Optimal Uncertainty Quantification (OUQ)

• Optimal Uncertainty Quantification (OUQ) is a rigorous framework that computes mathematically sharp bounds by formulating UQ as an optimization problem over all admissible scenarios.
• It reduces infinite-dimensional measure optimization to finite-dimensional problems using discrete measures and canonical moments, ensuring tractable and precise evaluations.
• OUQ demonstrates practical applicability in areas like nuclear safety, crash analysis, and high-dimensional PDEs by providing certification guarantees through worst-case performance bounds.

Optimal Uncertainty Quantification (OUQ) is a rigorous mathematical and algorithmic framework for producing the sharpest possible bounds on quantities of interest in complex systems subject to epistemic and/or aleatory uncertainty. The central idea is to recast uncertainty quantification objectives (e.g., bounds on failure probabilities, quantiles, or expectations) as optimization problems over the set of all scenarios (models, distributions) consistent with explicit available information, without introducing any unsubstantiated or implicit distributional assumptions. This approach provably yields mathematically optimal (i.e., sharp and unattainable by any alternative approach) bounds under the information at hand (Owhadi et al., 2010, Miska et al., 2024).

1. Mathematical Foundations and Problem Formulation

OUQ formalizes uncertainty quantification objectives via worst- or best-case optimization over an admissible set of possibilities. For a performance function $G: \mathcal X \to \mathbb{R}$, an unknown measure $\mu$ over $\mathcal X$, and partial information such as support and moment constraints, the central OUQ task is: $$\text{Compute }\, \sup_{\mu \in \mathcal{A}} \mathbb{E}_\mu[G(X)] \text{ or } \sup_{\mu \in \mathcal{A}} \mathbb{P}_\mu[G(X)\in \mathcal Y_0],$$ where $\mathcal{A}$ encodes all known constraints: support, (in)equalities on moments, independence, Lipschitz/oscillation, and legacy data (Owhadi et al., 2010, Miska et al., 2024).

For example, when quantifying the probability of failure (PoF)—the probability that $G(X)$ falls in a failure domain—OUQ computes: $$P_{\text{fail}}^{+} = \sup_{\mu\in\mathcal{A}} \int \mathbf{1}_{\left\{g(\theta, y, \widehat y) \leq 0\right\}} \, d\mu(y, \widehat y),$$ where $g$ encodes the limit state and $\mathcal{A}$ the set of admissible measures characterized by support and moment constraints for epistemic variables and prescribed PDFs for aleatory variables (Miska et al., 2024).

2. Finite-Dimensional Reduction and Canonical Optimization

The original OUQ problem is infinite-dimensional, as it requires maximizing (or minimizing) over all probability measures in $\mathcal{A}$. Structural results—specifically, the OUQ reduction theorem—guarantee that the extrema are attained on discrete measures supported on at most $k+1$ points per constraint, where $k$ is the number of binding (in)equalities per marginal (Owhadi et al., 2010, McKerns et al., 2012, Miska et al., 2024).

In the case of moment constraints on each marginal over bounded support, any admissible $\mu_i$ can be written as: $$\mu^{(i)} = \sum_{j=1}^{k+1} w_j^{(i)} \delta(x - x_j^{(i)}), \quad w_j^{(i)} \ge 0,\, \sum_j w_j^{(i)} = 1,$$ with nonlinear equality/inequality constraints enforcing the given moments. This key reduction transforms the infinite-dimensional problem into a finite, though generally non-convex, nonlinear optimization over the weights and support points.

For the case of quantile or CDF bounds under moment constraints, the canonical moments framework recasts the problem as an unconstrained optimization over so-called "free" canonical-moment parameters, ensuring automatic satisfaction of moment and positivity constraints (Stenger et al., 2019, Stenger et al., 2018, Jin et al., 22 Dec 2025).

3. Algorithmic Realizations and Computational Strategies

Practical OUQ implementations follow a double-loop structure when embedded in design-optimization settings. The outer loop addresses the optimization of engineering design variables $\theta \in \Theta$, while the inner loop solves the OUQ measure optimization problem for each candidate $\theta$ (Miska et al., 2024). The inner loop optimization is typically handled using global optimization algorithms (e.g., Differential Evolution, LSHADE44) acting on the reduced parameterization of the admissible measures:

• Weights and support locations of Dirac atoms for epistemic uncertainties.
• Numerical integration (quadrature, Monte Carlo, combination-line sampling) for aleatory (parametric) uncertainties.

For high-dimensional or computationally expensive cases, surrogate models (e.g., neural networks trained on deterministic simulations) or inverse transform sampling are employed to accelerate the evaluation of the system response and the PoF estimation (Jin et al., 22 Dec 2025, Sun et al., 2022, Miska et al., 2024).

4. Treatment of Aleatory and Epistemic Uncertainty

OUQ systematically distinguishes aleatory (irreducible stochastic) and epistemic (model/form) uncertainties:

• Aleatory uncertainty is modeled by fixed, fully specified probability distributions on the relevant input variables; in optimization, these PDFs are held fixed and not varied.
• Epistemic uncertainty is encoded by bounded supports and (in)equalities on moments or by explicit data constraints, with the optimization considering all possible distributions (within these bounds) as candidate input laws (Miska et al., 2024, Owhadi et al., 2010).

The admissible product measure for system inputs becomes: $$\mu(y, \widehat y) = \left( \bigotimes_{m=1}^{q-s} \mu^{(m)} \right) \otimes \left( \bigotimes_{s=1}^r f^{(s)}(\widehat y^{(s)}) d\widehat y^{(s)} \right),$$ where $\mu^{(m)}$ are optimized subject to epistemic constraints and $f^{(s)}$ are fixed PDFs for aleatory variables (Miska et al., 2024).

5. Mathematical Sharpness and Certification Guarantees

The bounds produced by OUQ are mathematically sharp: they are the largest (supremal) or smallest (infimal) values consistent with the admissible set of models and measures. If, for instance, a failure probability upper bound $P_{\text{fail}}^{+}$ is below the design threshold, certification is guaranteed under all scenarios consistent with the specified information. If $P_{\text{fail}}^{-}$ exceeds the threshold, the design is uncertifiable. Otherwise, further information (data, constraints) is needed for resolution (Owhadi et al., 2010, Miska et al., 2024).

This feature is critical for rigorous safety-critical certification because it does not implicitly rely on tubular assumptions (e.g., normality, independence) absent in the information set. Adding more constraints (e.g., higher-order moments, finer subdomain partitions) systematically reduces the feasible set and tightens the bound interval (Jin et al., 22 Dec 2025, Stenger et al., 2018).

6. Extended and Specialized OUQ Methodologies

Recent work extends canonical OUQ to accommodate further real-world complexities:

• Generalized moment constraints on input subdomains: Partitioning domains and prescribing local moments enhances bound tightening while maintaining tractability via a free canonical-moment parameterization, with the equivalence to evidence theory when only zeroth-order moments are known (Jin et al., 22 Dec 2025).
• Learning-based surrogates and neural approximators: Deep neural classifiers as surrogates combined with stochastic optimization facilitate tractable OUQ in high-dimensional, simulation-driven domains (Sun et al., 2022).
• OUQ in reliability-based design optimization (RBDO): Embedding OUQ within a double-loop RBDO framework yields designs provably optimized for worst-case performance and reliability under polymorphic uncertainties (both aleatory and epistemic) (Miska et al., 2024).

7. Practical Applications and Performance

OUQ has demonstrated practical viability on complex industrial and infrastructure systems:

• Thermal-hydraulic nuclear-safety codes: worst-case quantile bounds under limited moment data (Stenger et al., 2019).
• Car crash structures and Euler buckling columns: RBDO incorporating extended OUQ for polymorphic uncertainties; design decisions and safety margins reflect mathematically sharp bounds instead of potentially misleading heuristic UQ (Miska et al., 2024).
• High-dimensional PDE-governed systems: model validation, certification, and optimal experimental design under partial knowledge (McKerns et al., 2020).
• Ballistic plate impact using machine-learning OUQ surrogates: efficient certification maps for safety design (Sun et al., 2022).

In these applications, increasing available information (more moments, tighter bounds, finer partitioning) monotonically strengthens the tightness of the output interval, providing guidance for experimental design and uncertainty reduction efforts (Jin et al., 22 Dec 2025, Stenger et al., 2019).

• (Miska et al., 2024) "Reliability-Based Design Optimization Incorporating Extended Optimal Uncertainty Quantification"
• (Jin et al., 22 Dec 2025) "Optimal Uncertainty Quantification under General Moment Constraints on Input Subdomains"
• (Stenger et al., 2019) "Optimal Uncertainty Quantification of a risk measurement from a thermal-hydraulic code using Canonical Moments"
• (Stenger et al., 2018) "Optimal Uncertainty Quantification on moment class using canonical moments"
• (Sun et al., 2022) "A Learning-Based Optimal Uncertainty Quantification Method and Its Application to Ballistic Impact Problems"
• (Owhadi et al., 2010) "Optimal Uncertainty Quantification"
• (McKerns et al., 2020) "Optimal Bounds on Nonlinear Partial Differential Equations in Model Certification, Validation, and Experimental Design"
• (McKerns et al., 2012) "The Optimal Uncertainty Algorithm in the Mystic Framework"