Structural and Multidisciplinary Optimization

, Volume 51, Issue 4, pp 919–940 | Cite as

Design optimization using hyper-reduced-order models

  • David AmsallemEmail author
  • Matthew Zahr
  • Youngsoo Choi
  • Charbel Farhat


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.


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



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.


  1. Alexandrov NM, Dennis J E Jr, Lewis RM (1998) A trust-region framework for managing the use of approximation models in optimization. Structural Optimization 15:16–23CrossRefGoogle Scholar
  2. Amsallem D, Farhat C (2008) Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J 46 (7):1803–1813CrossRefGoogle Scholar
  3. Amsallem D, Farhat C (2011) An online method for interpolating linear parametric reduced-order models. SIAM J Sci Comput 33 (5):2169–2198zbMATHMathSciNetCrossRefGoogle Scholar
  4. Amsallem D, Zahr MJ, Farhat C (2012) Nonlinear model order reduction based on local reduced-order bases. Int J Numer Methods Eng 92(10):891–916MathSciNetCrossRefGoogle Scholar
  5. Antoulas A (2005) Approximation of large-scale dynamical systems. SIAMGoogle Scholar
  6. Astrid P, Weiland S, Willcox K (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53 (10):2237–2251MathSciNetCrossRefGoogle Scholar
  7. (2003). In: Biegler L T, Ghattas O, Heinkenschloss M, van Bloemen Waanders B (eds) Large-scale PDE-constrained optimization. SpringerGoogle Scholar
  8. (2007). In: Biegler L T, Ghattas O, Heinkenschloss M, Keyes D, van Bloemen Waanders B (eds) Real-time PDE-constrained optimization. SIAMGoogle Scholar
  9. 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):687–713zbMATHMathSciNetCrossRefGoogle Scholar
  10. 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):714–739zbMATHMathSciNetCrossRefGoogle Scholar
  11. 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, IllinoisGoogle Scholar
  12. Bui-Thanh T, Willcox K, Ghattas O (2008) Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications. AIAA J 46(10):2520–2529CrossRefGoogle Scholar
  13. 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):381–402MathSciNetCrossRefGoogle Scholar
  14. 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):155–181zbMATHMathSciNetCrossRefGoogle Scholar
  15. 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):623–647zbMATHMathSciNetCrossRefGoogle Scholar
  16. Chaturantabut S, Sorensen D (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764zbMATHMathSciNetCrossRefGoogle Scholar
  17. Dihlmann M, Haasdonk B (2013) Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems. Submitted to J Comput Optim ApplGoogle Scholar
  18. Everson R, Sirovich L (1995) Karhunen–Loeve procedure for gappy data. J Opt Soc Am A 12(8):1657–1664CrossRefGoogle Scholar
  19. 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 268–280CrossRefGoogle Scholar
  20. Fasshauer GE (2007) Meshfree approximation methods with MATLAB. World ScientificGoogle Scholar
  21. Forrester A, Keane A (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1-3):50–79CrossRefGoogle Scholar
  22. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic PressGoogle Scholar
  23. Gogu C, Passieux JC (2013) Efficient surrogate construction by combining response surface methodology and reduced order modeling. Struct Multidiscip Optim 47(6):821–837CrossRefGoogle Scholar
  24. Gunzburger M (2003) Perspectives in Flow Control and Optimization. SIAMGoogle Scholar
  25. Haftka RT (1985) Simultaneous analysis and design. AIAA J 23(7):1099–1103zbMATHMathSciNetCrossRefGoogle Scholar
  26. 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):1327–1352zbMATHMathSciNetCrossRefGoogle Scholar
  27. Hömberg D, Volkwein S (2003) Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition. Math Comput Model 38(10):1003–1028zbMATHCrossRefGoogle Scholar
  28. Jameson A (1988) Aerodynamic design via control theory. J Sci Comput 3(3):233–260zbMATHCrossRefGoogle Scholar
  29. 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, COGoogle Scholar
  30. Manzoni A (2011) Shape optimization for viscous flows by reduced basis method and free form deformation. Int J Num Meth Eng, Rozza GGoogle Scholar
  31. Nocedal J, Wright SJ (2006) Numerical optimization. SpringerGoogle Scholar
  32. 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 EngGoogle Scholar
  33. Queipo NV, Haftka RT, Shyy W, Goel T (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41:1–28CrossRefGoogle Scholar
  34. 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):193–210zbMATHMathSciNetCrossRefGoogle Scholar
  35. Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202(1):346–366zbMATHMathSciNetCrossRefGoogle Scholar
  36. Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423zbMATHMathSciNetCrossRefGoogle Scholar
  37. Sirovich L (1987) Turbulence and the dynamics of coherent structures. Part I: coherent structures. Q Appl Math 45(3):561–571zbMATHMathSciNetGoogle Scholar
  38. 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):773–788zbMATHMathSciNetCrossRefGoogle Scholar
  39. 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):599–611CrossRefGoogle Scholar
  40. 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):129–136CrossRefGoogle Scholar
  41. 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 17–43Google Scholar
  42. Yue Y, Meerbergen K (2013) Accelerating optimization of parametric linear systems by model order reduction. SIAM J Optim 23(2):1344–1370zbMATHMathSciNetCrossRefGoogle Scholar
  43. Zahr MJ, Farhat C (2014) Progressive construction of a parametric reduced-order model for PDE-constrained optimization. Accepted for publication, Int J Num Meth EngGoogle Scholar
  44. 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, VirginiaGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • David Amsallem
    • 1
    Email author
  • Matthew Zahr
    • 2
  • Youngsoo Choi
    • 1
  • Charbel Farhat
    • 1
    • 2
    • 3
  1. 1.Department of Aeronautics and AstronauticsStanford UniversityStanfordUSA
  2. 2.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  3. 3.Department of Mechanical EngineeringStanford UniversityStanfordUSA

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