regarding the convergence
rates of damped additive Schwarz methods compared to what is already shown in [13] (in the revised manuscript). Therefore, unfortunately, I cannot recommend publication of this manuscript. There might be some value in showing convergence of damped additive Schwarz for constrained variational inequalities, assuming this has not been done before. If the author believes this by itself to be a significant result, I would suggest to submit a new manuscript where this is presented as the main result.
Reviewer 3 Report
The revised paper is satisfactory.
