Coursework

 

Numerical Analysis

  • 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).

Probability

  • 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.

Miscellaneous

  • 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.