The class presents and ties together important notions of computational mathematics for scientists and engineers. It sheds a second light on linear algebra and differential equations. The focus is not on any particular application -- many will be covered from mechanical to electrical systems, graphs, networks, etc. -- but rather on the common mathematical framework that underlies most of them. Both modeling and computation will be covered. The class is suitable for master students, advanced undergraduates, or anyone interested in building a foundation in CSE.

Topics:

• Part 1: Applied Linear Algebra
  • Difference matrices and boundary conditions
  • Elimination, inverses, eigenvalues
  • Positive definite matrices
  • Fundamental subspaces and matrix decompositions (QR, SVD)
• Part 2: A Framework for Applied Mathematics
  • Stiffness matrices and oscillations
  • Least-squares
  • Kirchhoff's laws, graphs, etc.
• Part 3: Boundary-Value Problems
  • Gradient, divergence, Poisson's equation
  • Splines and finite elements
  • Finite differences and fast iterative methods
• Part 4: Fourier Series and Integrals
  • Periodic functions
  • The discrete Fourier transform and the FFT
  • Convolution and deconvolution