Optimal Uncertainty Quantification under General Moment Constraints on Input Subdomains

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.