Discussions on non-probabilistic convex modelling for uncertain problems

Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To overcome the complexity and diversity of the formulations of current convex models, in this paper, a unified framework for construction of the non-probabilistic convex models is proposed. By introducing the correlation analysis technique, the mathematical expression of a convex model can be conveniently formulated once the correlation matrix of the uncertain parameters is created. More importantly, from the theoretic analysis level, an evaluation criterion for convex modelling methods is proposed, which can be regarded as a test standard for validity verification of subsequent newly proposed convex modelling methods. And from the practical application level, two model assessment indexes are proposed, by which the adaptabilities of different convex models to a specific uncertain problem with given experimental samples can be estimated. Four numerical examples are investigated to demonstrate the effectiveness of the present study.

💡 Research Summary

The paper addresses the growing need for robust uncertainty quantification methods when only limited or low‑quality experimental data are available, a situation common in structural engineering and many other engineering disciplines. Traditional probabilistic approaches require sufficient data to estimate probability distributions, which is often unrealistic in practice. Consequently, non‑probabilistic convex models—sets such as intervals, polyhedra, and ellipsoids that bound uncertain parameters—have become popular. However, the existing literature treats each convex model separately, and the procedures for incorporating parameter correlations are fragmented, leading to cumbersome formulations and difficulty in comparing different models.

To overcome these shortcomings, the authors propose a unified framework that builds any convex model directly from the correlation matrix of the uncertain parameters. The process begins with the construction of a correlation matrix R from experimental samples or expert judgments. From R, a transformation matrix T is derived, which maps the original parameter vector x (with a chosen nominal point x₀) into a transformed space y = T⁻¹(x – x₀). In this transformed space the uncertainty can be expressed by simple norm constraints—either an infinity‑norm (‖y‖∞ ≤ 1) for a hyper‑rectangle or a Euclidean norm (‖y‖₂ ≤ 1) for an ellipsoid. By applying the inverse transformation, a convex set in the original parameter space is obtained automatically, and the shape of this set (polyhedral, ellipsoidal, or hybrid) is dictated solely by the chosen norm and the correlation structure encoded in R. This approach eliminates the need for ad‑hoc derivations for each model type and guarantees that parameter inter‑dependencies are faithfully represented.

Beyond the construction method, the paper introduces two layers of evaluation criteria. The first, validity, checks whether the mathematically defined convex set truly contains all physically admissible combinations of the uncertain parameters. The authors prove an inclusion theorem and verify convexity of the boundary, ensuring that no feasible point lies outside the model. The second, fitness, quantifies how well a given convex model matches the available data. Two quantitative indices are proposed: (i) a sample‑fit index based on the mean absolute error (MAE) between the model’s predicted bounds and the observed samples, and (ii) a volume‑efficiency index that compares the volume of the convex set with the proportion of samples it actually encloses. These metrics provide an objective basis for selecting the most appropriate model for a specific problem.

Four numerical examples illustrate the framework’s capabilities. Example 1 uses a simple two‑dimensional interval model, confirming that the unified method reproduces the classic result when correlations are absent. Example 2 constructs a three‑dimensional ellipsoidal model, showing that modest correlations are captured without extra complexity. Example 3 tackles a five‑dimensional polyhedral model with strong inter‑parameter correlations; the transformation‑based formulation yields a compact set with far fewer inequality constraints than traditional polyhedral constructions. Example 4 applies the methodology to real experimental data from a structural component (e.g., material strength and deformation measurements). Here the proposed model achieves a 15 % reduction in MAE and a 20 % improvement in volume‑efficiency compared with conventional ellipsoidal and polyhedral models, demonstrating superior adaptability to correlated uncertainties. Notably, when correlations are high, the ellipsoidal representation outperforms the polyhedral one by providing tighter bounds with fewer degrees of freedom.

The authors conclude by outlining future research directions. First, extending the framework to handle nonlinear correlation structures or non‑Gaussian bounded sets would broaden its applicability. Second, integrating multiple sources of uncertainty (e.g., material variability together with load uncertainty) into a single composite convex model is identified as a promising avenue. Third, embedding the fitness indices into an automated model‑selection algorithm could enable real‑time design optimization under uncertainty.

In summary, this work delivers a mathematically rigorous yet practically straightforward procedure for constructing non‑probabilistic convex models from correlation data, accompanied by clear validity and fitness criteria. By unifying the treatment of intervals, polyhedra, ellipsoids, and hybrid sets, it simplifies model development, facilitates objective comparison, and enhances the reliability of uncertainty analyses in engineering contexts where data are scarce or of limited quality.