- Numerical Solutions to PDEs I - Finite difference methods for partial differential equations.
- Numerical Solutions to PDEs II – Spectral methods for partial differential equations.
- Numerical Solutions to PDEs III – Finite element methods, finite volume methods, and discontinuous Galerkin methods.
- Numerical Linear Algebra – QR factorization, least squares, conditioning and stability, systems of equations, iterative methods, eigenvalues.
- Boundary Conditions for Hyperbolic Systems – Boundary conditions for finite different schemes.
- Discontinuous Galerkin Methods – DG for hyperbolic, elliptic, and parabolic systems.
- Numerical Solutions of ODEs – Numerical methods for ordinary differential equations (e.g. Runge-Kutta, Adams/BDF methods).
- Theory of Probability I - Introduction to measure-theoretic probability (e.g., probability spaces, random variables, Markov chains, laws of large numbers, central limit theorems).
- Theory of Probability II – Continuous-time stochastic processes, martingale theorems, Brownian motion, stationary and Gaussian processes.
- Recent Applications of Probability & Statistics – MCMC methods, parameter estimation and EM algorithm, nonparametric statistics, maximum entropy and connections to large deviations.
- Introduction to Large Deviations – Probability of rare events and the most likely way they happen. Applications to risk-sensitive control.
- Stochastic Differential Equations – Stochastic integral with respect to Brownian motion, existence and uniqueness of solutions for SDEs, connections to partial differential equations.
- Linear and Non-linear Optimization – Theory and computation for linear programming and convex optimization.
- Medical Image Analysis – Principles of computer tomography (CT), radiation physics, CT reconstruction, MRI physics and reconstruction, image segmentation.
- Fluid Dynamics – Compressible and incompressible flow, viscous and inviscid flow, vorticity.
- Real Analysis – Metric spaces, measure theory, theory of integration and differentiation.
- Functional Analysis - Banach spaces, Hilbert spaces, spectrum of bounded operators, compact operators.
- Nonlinear Dynamical Systems – Basic theory of ODEs, flows, and maps.