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This study presents a mathematical framework for optimal uncertainty quantification (OUQ) that calculates rigorous upper and lower bounds on system failure probability when input parameters are only partially known through moment constraints over specific subdomains. The researchers developed a computational method that transforms infinite-dimensional optimization problems into tractable finite-dimensional ones and incorporated inverse transform sampling to handle high-dimensional cases efficiently. Numerical demonstrations show the framework can reduce computational costs by up to two orders of magnitude while maintaining less than 1% relative error, and that increasing the number of subdomains or using higher-order moments systematically tightens the probability bounds.
Why it matters
This framework provides a rigorous mathematical basis for certifying system safety in engineering applications where complete probabilistic information about uncertain inputs is unavailable. The method's computational efficiency in high-dimensional problems makes it practically applicable to complex real-world systems requiring reliability assessment under partial uncertainty information.
arXiv:2512.19572v2 Announce Type: replace
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.
Source: Optimal Uncertainty Quantification under General Moment Constraints on Input Subdomains