Robust design with arbitrary distributions using Gauss-type quadrature formula

S. H. Lee, W. Chen*, B. M. Kwak

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

80 Scopus citations


In this paper, we present the Gauss-type quadrature formula as a rigorous method for statistical moment estimation involving arbitrary input distributions and further extend its use to robust design optimization. The mathematical background of the Gauss-type quadrature formula is introduced and its relation with other methods such as design of experiments (DOE) and point estimate method (PEM) is discussed. Methods for constructing one dimensional Gauss-type quadrature formula are summarized and the insights are provided. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed for two different multi-dimensional integration methods, the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. Through several examples, it is shown that the Gauss-type quadrature formula can be effectively used in robust design involving various non-normal distributions. The proposed design sensitivity analysis significantly reduces the number of function calls of robust optimization using the TPQ formulae, while using the UDR method, the savings of function calls are observed only in limited situations.

Original languageEnglish (US)
Pages (from-to)227-243
Number of pages17
JournalStructural and Multidisciplinary Optimization
Issue number3
StatePublished - Sep 2009


  • Analytical design sensitivity analysis
  • Design optimization
  • Gauss-type quadrature formula
  • Robust design
  • Tensor product quadrature
  • Univariate dimension reduction method

ASJC Scopus subject areas

  • Software
  • Control and Optimization
  • Control and Systems Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design


Dive into the research topics of 'Robust design with arbitrary distributions using Gauss-type quadrature formula'. Together they form a unique fingerprint.

Cite this