Taming Uncertainty in a Complex World: The Rise of Uncertainty Quantification - A Tutorial for Beginners

British statistician, wrote the famous aphorism, "All models are wrong, but some are useful."The aphorism acknowledges that models, regardless of whether they are qualitative, quantitative, dynamical, or statistical, always fall short of the complexities of reality.The ubiquitous imperfections of models come from various sources, including the lack of a perfect understanding of nature, limited spatiotemporal model resolutions due to computational power, inaccuracy in the initial and boundary conditions, etc.Therefore, uncertainty quantification (UQ), which quantitatively characterizes and estimates uncertainties, is essential to identify the usefulness of the model.UQ also defines the range of possible outcomes when certain aspects of the system are not precisely known.A key objective of UQ is to explore how uncertainties propagate, both through time evolution and across different quantities via complex nonlinear dependencies. Characterizing UncertaintiesWhen uncertainty appears in the model and the input, the output can potentially take different values with typically unequal chances.Therefore, it is natural to characterize the output as a random variable, and the UQ of the model output can be based on the associated probability density function (PDF).