Structural and Multidisciplinary Optimization

, Volume 60, Issue 4, pp 1571–1582 | Cite as

Uncertainty-based design optimization approach based on cumulative distribution matching

  • Xingzhi HuEmail author
  • Xiaoqian Chen
  • Geoffrey T. Parks
  • Fulin Tong
Research Paper


Uncertainty-based design optimization (UBDO) is becoming increasingly important with great attention to critical system reliability and robustness. However, most traditional UBDO methods only consider low-order response moments and a limited number of aleatory uncertainties, which desiderates to be resolved. In this study, an efficient UBDO approach based on cumulative distribution matching is presented to address all response moments under mixed aleatory and epistemic uncertainties. The mathematical formulation of cumulative distribution function (CDF) matching is systematically derived including the distance metric definition and numerical computation process. Uncertainty analysis within the CDF matching optimization loop is also investigated, including the identification of active subspaces and the propagation of constructed surrogate model in reduced dimensions. Extended active subspaces are furthered discussed for mixed uncertainty analysis to obtain the bounds of CDF efficiently, instead of the time-consuming full-dimensional simulation. The proposed approach is illustrated by three typical optimization examples, i.e., response function matching, standard NASA test problem, and satellite system design in aerospace engineering. The UBDO results based on the CDF matching are discussed and compared with the probability density function matching, which exhibits higher efficiency and better convergence in tackling constrained optimization design problems subject to mixed uncertainties.


Uncertainty-based design optimization Density/distribution matching Cumulative distribution function Uncertainty analysis Reduced subspace 



Objective response


Response CDF


Designer-given target CDF


Distance metric between target and system response


Overall uncertainty


Uncertain design variables


System uncertainty parameters


Optimal design under uncertainty


Aleatory uncertainty


Epistemic uncertainty


ith input uncertainty


ith integration weight


Kernel function


Required constraint

\( {\widehat{\mathbf{W}}}_1 \)

Estimated eigenvectors for active subspace


Reduced coordinate


Interval reduced coordinate

Response surface surrogate model


Number of quadrature points


Number of KDE samples


Probability density function


Cumulative distribution function


Cumulative belief function


Cumulative plausibility function



The authors would like to thank Dr. Pranay Seshadri and Prof. Wen Yao for their valuable comments and suggestions for improving this paper.

Funding information

This work was supported by National Nature Science Foundation of China (Grant Nos. 11702305 and 11725211) and Pre-research Generic Technology Project (Grant No. 41406030102).

Compliance with Ethical Standards

Conflict of interests

The authors declared that they have no conflicts of interest to this work.


  1. Adamson RD, Gill PMW, Pople JA (1998) Empirical Density Function (EDF). Chem Phys Lett 284:6–11CrossRefGoogle Scholar
  2. Allen M, Maute K (2004) Reliability-based design optimization of aeroelastic structures. Struct Multidiscip Optim 27:228–242. CrossRefGoogle Scholar
  3. Alonso JJ, Eldred MS, Constantine P, Duraisamy K, Farhat C, Iaccarino G, Jakeman J (2017) Scalable environment for quantification of uncertainty and optimization in industrial applications (SEQUOIA). In: Aiaa Non-Deterministic Approaches ConferenceGoogle Scholar
  4. Brévault L, Balesdent M, Bérend N, Le Riche R (2013) Challenges and future trends in Uncertainty-Based Multidisciplinary Design Optimization for space transportation system design. In: 5th European Conference for Aeronautics and Space Sciences (EUCASS 2013)Google Scholar
  5. Constantine PG (2015) Active subspaces: emerging ideas for dimension reduction in parameter studies vol 2. SIAMGoogle Scholar
  6. Cook LW, Jarrett JP (2018) Horsetail matching: a flexible approach to optimization under uncertainty. Engineering Optimization 50(4):549-567Google Scholar
  7. Cook LW, Jarrett JP, Willcox KE (2017) Horsetail matching for optimization under probabilistic, interval and mixed uncertainties. In: 19th AIAA Non-Deterministic Approaches Conference. AIAA SciTech Forum. American Institute of Aeronautics and Astronautics.
  8. Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Multidiscip Optim 14:63–69CrossRefGoogle Scholar
  9. Eldred MS, Swiler LP, Tang G (2011) Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation. Reliab Eng Syst Saf 96:1092–1113CrossRefGoogle Scholar
  10. Ferson S, Kreinovich V, Hajagos J, Oberkampf W, Ginzburg L (2007) Experimental uncertainty estimation and statistics for data having interval uncertainty. Sandia National Laboratories, AlbuquerqueGoogle Scholar
  11. Frangopol DM, Maute K (2003) Life-cycle reliability-based optimization of civil and aerospace structures. Comput Struct 81:397–410. CrossRefGoogle Scholar
  12. Helton JC, Davis FJ (2003) Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab Eng Syst Saf 81:23–69CrossRefGoogle Scholar
  13. Hu X, Chen X, Parks GT, Yao W (2016a) Review of improved Monte Carlo methods in uncertainty-based design optimization for aerospace vehicles. Prog Aerosp Sci 86:20–27CrossRefGoogle Scholar
  14. Hu XZ, Parks GT, Chen XQ, Seshadri P (2016b) Discovering a one-dimensional active subspace to quantify multidisciplinary uncertainty in satellite system design. Adv Space Res 57:1268–1279. CrossRefGoogle Scholar
  15. Hu X, Zhou Z, Chen X, Parks GT (2018) Chance-constrained optimization approach based on density matching and active subspaces. AIAA J 56:1158–1169. CrossRefGoogle Scholar
  16. James RW, Wiley JL (1999) Space mission analysis and design Space Technology LibraryGoogle Scholar
  17. Li P, Arellano-Garcia H, Wozny G (2008) Chance constrained programming approach to process optimization under uncertainty. Comput Chem Eng 32:25–45. CrossRefGoogle Scholar
  18. Lillacci G, Khammash M (2012) A distribution-matching method for parameter estimation and model selection in computational biology. Int J Robust Nonlinear Control 22:1065–1081MathSciNetCrossRefGoogle Scholar
  19. Messac A, Ismail-Yahaya A (2002) Multiobjective robust design using physical programming. Struct Multidiscip Optim 23:357–371. CrossRefGoogle Scholar
  20. Park H-U, Lee J-W, Chung J, Behdinan K (2015) Uncertainty-based MDO for aircraft conceptual design. Aircr Eng Aerosp Tec 87:345–356CrossRefGoogle Scholar
  21. Ray T (2003) Golinski’s speed reducer problem revisited. AIAA J 41:556–558. CrossRefGoogle Scholar
  22. Royset JO, Kiureghian AD, Polak E (2001) Reliability-based optimal structural design by the decoupling approach. Reliab Eng Syst Saf 73:213–221CrossRefGoogle Scholar
  23. Russi TM (2010) Uncertainty quantification with experimental data and complex system models. University of California, BerkeleyGoogle Scholar
  24. Sankararaman S, Mahadevan S (2011) Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data. Reliab Eng Syst Saf 96:814–824. CrossRefGoogle Scholar
  25. Scott DW (2015) Multivariate density estimation: theory, practice, and visualization. Wiley, HobokenCrossRefGoogle Scholar
  26. Seshadri P, Constantine P, Iaccarino G, Parks G (2016) A density-matching approach for optimization under uncertainty. Comput Methods Appl Mech Eng 305:562–578MathSciNetCrossRefGoogle Scholar
  27. Tang G (2013) Methods for high dimensional uncertainty quantification: regularization, sensitivity analysis, and derivative enhancement. Stanford University, StanfordGoogle Scholar
  28. Tripathy R, Bilionis I, Gonzalez M (2016) Gaussian processes with built-in dimensionality reduction: applications to high-dimensional uncertainty propagation. J Comput Phys 321:191–223MathSciNetCrossRefGoogle Scholar
  29. Wang X, Qiu Z, Söffker D (2013) Uncertainty-based design optimization in engineering: model, algorithm, and application. J Appl Math 2013:1–2Google Scholar
  30. Yao W (2011) Research on uncertainty-based multidisciplinary design optimization for aerospace vehicle system design PhD diss. National University of Defense Technology, ChangshaGoogle Scholar
  31. Yao W, Chen XQ, Huang YY, Gurdal Z, van Tooren M (2013) Sequential optimization and mixed uncertainty analysis method for reliability-based optimization. AIAA J 51:2266–2277. CrossRefGoogle Scholar
  32. Youn BD, Choi KK (2004) A new response surface methodology for reliability-based design optimization. Comput Struct 82:241–256CrossRefGoogle Scholar
  33. Zaman K, Rangavajhala S, Mcdonald MP, Mahadevan S (2011) A probabilistic approach for representation of interval uncertainty. Reliab Eng Syst Saf 96:117–130. CrossRefGoogle Scholar
  34. Zhang XD, Huang HZ (2010) Sequential optimization and reliability assessment for multidisciplinary design optimization under aleatory and epistemic uncertainties. Struct Multidiscip Optim 40:165–175. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.China Aerodynamics Research & Development CenterMianyangChina
  2. 2.Chinese Academy of Military ScienceBeijingChina
  3. 3.University of CambridgeCambridgeUK

Personalised recommendations