DR. GODELA SCHERER
Departamento de Cómputo Científico y Estadística
Universidad Simón Bolívar, Venezuela
and Dept. of Mathematics,
Reading University, U.K.
godela@compuserve.com

Least Squares Data Fitting with Applications

Aim: The course is a survey and analysis of the basic techniques for the numerical solution of linear and non-linear least squares problems, as well as an introduction to the treatment of large and ill conditioned problems. The theory will be illustrated with examples from environment, geophysics, and engineering applications.

Contents:
1. Linear least squares (2 lectures).

• Brief review of the basic mathematical aspects and numerical algorithms for least squares: singular value decomposition (SVD), QR and Lanczos methods.
  
• Ill conditioning: characterization and examples. Numerical rank. Regularization methods: TSVD and projection methods. Parameter selection.

2. Non-linear least squares (3 lectures)

• Formulation. Survey of the classical methods: Gauss-Newton, trust region, and Newton-type methods.
• Separable non-linear problems, variable projection algorithm.
• Methods for large and for ill conditioned problems, block decomposition and Tikhonov regularization.

Recommended background: Basic knowledge of numerical linear algebra and computing skills.

Optional Evaluation: A take-home test.

Course Material: There will be course notes.
Good reference for parts of the course are the following books:
[1] Björck, Å.; "Numerical Methods for Least Squares Problems", SIAM 1996.
[2] Hansen, P.H.; "Rank deficient and Discrete Ill-posed Problems", SIAM 1998.
[3] Nocedal J. and Wright S.; "Numerical Optimization", Springer Verlag, 1999.
[4] Golub, G. and Pereyra, V.; "Separable nonlinear least squares: the variable projection method and its applications." Topical Review, Inverse Problems 19:R1-R26 (2003).