Skip to main content

Design optimization using hyper-reduced-order models


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21


  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 .


  • Alexandrov NM, Dennis J E Jr, Lewis RM (1998) A trust-region framework for managing the use of approximation models in optimization. Structural Optimization 15:16–23

    Article  Google Scholar 

  • Amsallem D, Farhat C (2008) Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J 46 (7):1803–1813

    Article  Google Scholar 

  • Amsallem D, Farhat C (2011) An online method for interpolating linear parametric reduced-order models. SIAM J Sci Comput 33 (5):2169–2198

    Article  MATH  MathSciNet  Google Scholar 

  • Amsallem D, Zahr MJ, Farhat C (2012) Nonlinear model order reduction based on local reduced-order bases. Int J Numer Methods Eng 92(10):891–916

    Article  MathSciNet  Google Scholar 

  • Antoulas A (2005) Approximation of large-scale dynamical systems. SIAM

  • Astrid P, Weiland S, Willcox K (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53 (10):2237–2251

    Article  MathSciNet  Google Scholar 

  • (2003). In: Biegler L T, Ghattas O, Heinkenschloss M, van Bloemen Waanders B (eds) Large-scale PDE-constrained optimization. Springer

  • (2007). In: Biegler L T, Ghattas O, Heinkenschloss M, Keyes D, van Bloemen Waanders B (eds) Real-time PDE-constrained optimization. SIAM

  • Biros G, Ghattas O (2005a) Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: the Krylov–Schur solver. SIAM J Sci Comput 27(2):687–713

    Article  MATH  MathSciNet  Google Scholar 

  • Biros G, Ghattas O (2005b) Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part II: the Lagrange–Newton solver and its application to optimal control of steady viscous flows. SIAM J Sci Comput 27(2):714–739

    Article  MATH  MathSciNet  Google Scholar 

  • Buffoni M, Willcox K (2010) Projection-based model reduction for reacting flows. AIAA Paper 2010-5008, 40th Fluid Dynamics Conference and Exhibit, 28 June - 1 July 2010, Chicago, Illinois

  • Bui-Thanh T, Willcox K, Ghattas O (2008) Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications. AIAA J 46(10):2520–2529

    Article  Google Scholar 

  • Carlberg K, Farhat C (2010) A low-cost, goal-oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems. Int J Numer Methods Eng 86(3):381–402

    Article  MathSciNet  Google Scholar 

  • Carlberg K, Bou-Mosleh C, Farhat C (2011) Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. Int J Numer Methods Eng 86(2):155–181

    Article  MATH  MathSciNet  Google Scholar 

  • Carlberg K, Farhat C, Cortial J, Amsallem D (2013) The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J Comput Phys 242(C):623–647

    Article  MATH  MathSciNet  Google Scholar 

  • Chaturantabut S, Sorensen D (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764

    Article  MATH  MathSciNet  Google Scholar 

  • Dihlmann M, Haasdonk B (2013) Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems. Submitted to J Comput Optim Appl

  • Everson R, Sirovich L (1995) Karhunen–Loeve procedure for gappy data. J Opt Soc Am A 12(8):1657–1664

    Article  Google Scholar 

  • Fahl M, Sachs EW (2003) Reduced order modelling approaches to PDE-constrained optimization based on proper orthogonal decomposition. In: Biegler L T, Ghattas O, Heinkenschloss M, van Bloemen Waanders B (eds) Large-scale PDE-constrained optimization. Springer, Berlin, pp 268–280

    Chapter  Google Scholar 

  • Fasshauer GE (2007) Meshfree approximation methods with MATLAB. World Scientific

  • Forrester A, Keane A (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1-3):50–79

    Article  Google Scholar 

  • Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press

  • Gogu C, Passieux JC (2013) Efficient surrogate construction by combining response surface methodology and reduced order modeling. Struct Multidiscip Optim 47(6):821–837

    Article  Google Scholar 

  • Gunzburger M (2003) Perspectives in Flow Control and Optimization. SIAM

  • Haftka RT (1985) Simultaneous analysis and design. AIAA J 23(7):1099–1103

    Article  MATH  MathSciNet  Google Scholar 

  • Hay A, Borggaard JT, Akhtar I, Pelletier D (2010) Reduced-order models for parameter dependent geometries based on shape sensitivity analysis. J Comput Phys 229(4):1327–1352

    Article  MATH  MathSciNet  Google Scholar 

  • Hömberg D, Volkwein S (2003) Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition. Math Comput Model 38(10):1003–1028

    Article  MATH  Google Scholar 

  • Jameson A (1988) Aerodynamic design via control theory. J Sci Comput 3(3):233–260

    Article  MATH  Google Scholar 

  • LeGresley PA, Alonso JJ (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. AIAA Paper 2000-2545 Fluids 2000 Conference and Exhibit, Denver, CO

  • Manzoni A (2011) Shape optimization for viscous flows by reduced basis method and free form deformation. Int J Num Meth Eng, Rozza G

  • Nocedal J, Wright SJ (2006) Numerical optimization. Springer

  • Paul-Dubois-Taine A, Amsallem D (2014) An adaptive and efficient greedy procedure for the optimal training of parametric reducedorder models. Int J Num Meth Eng

  • Queipo NV, Haftka RT, Shyy W, Goel T (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41:1–28

    Article  Google Scholar 

  • Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2/3):193–210

    Article  MATH  MathSciNet  Google Scholar 

  • Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202(1):346–366

    Article  MATH  MathSciNet  Google Scholar 

  • Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423

    Article  MATH  MathSciNet  Google Scholar 

  • Sirovich L (1987) Turbulence and the dynamics of coherent structures. Part I: coherent structures. Q Appl Math 45(3):561–571

    MATH  MathSciNet  Google Scholar 

  • Veroy K, Patera AT (2005) Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int J Numer Methods Fluids 47(8–9):773–788

    Article  MATH  MathSciNet  Google Scholar 

  • Weickum G, Eldred MS, Maute K (2009) A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization. Struct Multidiscip Optim 38(6):599–611

    Article  Google Scholar 

  • Xiao M, Breitkopf P, Filomeno Coelho R, Knopf-Lenoir C, Villon P (2012) Enhanced POD projection basis with application to shape optimization of car engine intake port. Struct Multidiscip Optim 46(1):129–136

    Article  Google Scholar 

  • Young DP, Huffman WP, Melvin RG, Hilmes CL, Johnson FT (2003) Nonlinear elimination in aerodynamic analysis and design optimization. In: Large-scale PDE-constrained optimization. Springer, Berlin, Heidelberg, pp 17–43

    Google Scholar 

  • Yue Y, Meerbergen K (2013) Accelerating optimization of parametric linear systems by model order reduction. SIAM J Optim 23(2):1344–1370

    Article  MATH  MathSciNet  Google Scholar 

  • Zahr MJ, Farhat C (2014) Progressive construction of a parametric reduced-order model for PDE-constrained optimization. Accepted for publication, Int J Num Meth Eng

  • Zahr MJ, Amsallem D, Farhat C (2013). Construction of parametrically-robust CFD-based reduced-order models for PDE-constrained optimization. In: AIAA Paper 2013–2845, 21st AIAA Computational Fluid Dynamics Conference, San Diego, CA, June 26–29, 2013, American Institute of Aeronautics and Astronautics, Reston, Virginia

Download references


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


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.

figure h

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Amsallem, D., Zahr, M., Choi, Y. et al. Design optimization using hyper-reduced-order models. Struct Multidisc Optim 51, 919–940 (2015).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • PDE-constrained optimization
  • Surrogate modeling
  • Parametric model reduction
  • Hyper-reduction
  • Discrete empirical interpolation method
  • Shape optimization