Uncertainty quantification

Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if we exactly knew the speed, 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. Computer simulation modeling is the most commonly used approach to study problems in uncertainty quantification (UQ).

Reasons for uncertainties

Uncertainty can enter numerical or mathematical models in various contexts. For example:

1. The model structure, i.e., how accurately a mathematical model describes the true system for a real-life situation, may only be known approximately. Models are almost always only approximations to reality. For example, the Maxwell equations describe electromagnetic fields very well; yet, it is known that quantum electrodynamics is the correct description if field strengths become large.
2. The numerical approximation, i.e., how appropriately a numerical method is used in approximating the operation of the system. Most models are too complicated to solve exactly. For example the finite element method may be used to approximate the solution of a partial differential equation, but this introduces an error (the difference between the exact and the numerical solution).
3. Input and/or model parameters may only be known approximately. For example, simulating the take-off of an airplane would require us to know the exact wind speed everywhere along the runway, but we may only have data for a few individual locations.
4. Input and/or model parameters may vary between different instances of the same object for which predictions are sought. As an example, the wings of two different airplanes of the same type may have been fabricated to the same specifications, but will nevertheless differ by small amounts due to fabrication process differences. Computer simulations therefore almost always consider only idealized situations.

Uncertainties can be classified into different categories:

1. Aleatoric or statistical uncertainties are unknowns that differ each time we run the same experiment. In the example above, even if we could exactly control the wind speeds along the run way, if we let 10 planes of the same make start their trajectories would still differ due to fabrication differences. Similarly, if all we knew is that the average wind speed is the same, letting the same plane start 10 times would still yield different trajectories because we do not know the exact wind speed at every point of the runway, only its average. Statistical uncertainties are therefore something an experimenter can not do anything about: they exist, and they can not be suppressed by more accurate measurements.
2. Epistemic or systematic uncertainties are due to things we could in principle know but don't in practice. This may be because we have not measured a quantity sufficiently accurately, or because our model neglects certain effects, or because particular data are deliberately hidden.

In real life applications, both kinds of uncertainties are often present. Uncertainty quantification intends to work toward reducing type 2 uncertainties to type 1. The quantification for the type 1 uncertainty is relatively straightforward to perform. Techniques such as Monte Carlo methods are frequently used. Pdf can be represented by its moments (in the Gaussian case,the mean and covariance suffice), or more recently, by techniques such as Karhunen–Loève and polynomial chaos expansions. To evaluate type 2 and 3 uncertainties, the efforts are made to gain better knowledge of the system, process or mechanism. Methods such as fuzzy logic or evidence theory (Dempster–Shafer theory – generalization of Bayes theory) are used.