Hengduan Mountains
copyright@planets_for
copyright@planets_for
“The only thing certain in life is uncertainty.” Uncertainty Quantification (UQ) is an interdisciplinary field about quantitative characterization, propagation, and reduction of uncertainties, as well as ultimate decision-making under uncertainties that are ubiquitous not only in life but in essentially all science and engineering fields. This course introduces the basic concepts, examples, problem formulations, computational models and algorithms, mathematical and numerical analyses, as well as practical applications of UQ in various scientific and engineering fields. Broad and introductory materials will be covered on (1) characterization of different types of uncertainties using probability models, including random variables, random fields, and stochastic processes with different probability distributions, (2) propagation of the input uncertainties of mathematical models to their output solutions or quantities of interests by computational methods, including Monte Carlo, polynomial chaos, reduced order models, and deep neural networks, (3) reduction of the uncertainties by observational or experimental data, using statistical or Bayesian inference methods such as Markov chain Monte Carlo and variational inference methods, (4) sensitivity analysis of system outputs with respect to the uncertain inputs and risk analysis of system outputs under uncertain inputs, (5) decision-making under uncertainty in the framework of optimal control and design of systems by risk-averse and model-constrained stochastic optimization.
In the opening of his book, The nature of mathematical modeling, Neil Gershenfeld poses the following questions:
How would you understand:
* How the sound of a violin works? Synthesize the sound of a violin?
* Why traffic jams occur on a highway? Relieve traffic jams on a highway?
* Why it's raining today? Predict whether it will rain tomorrow?
* How people move through a supermarket checkout faster?
* The speed of misinformation in a social network? Mitigate that spread?
* How a fish moves through water? Optimize the body of a fish so it swims faster?
This class is a survey of elementary techniques that aim to help us pose and answer these kinds of questions. The framework involves building an abstract model of a system and then simulating that model on a computer. Together, the techniques form the foundations of computer-based modeling and simulation (M&S).