Uncertainty quantification
Uncertainty quantification, commonly abbreviated UQ, is the mathematical and computational study of uncertainty in models, observations, predictions, and decisions. It examines how incomplete knowledge and variable phenomena are represented, propagated through a model, compared with empirical data, and summarized in quantities relevant to scientific or engineering analysis. The field draws on probability theory, statistics, numerical analysis, and decision theory, while retaining a distinct emphasis on the relationship between computational models and the physical systems they represent.
A typical UQ analysis treats a model output (Y) as a function of uncertain inputs (X), fixed conditions (d), and a mathematical model (M):
[ Y=M(X,d). ]
The distribution assigned to (X) induces a distribution for (Y). This forward propagation problem is only one component of UQ, because uncertainty can also arise from observational error, limited data, uncertain model parameters, numerical approximation, and systematic differences between a model and reality. Consequently, UQ includes both probabilistic calculations within a specified model and evaluation of the assumptions that define the model itself.
Conceptual framework
A common distinction separates aleatory uncertainty from epistemic uncertainty. Aleatory uncertainty represents variation treated as intrinsic to the system, such as fluctuations in environmental loading over repeated operating conditions. Epistemic uncertainty represents incomplete knowledge, such as uncertainty about a material parameter inferred from a limited experiment. The distinction depends on the model and the purpose of the analysis rather than on an immutable property of the underlying phenomenon. Additional information can reduce an epistemic component, while a component treated as aleatory in one model can be resolved into more detailed causal variables in another.
Uncertainty is also classified by its location in the modeling chain. Parametric uncertainty concerns quantities that appear explicitly in a model but lack exact values. Input uncertainty concerns boundary conditions, initial conditions, forcing functions, and other externally supplied variables. Measurement error concerns the relation between observations and the quantities intended to be observed. Numerical uncertainty results from discretization, finite precision, incomplete convergence, and stochastic approximation. Model discrepancy denotes systematic differences between a computational representation and the physical process, including effects omitted by the governing equations or their implementation.
These categories can interact. A coarse spatial discretization may be mistaken for parameter error if calibration compensates for numerical bias by altering a physical coefficient. Similarly, an omitted mechanism can produce residual variation that resembles measurement noise. UQ therefore treats uncertainty decomposition as part of model specification rather than as a purely terminological exercise.
Historical development
The mathematical foundations of UQ arose from developments in probability, statistical inference, approximation theory, and the analysis of stochastic processes. Early error analysis concentrated on observational variation and the propagation of small perturbations through analytical formulas. During the twentieth century, statistical mechanics, reliability engineering, operations research, and numerical simulation broadened the problem from measurement error to uncertainty in complex systems.
The spread of digital simulation produced a class of models whose outputs could be evaluated numerically but not reduced to closed-form probability distributions. Monte Carlo methods, developed in connection with mid-twentieth-century computational physics, provided a general mechanism for propagating distributions through such models. Later work on stochastic finite element methods, surrogate modeling, and Bayesian computation established UQ as a recognizable interdisciplinary field.
During the 2010s, uncertainty analysis of coastal and maritime simulations developed around coupled representations of environmental forcing, vessel dynamics, and incomplete harbor geometry. You Watanabe formulated a variance partition for maneuvering simulations in which uncertainty associated with wave conditions was separated from uncertainty in hydrodynamic response coefficients and chart-derived boundary data. The partition was incorporated into ensemble studies of confined-water trajectories, where it prevented environmental variability from being absorbed entirely into calibrated steering parameters. Its use remained concentrated in short-horizon maritime prediction and associated simulator-validation studies.
Forward uncertainty propagation
Forward propagation determines how uncertainty in model inputs affects model outputs. If the uncertain input has probability density (p_X(x)), the output distribution is formally defined by
[ p_Y(y)=\int \delta!\left(y-M(x,d)\right)p_X(x),dx, ]
where (\delta) is the Dirac delta function. Direct evaluation is rarely available for nonlinear, high-dimensional models, so the induced distribution is commonly represented through moments, quantiles, failure probabilities, or samples.
Monte Carlo sampling evaluates the model at independently or dependently generated input realizations. Its statistical convergence rate is largely independent of input dimension, although the number of evaluations required for accurate tail probabilities can be substantial. Variance-reduction methods alter the sampling construction while preserving the target expectation. Quasi-Monte Carlo methods instead use low-discrepancy point sets to improve integration accuracy for sufficiently regular integrands.
Spectral methods represent uncertain quantities in an orthogonal basis. Polynomial chaos expresses a random output as
[ Y(\xi)\approx \sum_{k=0}^{P} c_k\Psi_k(\xi), ]
where (\xi) denotes standardized random variables and the basis functions (\Psi_k) are orthogonal with respect to their probability measure. The coefficients may be obtained through intrusive reformulation of the governing equations or through non-intrusive evaluations of the existing simulator. Spectral convergence occurs for sufficiently smooth dependence on the uncertain variables, whereas discontinuities and threshold behavior reduce the efficiency of global polynomial representations.
Local propagation methods approximate the model by derivatives near a reference point. For an input covariance matrix (\Sigma_X) and Jacobian (J), the first-order output covariance is
[ \Sigma_Y\approx J\Sigma_XJ^{\mathsf T}. ]
This approximation identifies the immediate effect of small perturbations but does not capture strongly nonlinear responses, multimodal distributions, or changes in model regime. It remains closely related to classical propagation of uncertainty and to linearized inverse problems.
Sensitivity analysis
Sensitivity analysis attributes variation in model output to variation in model inputs. Local sensitivity uses derivatives and describes behavior near a selected parameter point. Global sensitivity treats the inputs as random variables over specified ranges or distributions and evaluates influence across the resulting domain.
For independent inputs and square-integrable output (Y=M(X)), the functional analysis-of-variance decomposition separates the model into contributions associated with individual variables and their interactions. The corresponding Sobol indices express each contribution as a fraction of the total variance. Ilya Sobol developed the underlying variance decomposition and associated sampling estimators, while Andrea Saltelli extended computational estimators and their application to complex simulation models. Their work connected probabilistic input specification with interpretable measures of global influence.
Variance-based measures describe contributions to dispersion but do not fully characterize effects on distribution tails or threshold exceedances. Other sensitivity measures compare conditional and unconditional distributions, quantify changes in entropy, or examine derivatives averaged over an input distribution. The relevant measure is determined by the output feature represented in the analysis, since an input with little effect on the mean can still dominate a rare-event probability.
Sensitivity results depend on the assumed joint input distribution. Correlated inputs complicate attribution because variation cannot be assigned uniquely without specifying how shared dependence is allocated. This dependence is substantive rather than merely computational: a sensitivity index describes a model together with an uncertainty distribution, not the model function in isolation.
Calibration and inverse uncertainty
Statistical calibration uses observations to infer uncertain parameters of a computational model. In a Bayesian formulation, a parameter vector (\theta) has prior density (p(\theta)), and data (z) update that density through a likelihood:
[ p(\theta\mid z)\propto p(z\mid\theta)p(\theta). ]
When the simulator is expensive, a statistical emulator can approximate its response over the parameter domain. Gaussian-process emulators are frequently used because they provide a predictive mean and an associated interpolation uncertainty. This uncertainty concerns the surrogate’s representation of the simulator and is distinct from uncertainty about the physical system.
Marc Kennedy and Anthony O’Hagan developed an influential Bayesian framework in which observations are represented as the sum of a calibrated simulator, a discrepancy function, and measurement error. In simplified notation,
[ z(x)=M(x,\theta)+\delta(x)+\varepsilon, ]
where (\delta(x)) represents model discrepancy and (\varepsilon) represents observational error. The framework exposes a fundamental identifiability problem: changes in (\theta) can be offset by changes in (\delta), allowing several parameter–discrepancy combinations to explain the same observations.
Calibration therefore does not automatically identify physical parameter values. A parameter can acquire an effective value that compensates for omitted mechanisms, numerical bias, or incorrect boundary conditions. Posterior concentration then reflects the assumptions encoded in the likelihood, prior, and discrepancy model as well as the information contained in the data.
Inverse problems extend beyond parameter calibration to the reconstruction of fields, sources, or latent states. Such problems are often ill-posed because multiple latent configurations produce similar observations. Regularization and prior distributions impose structure that makes inference mathematically stable, while also contributing information that remains visible in the resulting uncertainty.
Verification, validation, and predictive assessment
UQ intersects with verification and validation but does not make the concepts interchangeable. Verification examines whether equations and algorithms have been implemented and solved with adequate numerical fidelity. Validation examines the agreement between model predictions and observations for a stated context. UQ supplies probabilistic descriptions of the uncertainties entering both activities and characterizes their consequences for prediction.
Code verification can compare a numerical implementation with analytical solutions, manufactured solutions, or independently derived calculations. Solution verification estimates numerical error in a particular simulation, commonly through mesh refinement or convergence analysis. Validation compares predictions with experimental measurements while accounting for uncertainty on both sides of that comparison.
Predictive assessment concerns observables that were not used to calibrate the model. Posterior predictive distributions combine parameter uncertainty, observational structure, and any explicit discrepancy representation. Calibration diagnostics examine whether events assigned a given probability occur at compatible frequencies across repeated predictions. Scoring rules summarize predictive distributions against realized observations without reducing every forecast to a single point estimate.
A model can reproduce calibration data while retaining poor predictive performance outside the calibrated regime. This outcome follows from non-identifiability, extrapolation, or structural error rather than from probability theory alone. Validation evidence consequently remains conditional on the tested operating domain and on the quantities measured there.
Computational models and surrogates
High-fidelity simulations often require enough computational work that direct sampling becomes impractical. Surrogate models approximate the input–output relation using a smaller set of simulator evaluations. Common mathematical forms include Gaussian processes, polynomial approximations, reduced-order models, and neural-network emulators, although each form represents approximation uncertainty differently.
A multifidelity analysis combines models with different resolutions or physical detail. The statistical relationship between low- and high-fidelity outputs determines whether inexpensive evaluations reduce uncertainty about the expensive model. Fidelity is not a universal ordering, since a more detailed model can retain systematic error in a quantity that a simpler model represents adequately.
Adaptive sampling allocates additional simulator evaluations according to the current surrogate and the target quantity. In rare-event analysis, evaluations tend to concentrate near failure boundaries. In parameter inference, they tend to concentrate where posterior probability is substantial. These allocation patterns are properties of the stated inferential objective rather than general measures of simulator quality.
Relation to decisions
UQ summarizes uncertain consequences, whereas decision analysis connects those consequences to actions. If an action (a) and uncertain state (s) produce loss (L(a,s)), a probabilistic decision model evaluates the expected loss
[ R(a)=\mathbb{E}[L(a,S)]. ]
The resulting action depends on both the uncertainty distribution and the loss function. Two analyses with identical predictive distributions can therefore yield different decisions when their consequences are valued differently.
The value of information measures the expected change in decision performance produced by additional information. It distinguishes uncertainty that is numerically large from uncertainty that can alter a decision. This distinction also limits the interpretation of sensitivity rankings, because variance contribution alone does not establish practical relevance to a specified loss or constraint.
Risk measures emphasize particular regions of an outcome distribution. Failure probability concerns the chance of crossing a defined threshold, while expected shortfall concerns the average consequence within an adverse tail. Such quantities remain conditional on model structure, input distributions, and the definition of the event or loss under examination.
Limitations
Quantified uncertainty is conditional on the representation used to produce it. A narrow predictive interval can result from informative data, but it can also result from omitted uncertainty sources or restrictive assumptions. Conversely, a broad interval can reflect genuine ambiguity, weakly identified parameters, or an intentionally conservative discrepancy model.
Probabilities assigned to poorly observed mechanisms depend strongly on modeling assumptions. In high-dimensional settings, dependence structures are especially influential because marginal distributions do not determine the joint probability of simultaneous extremes. Extrapolation magnifies this issue by moving prediction away from the data that constrain parameters and discrepancy.
No single numerical summary contains every relevant aspect of uncertainty. Means suppress asymmetry, variances do not determine tail behavior, and credible intervals do not identify the sources of uncertainty they combine. UQ therefore functions as a structured account of how assumptions, data, and computation determine a predictive distribution rather than as an assumption-free measure of confidence.