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Design optimization using hyper-reduced-order models

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Abstract

Solving large-scale PDE-constrained optimization problems presents computational challenges due to the large dimensional set of underlying equations that have to be handled by the optimizer. Recently, projection-based nonlinear reduced-order models have been proposed to be used in place of high-dimensional models in a design optimization procedure. The dimensionality of the solution space is reduced using a reduced-order basis constructed by Proper Orthogonal Decomposition. In the case of nonlinear equations, however, this is not sufficient to ensure that the cost associated with the optimization procedure does not scale with the high dimension. To achieve that goal, an additional reduction step, hyper-reduction is applied. Then, solving the resulting reduced set of equations only requires a reduced dimensional domain and large speedups can be achieved. In the case of design optimization, it is shown in this paper that an additional approximation of the objective function is required. This is achieved by the construction of a surrogate objective using radial basis functions. The proposed method is illustrated with two applications: the shape optimization of a simplified nozzle inlet model and the design optimization of a chemical reaction.

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Notes

  1. For simplicity, additional equality constraints are here embedded in k(⋅,⋅)≤0 as an equality constraint can always be written as two inequality constraints

  2. A weighted norm can be involved in place of the Euclidian norm when the entries in w r and p are of different scales

  3. The ROB V can be also updated by including the additional information obtained at \(\mathbf {p}_{N_{s}+1}\)

  4. For simplicity, in this complexity analysis, it is assumed that the same class of RBFs ϕ 𝜖 can be used to interpolate the objective function and each of the N i inequality constraints in k r .

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Acknowledgments

The authors acknowledge partial support by the Army Research Laboratory through the Army High Performance Computing Research Center under Cooperative Agreement W911NF-07-2-0027, and partial support by the Office of Naval Research under grant no. N00014-11-1-0707. This document does not necessarily reflect the position of these institutions, and no official endorsement should be inferred.

Author information

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Stanford University, Stanford, CA, 94305, USA

    David Amsallem, Youngsoo Choi & Charbel Farhat

  2. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, 94305, USA

    Matthew Zahr & Charbel Farhat

  3. Department of Mechanical Engineering, Stanford University, Stanford, CA, 94305, USA

    Charbel Farhat

Authors
  1. David Amsallem
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  2. Matthew Zahr
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  3. Youngsoo Choi
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  4. Charbel Farhat
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Corresponding author

Correspondence to David Amsallem.

Additional information

A preliminary version of this work was presented at the World Congress on Structural and Multidisciplinary Optimization in 2013.

Appendix: cross-validation procedure for choosing the radial basis function parameter

Appendix: cross-validation procedure for choosing the radial basis function parameter

The cross-validation procedure determines the optimal RBF parameter 𝜖>0 as in the work of (Rippa 1999). For that purpose, the sample set \(\left \{ \mathbf {x}_{rj} \right \}_{j=1}^{N_{s}}\) is partitioned into K non-overlapping subsets \(\{ \mathcal {S}_{j}\}_{j=1}^{K}\), and a series of candidate parameters 𝜖 i , i=1,⋯ ,N c is proposed. The generalization error associated with each candidate parameter is then estimated by building surrogates using K−1 subsets and testing its accuracy on the remaining subset. The optimal parameter 𝜖 ⋆ is then selected as being the one minimizing the generalization error. The procedure is summarized in Algorithm 5.

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Amsallem, D., Zahr, M., Choi, Y. et al. Design optimization using hyper-reduced-order models. Struct Multidisc Optim 51, 919–940 (2015). https://doi.org/10.1007/s00158-014-1183-y

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