Course web page for Fall 2015 (T. von Petersdorff)

Syllabus for AMSC/CMSC 666

• Approximation Theory (3 weeks)
  • Vector, Matrix and Functional Norms
  • Least Squares, QR, SVD
  • Orthogonal Polynomials
  • Chebyshev Expansions
  • Gaussian Quadrature
  Refs: 1, 2, 3
• Numerical Solution of Initial-Value Problems (3 weeks)
  • Consistency, Stability, and Convergence Analysis
  • One-Step Methods, Runge-Kutta Methods
  • Multistep Methods
  • Methods for Stiff Problems
  • Error Estimation and Adaptivity*
  Refs: 4, 5, 6
• Iterative Methods for Linear Algebraic Systems (3 weeks)
  • Motivation: Boundary-Value Problems for Elliptic PDEs
  • Conjugate Gradient Method
  • Preconditioning
  • Classic Iterative Methods
  • Spectral Convergence Analysis
  • Multigrid Methods*
  • GMRES*
  Refs: 7, 8, 9, 10
• Optimization (3 weeks)
  • Steepest Descent, Newton and Quasi-Newton Methods
  • Line Search Methods and Trust Region Methods
  • Rates of Convergence
  • Nonlinear Conjugate Gradient Method
  • Nonlinear Least Squares Problem* (Gauss-Newton and Levenberg-Marguardt)
  Refs: 9, 10, 11

Books used in previous years:

1. James Demmel, Applied Numerical Linear Algebra, SIAM, 1997
   freely available online via UMD Library
2. Rivlin, T. J., an Introduction to the Approximation of Functions, Dover, 1969
3. Gil, A., Segura, J., Temme, N., Numerical Methods for Special Functions, SIAM, 2007
   chapter on Chebyshev Expansions is freely available online
4. Hairer, E., Norsett, S.P., Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, Second Revised Edition, Springer 1993
5. Hairer, E., Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Second Revised Edition, Springer 1993
6. Deuflhard, P., Bornemann, F., Scientific Computing with Ordinary Differential Equations, Springer, 2002
7. Elman, H., Silvester, D., Wathen, A., Finite Elements and Fast Iterative Solvers, Second Edition, Oxford Science Publications, 2014
8. Morton, K. W., Mayers, D.F., Numerical Solutions of Partial Differential Equations, Second Edition, Cambridge, 2005
9. Nocedal, J., Wright, S., Numerical Optimization, Second Edition, Springer 2006
   freely available online via UMD Library
10. Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations, SIAM 1995
11. Kelley, C. T., Iterative Methods for Optimization, SIAM 1999
    freely available online via UMD Library