ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization

Round 1

Reviewer 1 Report

Dear Authors, 

In this paper, you considered the application of model order reduction techniques to solve a multi-objective optimisation problem. The reduction approach applied is the reduced basis one along with a hyper reduction using the Discrete Empirical Interpolation Method. The optimisation algorithm considered is based on the combination of the descent direction with inexact gradients and, the subdivision algorithm. This latter is improved using concepts from the Greedy algorithm in the reduced basis technique. 

The paper is clear and well written. Mathematical details are considered within all the sections of the manuscript. However, some remaining issues in what follows need to be clarified and considered:  

1)The alternative descent directions mentioned on tables 3 and 5 are not clearly defined within the paper. Please clarify this point. 

2)The practical application of the original subdivision algorithm is not detailed. Only the remark number 4 is considered. Please detail the practical use of this technique at least within the numerical experiments.  

3) The cost of the new proposed subdivision algorithm along with the reduced basis and DEIM approaches might be prohibitive for mutli-parametric cases. It is interesting to discuss a priori or a posteriori efficient error estimates in this field. There is a wide litterature concerning the development of certified a posteriori error estimates for linear and non-linear elliptic partial differential equations. This might be naturally applied for both the reference model equations and the adjoint ones. I advise to add a detailed discussion concerning this domain and the ability to apply these estimations to the present paper. 

4) Have you considered the application of the reduced basis - DEIM approximation only with the inexact gradients technique? What could be the advantages and drawbacks with respect to the technique proposed in this paper which considers also the subdivision algorithm?  

Author Response

Please see our attached pdf file.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper, multiobjective parameter optimization problems are solved in
form of semilinear elliptic PDEs by combining a reduced based approach and
discrete empirical interpolation with the set-oriented method based on inexact gradient evaluations. The error introduced by the surrogate model is monitored by deriving an additional condition for the descent direction. This allows that the errors for the objective functions are independent and a superset of the Pareto critical set is constructed. The error bounds are updated after each iteration step with a subdivision algorithm. Numerical results are presented with the subdivision and the modi ed algorithm. The di erent RB-based methods are four times faster than the full-order FEM solution.

The manuscript makes a valuable contribution to current research in opti-mization and model order reduction. It is clearly presented, and the method-
ology is described in sucient detail and supported by numerical results. The
paper is acceptable in the present form.

Author Response

Please see our attached pdf file.

Author Response File: Author Response.pdf

Reviewer 3 Report

I only have a few minor suggestions:

• line 75: remove "holds true"
• Gradients are usually column vectors so there appear to be transpose signs missing in line 83 and the unnumbered line directly above it. The same issue appears a number of times throughout the paper. 
• It could be mentioned that (4) is a convex problem.
• The transpose sign following the Jacobian D \hat J (step 3 of Algorithm 1) is incorrect. A Jacobian (not its transpose) gets multiplied by direction vectors (here: \nu) to yield a directional derivative. Same in line 5 of the algorithm and a number of times throughout the paper.
• Take particular note of eq. (8) where the transpose sign must follow the first occurrence of D \hat J, not the second. 
• After eq.(16) it is assumed that this equation has a unique solution. Isn't this clear from the assumptions on the PDE?
• In the displayed eq. above line 178, the factor 1/2 is missing.
• I suppose Fig. 5 is showing the norm of the difference of gradients? Please update the caption.
• Please correct reference 29. 

Author Response

Please see our attached pdf file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Dear authors,

Thank you for your point by point responses. I will suggest to the editorial office to accept this publication in the present form. 

Best regards. 

Reviewer 3 Report

All suggestions were taken into account.