Optimal Uncertainty Quantification under General Moment Constraints on Input Subdomains

TL;DR

This work extends Optimal Uncertainty Quantification (OUQ) to cases where input uncertainties are constrained by truncated moments defined on input subdomains, enabling rigorous upper and lower PoF bounds within an admissible set of probability measures. A reduction theorem reframes the problem as finite mixtures of Dirac measures, which are efficiently handled using a canonical-moment formulation that converts constrained optimization into an unconstrained search over higher-order canonical moments; supports and weights are recovered via Jacobi-matrix eigendecomposition, and PoF evaluation is accelerated with inverse-transform sampling (ITS) and variance reduction. The methodology is demonstrated through three numerical experiments, including a five-dimensional nonlinear problem and a ballistic-impact scenario with a neural surrogat e forward model; results show that increasing subdomain count or moment order tightens PoF bounds and that ITS reduces computational cost by up to two orders of magnitude while maintaining sub-percent accuracy. The framework also reveals a connection to evidence theory in the zeroth-moment case and provides practical guidance on where to invest uncertainty reduction efforts, with implications for safety certification under epistemic uncertainty. Together, these contributions yield a scalable, rigorous approach for safety assessment when only partial, localized statistical information is available.

Abstract

We present an optimal uncertainty quantification (OUQ) framework for systems whose uncertain inputs are characterized by truncated moment constraints defined over subdomains. Based on this partial information, rigorous optimal upper and lower bounds on the probability of failure (PoF) are derived over the admissible set of probability measures, providing a principled basis for system safety certification. We formulate the OUQ problem under general subdomain moment constraints and develop a high-performance computational framework to compute the optimal bounds. This approach transforms the original infinite-dimensional optimization problems into finite-dimensional unconstrained ones parameterized solely by free canonical moments. To address the prohibitive cost of PoF evaluation in high-dimensional settings, we incorporate inverse transform sampling (ITS), enabling efficient and accurate PoF estimation within the OUQ optimization. We also demonstrate that constraining inputs only by zeroth-order moments over subdomains yields a formulation equivalent to evidence theory. Three groups of numerical examples demonstrate the framework's effectiveness and scalability. Results show that increasing the number of subdomains or the moment order systematically tightens the bound interval. For high-dimensional problems, the ITS strategy reduces computational costs by up to two orders of magnitude while maintaining relative error below 1%. Furthermore, we identify regimes where optimal bounds are sensitive to subdomain partitioning or higher-order moments, guiding uncertainty reduction efforts for safety certification.

Figures (13)

• Figure 1: Illustration of certification in OUQ: (a) Rigorous decision criterion for system safety, and (b) Convergence of optimal bounds with increasing information.
• Figure 2: Illustration of ITS in OUQ: (a) PDF of a three-point Dirac measure, and (b) Corresponding CDF and ITS.
• Figure 3: Schematic illustration of the probability constraints for a one-dimensional random variable. The input domain is partitioned into subintervals $\mathcal{X}_{1,j}$, and the failure region $\mathcal{X}^\text{c}_1$ is highlighted for comparison. In this example, the upper and lower bounds on the PoF are $U=M_{1,2,0}+M_{1,3,0}+M_{1,4,0}+M_{1,5,0}$ and $L=M_{1,3,0}+M_{1,4,0}$, respectively.
• Figure 4: Truncated PDFs and corresponding failure regions for the four considered distributions: (a) truncated normal, (b) uniform, (c) truncated Weibull, and (d) truncated bimodal normal-mixture.
• Figure 5: Optimal PoF bounds as functions of the number of subdomains and moment constraints: (a) truncated normal, (b) uniform, (c) truncated Wei-bull, and (d) truncated bimodal normal-mixture.

Theorems & Definitions (3)

• Theorem B.1: Characterization of Supremum and Infimum
• Proposition B.2
• proof