Courses taught at the University of Tennessee Space Institute (UTSI) and the University of Tennessee, Knoxville (UTK).
Scientific computing can transform engineering systems, particularly when corresponding experiments are costly, hazardous, or otherwise challenging. However, computational simulations entail approximations, so predictions inevitably contain errors. Quantifying such errors, also referred to as uncertainties, is important for rational decision-making. This course introduces theory and methods for uncertainty quantification in computational predictions. The course begins with a review of probability theory and its interpretations. Bayes’ theorem arises as a natural framework to update degrees of confidence in hypotheses about the physical world. The course then covers Bayesian applications to parameter inference and model selection in detail. Bayesian inference can be computationally demanding, so approximations such as Kalman filters will be discussed. Next, the course focuses on prediction by discussing propagation of uncertainties, sensitivity analysis, and validation. Examples of how uncertainties are used in decision-making, such as for experimental design will be presented. The course covers necessary computational methods to solve UQ problems, such as quadrature rules, Monte Carlo methods, and automatic differentiation. The course ends with a discussion of emerging computational paradigms related to uncertainty quantification, such as digital twins and artificial intelligence.