Uncertainty Quantification — Polynomial Chaos and Kriging Surrogate
Uncertainty Quantification (Polynomial Chaos Expansion and Kriging Surrogate) · Also known as: UQ, polynomial chaos expansion, PCE, Kriging surrogate, Gaussian process surrogate, Belirsizlik Nicelleştirme (UQ — Polynomial Chaos, Kriging Surrogate)
Uncertainty Quantification (UQ) is a computational framework for systematically measuring how uncertainty in the inputs of a model propagates into uncertainty in its outputs. Building on Wiener's polynomial chaos theory (1938) and formalised for general stochastic problems by Xiu and Karniadakis (2002), UQ uses two primary strategies: Polynomial Chaos Expansion (PCE), which represents the model output as a series of orthogonal polynomials matched to the input distributions, and Kriging (Gaussian process) surrogates, which replace an expensive simulation with a fast statistical approximation fitted to a small set of carefully chosen runs.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
+5 more
When to use it
UQ is appropriate whenever a computational model has uncertain inputs — parameters measured with error, boundary conditions that are only approximately known, or material properties with natural variability — and the analyst needs to characterise how that uncertainty affects model outputs. It is especially valuable when the simulator is computationally expensive (minutes to hours per run), making direct Monte Carlo infeasible. Input uncertainties must be expressible as probability distributions; if only worst-case bounds are available without probabilistic information, interval or fuzzy methods may be more suitable. PCE is well-suited to moderate numbers of inputs (roughly up to 10–15 dimensions before the curse of dimensionality requires sparse variants); Kriging surrogates can handle somewhat higher dimensions but require careful design-of-experiments to keep the number of training runs manageable. Both approaches assume that the model is sufficiently smooth for a polynomial or Gaussian process approximation to be accurate.
Strengths & limitations
- Provides a full characterisation of the output distribution — not just a mean or worst case — enabling rigorous probabilistic statements about model predictions.
- Sobol sensitivity indices identify which uncertain inputs drive output variability, guiding experimental design and data collection priorities.
- Once the surrogate is built, propagation and sensitivity analysis are computationally negligible, making UQ practical even for expensive simulators.
- PCE delivers analytical expressions for mean, variance, and sensitivity indices, avoiding Monte Carlo sampling noise in those quantities.
- PCE suffers from the curse of dimensionality: the number of polynomial terms grows rapidly with the number of uncertain inputs and polynomial degree, requiring sparse PCE or dimensionality-reduction strategies for high-dimensional problems.
- The accuracy of a Kriging surrogate depends critically on the design of experiments; too few or poorly placed training points yield unreliable predictions between them.
- Both methods assume the simulator response is smooth enough to be approximated well by polynomials or a Gaussian process; discontinuities or sharp gradients in the response surface degrade accuracy.
- The quality of the UQ result is only as good as the input distributions chosen; poorly specified input distributions propagate incorrect uncertainty into the output.
Frequently asked
What is the difference between PCE and a Kriging surrogate?
PCE represents the model output as an explicit polynomial series in the uncertain inputs, chosen to be orthogonal with respect to the input distributions. Sobol sensitivity indices and moments follow analytically from the coefficients. Kriging treats the model output as a realisation of a Gaussian process and interpolates between training points, providing both a prediction and a prediction variance. PCE tends to be more efficient when the input dimension is low and the response is smooth and globally polynomial; Kriging is more flexible for irregular or locally varying responses and can be extended with active learning strategies such as EGO.
How many simulator runs are needed to build a reliable surrogate?
For PCE the minimum is roughly two to three times the number of polynomial basis terms, but more runs improve accuracy. For a Kriging surrogate a common rule of thumb is ten times the number of uncertain inputs for the initial design, with additional points added adaptively if budget allows. In both cases, validation against held-out runs is essential before the surrogate is used for decision-making.
What are Sobol indices and why do they matter?
Sobol indices are variance-based global sensitivity measures. The first-order index for an input reports the fraction of the total output variance attributable to that input alone. The total-order index includes all interactions between that input and others. Together they tell the analyst which inputs are worth measuring more precisely and which can be treated as fixed without significantly affecting the output uncertainty. Unlike local (derivative-based) sensitivity measures, Sobol indices are valid across the entire input distribution, not just near a nominal point.
Can UQ be applied when input distributions are unknown?
UQ in the probabilistic sense requires that input uncertainties be specified as probability distributions. When only ranges or bounds are available without distributional knowledge, uniform distributions are sometimes used as a conservative choice, but this assumption should be made explicit. If the analyst genuinely cannot assign a distribution — for example in deep uncertainty settings — scenario analysis or robust optimisation approaches may be more appropriate than probabilistic UQ.
Sources
- Xiu, D. & Karniadakis, G.E. (2002). The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2), 619–644. DOI: 10.1137/S1064827501387826 ↗
- Smith, R.C. (2013). Uncertainty Quantification: Theory, Implementation, and Applications. SIAM. ISBN: 978-1611973211
How to cite this page
ScholarGate. (2026, June 1). Uncertainty Quantification (Polynomial Chaos Expansion and Kriging Surrogate). ScholarGate. https://scholargate.app/en/simulation/uncertainty-quantification
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Bayesian OptimizationOptimization↔ compare
- Global Sensitivity AnalysisSimulation↔ compare
- KrigingSpatial analysis↔ compare
- Latin Hypercube SamplingSimulation↔ compare
- MONTE-CARLO-SIMULATIONDecision-making↔ compare
- Stochastic Differential EquationsSimulation↔ compare
- Surrogate-Based OptimizationOptimization↔ compare
- System DynamicsSimulation↔ compare
- Variance Reduction for Monte CarloSimulation↔ compare