16.940: Numerical Methods for Stochastic Modeling and Inference
Offered Spring 2010, Spring 2012, Spring 2014, Fall 2015, Fall 2018, Fall 2020
An “advanced introduction” to numerical methods for treating uncertainty in modeling and simulation. Motivation and examples are drawn from science and engineering applications, but our focus is on developing and understanding broadly applicable methodology. The underlying theme of this subject is uncertainty quantification, with a heavy dose of computational statistics. Part one of the subject focuses on uncertainty propagation and assessment, with foundations in Monte Carlo simulation and in approximation theory: Monte Carlo methods; variance reduction; global sensitivity analysis; polynomial approximation; Gaussian process regression and scattered data approximation; stochastic Galerkin and collocation methods; sparse grids, tensor decompositions, and other methods for high-dimensional approximation and integration. Part two of the subject focuses on the interaction of models with observational data, from a largely Bayesian statistical perspective: Bayesian modeling and inference; inverse problems; Markov chain Monte Carlo methods; sequential Monte Carlo methods; nonlinear filtering and data assimilation; model selection; model criticism and validation.
16.09: Statistics and Probability
Offered Spring 2019, 2020
Introduction to probability and statistics, with applications to aerospace engineering. In the probability section of the course, we introduce discrete and continuous random variables; probability distributions (including joint, marginal, and conditional distributions); expectation; transformations of random variables; and limit theorems. In the statistics section of the course, we cover Bayesian parameter inference, prior modeling, and aspects of Bayesian computation; then frequentist estimation theory, confidence intervals, hypothesis testing, ANOVA, and regression. Throughout the course, we emphasize hands-on computation whenever possible.
Offered Fall 2009, 2010, 2011, 2013, 2014, 2015, 2016, 2019
Extends fluid mechanic concepts from Unified Engineering to aerodynamic performance of wings and bodies in sub/supersonic regimes. Subject generally has four components: subsonic potential flows, including source/vortex panel methods; viscous flows, including laminar and turbulent boundary layers; aerodynamics of airfoils and wings, including thin airfoil theory, lifting line theory, and panel method/interacting boundary layer methods; and supersonic and hypersonic airfoil theory. Material may vary somewhat each year depending upon focus of design problem. Elementary usage of MATLAB and other computational tools expected.
16.90: Computational Methods in Aerospace Engineering
Offered Spring 2009, 2010, 2011
Introduction to computational techniques arising in aerospace engineering. Techniques include numerical integration of systems of ordinary differential equations; numerical discretization of partial differential equations; and probabilistic methods for quantifying the impact of variability. Specific emphasis will be given to finite volume methods in fluid mechanics, and energy and finite element methods in structural mechanics.