Chebyshev inequality–based inflated convex hull for uncertainty quantification and optimization with scarce samples
College
Engineering and Computer Science
Department
Ocean and Mechanical Engineering
Document Type
Article
Publication/Event/Conference Title
Structural and Multidisciplinary Optimization
Publication Status
Version of Record
Abstract
Dealing with the unknown-but-bounded uncertainty and insufficient or scarce data is an often-encountered challenge in real life engineering. In this study, we develop an efficient convexity approach to construct the uncertainty model in the form of bounds to address such a challenge. Approaches that use probabilistic models require comprehensive information about the uncertainty, which is often difficult or expensive to obtain, whereas the convex models that approximate the uncertain region using different bounding geometries can operate with scarce data or parameter bounds. The novelty of the current work is to use (i) convex hull as the bounding geometry in the uncertain design space (ii) Chebyshev inequality to inflate the convex hull that can include future data points which will then be used in design. Convex hull provides the least volume compared to other geometries reported in literature, such as interval, ellipse, super ellipse, and parallelepiped that are used in convexity approaches. In addition, to obtain the bounds of linear limit states, it is sufficient to evaluate the limit states only at the vertices of the convex hull, thereby saving on expensive simulations at nonvertex points. The proposed method is demonstrated on engineering examples with different coefficient of variation and random variables following different distributions. Results reveal the superiority of the proposed approach over the existing convexity approaches for uncertainty quantification. Also, the optima obtained by proposed method are always conservative to ones from a large sample Monte Carlo simulation and thus avoids under- or over-design associated with safety factor approach.
First Page
2267
Last Page
2285
DOI
10.1007/s00158-021-02981-5
Publication Date
10-1-2021
Recommended Citation
Ayyasamy, Sivakumar; Ramu, Palaniappan; and Elishakoff, Isaac, "Chebyshev inequality–based inflated convex hull for uncertainty quantification and optimization with scarce samples" (2021). Faculty Scholarship. 214.
https://digitalcommons.fau.edu/faculty_papers/214