21671 - Computational Linear Algebra
This is a survey of methods in computational linear algebra. Topics covered in this course focus around algorithms for solving (dense or large and sparse) linear systems. Regularization and underdetermined systems will be discussed in detail. Rather than assuming prior knowledge in numerical analysis or matrix theory, we will introduce standard methods or results when needed. In this way, much of the material is self-contained. Theoretical and experimental results will be covered accordingly, with an emphasis on cost, stability, and convergence.
The main computational topics include:
- QR (Gram-Schmidt, Modified Gram-Schmidt, Householder reflections, Givens rotations).
- Singular value decomposition and its application to reduced order modeling.
- Least squares problems (over-determined), regularization, and its application to data-driven inverse problems.
- Cholesky, incomplete methods, and its application to scientific computing.
- Algebraic eigenvalue problems.
- Krylov Methods (CG, etc.) and preconditioning.
- Least Squares problem (under-determined), sparsity, subspace approximations, LASSO, basis pursuit, with applications to model selection and parameter estimation.