Forward Uncertainty Propagation

Forward UQ pushes known input distributions through a model to compute the resulting distribution, moments, and failure probabilities of the output.

The forward problem

Given uncertain inputs described by probability distributions, forward uncertainty propagation asks: what is the distribution of the output? The goal may be the mean and variance, the full probability density, a quantile, or the probability that the output exceeds a safety threshold. It is the most common UQ task in engineering design.

Sampling methods

Kronos motion — pid vs model

Monte Carlo is the baseline: draw input samples from their distributions, evaluate the model for each, and build output statistics from the results. It is simple, dimension-independent in convergence rate, and robust, but its error shrinks only as one over the square root of the sample count, so it needs many runs. Quasi-Monte Carlo and Latin hypercube sampling improve this for smooth problems.

Spectral and surrogate methods

Polynomial chaos expansions give mean, variance, and Sobol indices in closed form for smooth responses
Gaussian-process emulators propagate uncertainty with error bars from few runs
Stochastic collocation evaluates on a sparse grid of quadrature points
The delta method uses a first-order Taylor expansion for small uncertainties

Moment propagation shortcut

For small input uncertainties, a first-order Taylor expansion propagates variance cheaply: output variance is approximately the sum over inputs of (partial derivative)^2 times input variance. This linearization is fast but fails for nonlinear responses or large uncertainties, where full sampling is required.

Rare events

When the quantity of interest is a small failure probability, plain Monte Carlo is inefficient because failures are rare. Importance sampling, subset simulation, and first- and second-order reliability methods concentrate effort near the failure boundary to estimate small probabilities with far fewer runs.

In design practice

Forward UQ turns a Kronos performance prediction into a distribution rather than a single figure. Uncertain physics inputs are propagated through surrogates of the machines to report expected performance with credible intervals, and to estimate the chance of crossing an operating limit - always distinguishing simulated ranges from any future measured result.