Selective dimension reduction method (DRM) to enhance accuracy and efficiency of most probable point (MPP)–based DRM

  • Jeong Woo Park
  • Hyunkyoo Cho
  • Ikjin LeeEmail author
Research Paper


To perform reliability-based design optimization (RBDO) in engineering systems, reliability analysis is required to calculate probability of failure (PF) for each performance function. Most probable point (MPP)–based dimensional reduction method (DRM) has been developed to accurately estimate PF using the Gaussian quadrature integration method. However, the existing MPP-based DRM is computationally expensive for highly nonlinear and/or high dimensional problems since it needs to increase the number of integration points in all directions to guarantee accuracy. In the proposed method, three statistical model selection methods—Akaike information criterion (AIC), AIC correction (AICc), and Bayesian information criterion (BIC)—are utilized to identify characteristic of performance functions for more efficient integration point allocation. Then, genetic algorithm and simplex optimization method are used to find the best models with the smallest AIC, AICc, and BIC values. No additional function evaluations are required for the model selection process since MPP candidate points are utilized. The best models obtained through optimization show where to allocate integration points which makes it possible not to allocate unnecessary integration points. Numerical study verifies that the proposed method can guide how to allocate integration points according to characteristic of performance functions: no integration points for almost linear performance functions, minimal additional integration points for mildly nonlinear performance functions, and more integration points for highly nonlinear functions.


Reliability-based design optimization (RBDO) Reliability analysis Dimension reduction method (DRM) Akaike information criterion (AIC) 



Vector of random variables; X = {X1, X2, ⋯, XN}T


Realization of X; x = {x1, x2, ⋯, xN}T


Vector of standard normal random variables; U = {U1, U2, ⋯, UN}T


Realization of U; u = {u1, u2, ⋯, uN}T


Vector of rotated standard normal random variables; V = {V1, V2, ⋯, VN}T


Realization of V; v = {v1, v2, ⋯, vN}T


Probability measure


Joint probability density function (PDF) of random variables


Performance function; considered failure if G(X) > 0


Standard normal cumulative distribution function (CDF)


kth candidate model used in statistical model selection method

\( {L}_{M_k} \)

Likelihood estimation value in candidate model Mk

θ = {ρ, λ, σ}

Hyperparameters used in calculation of \( {L}_{M_k} \)


Total number of random variables


Number of variables considered in the selected interaction terms


Number of times Vi is used in the selected interaction terms


Number of variables selected only in the axial direction


Number of selected interaction terms


Funding information

This research was supported by the development of reliability validation method on computational simulation of gas turbine funded by the Korea Electric Power Corporation (No. R17XA05-06).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Akaike H (1974) A new look at the statistical model identification. In Selected Papers of Hirotugu Akaike (pp. 215–222). Springer, New York, NYGoogle Scholar
  2. Bae HR, Alyanak E (2016) Sequential subspace reliability method with univariate revolving integration. AIAA J 54(7):2160–2170CrossRefGoogle Scholar
  3. Ben-Ari EN, Steinberg DM (2007) Modeling data from computer experiments: an empirical comparison of kriging with MARS and projection pursuit regression. Qual Eng 19(4):327–338CrossRefGoogle Scholar
  4. Bien J, Taylor J, Tibshirani R (2013) A lasso for hierarchical interactions. Ann Stat 41(3):1111MathSciNetCrossRefGoogle Scholar
  5. Breitung K (1984) Asymptotic approximations for multinormal integrals. J Eng Mech 110(3):357–366CrossRefGoogle Scholar
  6. Burnham KP, Anderson DR, Huyvaert KP (2011) AIC model selection and multimodel inference in behavioral ecology: some background, observations, and comparisons. Behav Ecol Sociobiol 65(1):23–35CrossRefGoogle Scholar
  7. Cho H, Choi KK, Gaul NJ, Lee I, Lamb D, Gorsich D (2016a) Conservative reliability-based design optimization method with insufficient input data. Struct Multidiscip Optim 54(6):1609–1630MathSciNetCrossRefGoogle Scholar
  8. Cho H, Choi KK, Lee I, Lamb D (2016b) Design sensitivity method for sampling-based RBDO with varying standard deviation. J Mech Des 138(1):011405CrossRefGoogle Scholar
  9. Denny M (2001) Introduction to importance sampling in rare-event simulations. Eur J Phys 22(4):403CrossRefGoogle Scholar
  10. Findley DF (1991) Counterexamples to parsimony and BIC. Ann Inst Stat Math 43(3):505–514MathSciNetCrossRefGoogle Scholar
  11. Geladi P, Kowalski BR (1986) Partial least-squares regression: a tutorial. Anal Chim Acta 185:1–17CrossRefGoogle Scholar
  12. Golub GH, Welsch JH (1969) Calculation of Gauss quadrature rules. Math Comput 23(106):221–230MathSciNetCrossRefGoogle Scholar
  13. Hao N, Feng Y, Zhang HH (2018) Model selection for high-dimensional quadratic regression via regularization. J Am Stat Assoc 113(522):615–625MathSciNetCrossRefGoogle Scholar
  14. Hao P, Wang Y, Ma R, Liu H, Wang B, Li G (2019) A new reliability-based design optimization framework using isogeometric analysis. Comput Methods Appl Mech Eng 345:476–501MathSciNetCrossRefGoogle Scholar
  15. Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111–121Google Scholar
  16. Jung Y, Cho H, Lee I (2019) MPP-based approximated DRM (ADRM) using simplified bivariate approximation with linear regression. Structural and multidisciplinary optimization 59(5):1761–1773. MathSciNetCrossRefGoogle Scholar
  17. Kang SB, Park JW, Lee I (2017) Accuracy improvement of the most probable point-based dimension reduction method using the hessian matrix. Int J Numer Methods Eng 111(3):203–217MathSciNetCrossRefGoogle Scholar
  18. Keshtegar B, Hao P (2018) Enriched self-adjusted performance measure approach for reliability-based design optimization of complex engineering problems. Appl Math Model 57:37–51MathSciNetCrossRefGoogle Scholar
  19. Keshtegar B, Hao P, Meng Z (2017) A self-adaptive modified chaos control method for reliability-based design optimization. Struct Multidiscip Optim 55(1):63–75MathSciNetCrossRefGoogle Scholar
  20. Lee I, Choi KK, Du L, Gorsich D (2008) Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput Methods Appl Mech Eng 198(1):14–27CrossRefGoogle Scholar
  21. Lee I, Choi KK, Zhao L (2011) Sampling-based RBDO using the stochastic sensitivity analysis and dynamic kriging method. Struct Multidiscip Optim 44(3):299–317MathSciNetCrossRefGoogle Scholar
  22. Lim M, Hastie T (2015) Learning interactions via hierarchical group-lasso regularization. J Comput Graph Stat 24(3):627–654MathSciNetCrossRefGoogle Scholar
  23. Lumley T, Scott A (2015) AIC and BIC for modeling with complex survey data. J Surv Stat Methodol 3(1):1–18CrossRefGoogle Scholar
  24. Madsen HO, Krenk S, Lind NC (2006) Methods of structural safety. Courier CorporationGoogle Scholar
  25. Park JW, Lee I (2018) A study on computational efficiency improvement of novel SORM using the convolution integration. J Mech Des 140(2):024501CrossRefGoogle Scholar
  26. Penmetsa RC, Grandhi RV (2003) Adaptation of fast Fourier transformations to estimate structural failure probability. Finite Elem Anal Des 39(5–6):473–485CrossRefGoogle Scholar
  27. Rahman S, Wei D (2006) A univariate approximation at most probable point for higher-order reliability analysis. Int J Solids Struct 43(9):2820–2839CrossRefGoogle Scholar
  28. Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probabilist Eng Mech 19(4):393–408CrossRefGoogle Scholar
  29. Rosenblueth E (1981) Two-point estimates in probabilities. Appl Math Model 5(5):329–335MathSciNetCrossRefGoogle Scholar
  30. Rubinstein RY, Kroese DP (2016) Simulation and the Monte Carlo method (Vol. 10). John Wiley & SonsGoogle Scholar
  31. Shin J, Lee I (2014) Reliability-based vehicle safety assessment and design optimization of roadway radius and speed limit in windy environments. J Mech Des 136(8):081006CrossRefGoogle Scholar
  32. Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1–3):271–280MathSciNetCrossRefGoogle Scholar
  33. Sues R, Aminpour M, Shin Y (2000) Reliability based MDO for aerospace systems. In 19th AIAA Applied Aerodynamics Conference (p. 1521)Google Scholar
  34. Sugiura N (1978) Further analysts of the data by Akaike’s information criterion and the finite corrections: further analysts of the data by Akaike’s. Commun Stat-Theor M 7(1):13–26CrossRefGoogle Scholar
  35. Süli E, Mayers DF (2003) An introduction to numerical analysis. Cambridge university pressGoogle Scholar
  36. Tu J, Choi KK, Park YH (2001) Design potential method for robust system parameter design. AIAA J 39(4):667–677CrossRefGoogle Scholar
  37. Tvedt L (1990) Distribution of quadratic forms in normal space—application to structural reliability. J Eng Mech 116(6):1183–1197CrossRefGoogle Scholar
  38. Valdebenito MA, Schuëller GI (2010) A survey on approaches for reliability-based optimization. Struct Multidiscip Optim 42(5):645–663MathSciNetCrossRefGoogle Scholar
  39. Venables WN, Ripley BD (2013) Modern applied statistics with S-PLUS. Springer Science & Business MediaGoogle Scholar
  40. Walker JR (1986) The practical application of variance reduction techniques in probabilistic assessmentsGoogle Scholar
  41. Wu YT, Shin Y, Sues R, Cesare M (2001) Safety-factor based approach for probability-based design optimization. In 19th AIAA applied aerodynamics conference (p. 1522)Google Scholar
  42. Xu H, Rahman S (2003) A moment-based stochastic method for response moment and reliability analysis. In Computational fluid and solid mechanics 2003 (pp. 2402-2404). Elsevier Science Ltd.Google Scholar
  43. Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Methods Eng 61(12):1992–2019CrossRefGoogle Scholar
  44. Youn BD, Choi KK, Yang RJ, Gu L (2004) Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multidiscip Optim 26(3–4):272–283CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringKorea Advanced Institute of Science and TechnologyDaejeon 34141South Korea
  2. 2.Department of Mechanical EngineeringMokpo National UniversityMuan-gunSouth Korea

Personalised recommendations