Uncertainty quantification
An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc., will lead to different results that can only be predicted in a statistical sense.
Many problems in the natural sciences and engineering are also rife with sources of uncertainty.
[1][2][3][4][5][6] Uncertainty can enter mathematical models and experimental measurements in various contexts.
[10] In real life applications, both kinds of uncertainties are present.
The quantification for the aleatoric uncertainties can be relatively straightforward, where traditional (frequentist) probability is the most basic form.
A probability distribution can be represented by its moments (in the Gaussian case, the mean and covariance suffice, although, in general, even knowledge of all moments to arbitrarily high order still does not specify the distribution function uniquely), or more recently, by techniques such as Karhunen–Loève and polynomial chaos expansions.
To evaluate epistemic uncertainties, the efforts are made to understand the (lack of) knowledge of the system, process or mechanism.
In mathematics, uncertainty is often characterized in terms of a probability distribution.
From that perspective, epistemic uncertainty means not being certain what the relevant probability distribution is, and aleatoric uncertainty means not being certain what a random sample drawn from a probability distribution will be.
There has been a proliferation of research on the former problem and a majority of uncertainty analysis techniques were developed for it.
On the other hand, the latter problem is drawing increasing attention in the engineering design community, since uncertainty quantification of a model and the subsequent predictions of the true system response(s) are of great interest in designing robust systems.
It focuses on the influence on the outputs from the parametric variability listed in the sources of uncertainty.
The targets of uncertainty propagation analysis can be: Given some experimental measurements of a system and some computer simulation results from its mathematical model, inverse uncertainty quantification estimates the discrepancy between the experiment and the mathematical model (which is called bias correction), and estimates the values of unknown parameters in the model if there are any (which is called parameter calibration or simply calibration).
Generally this is a much more difficult problem than forward uncertainty propagation; however it is of great importance since it is typically implemented in a model updating process.
The general model updating formula for bias correction is: where
A prediction confidence interval is provided with the updated model as the quantification of the uncertainty.
denotes the true values of the unknown parameters in the course of experiments.
Its cornerstone is the calculation of probability density functions for sampling statistics.
In regression analysis and least squares problems, the standard error of parameter estimates is readily available, which can be expanded into a confidence interval.
Several methodologies for inverse uncertainty quantification exist under the Bayesian framework.
The most complicated direction is to aim at solving problems with both bias correction and parameter calibration.
The challenges of such problems include not only the influences from model inadequacy and parameter uncertainty, but also the lack of data from both computer simulations and experiments.
A common situation is that the input settings are not the same over experiments and simulations.
Another common situation is that parameters derived from experiments are input to simulations.
For computationally expensive simulations, then often a surrogate model, e.g. a Gaussian process or a Polynomial Chaos Expansion, is necessary, defining an inverse problem for finding the surrogate model that best approximates the simulations.
Apart from the current available data, a prior distribution of unknown parameters should be assigned.
Similarly with the first module, the discrepancy function is replaced with a GP model where Together with the prior distribution of unknown parameters, and data from both computer models and experiments, one can derive the maximum likelihood estimates for
Bayes' theorem is applied to calculate the posterior distribution of the unknown parameters: where
Fully Bayesian approach requires that not only the priors for unknown parameters
It follows the following steps:[18] However, the approach has significant drawbacks: The fully Bayesian approach requires a huge amount of calculations and may not yet be practical for dealing with the most complicated modelling situations.
