This course will present a general overview of Numerical Linear Algebra
(NLA), including dense linear algebra, a brief intro to sparse direct
methods, in depth coverage of Krylov methods, preconditioning (ILU and
Domain Decomposition, API), Krylov methods for eigenvalue computation,
large scale matrix equations (Lyapunov, Sylvester, and possibly Riccati
equations). The course will also discuss useful criteria to choose the
most adequate technique depending on the problem.
The breakdown will be:
Brief
course outline, brief review of Dense LA, Sensitivity and Condition,
assessment of accuracy of solution, intro to Sparse Direct methods
Additional
material on Sparse Direct Methods, introduce basic single vector
iterative schemes, introduce Krylov projection as a natural extension
to extract more info from power sequence, discuss convergence and
optimality properties related to polynomial spaces
A little
matrix theory (Schur Decomposition, characteristic and minimal
polynomials, Cayley Hamilton, non-normality), Effects of non-normality,
motivation for pre-conditioning, Pre-conditioning techniques (ILU,
Domain Decomp, Approx. Inverse)
Eigenvalue
Computation: The power method and inverse iteration, brief intro
to implicitly shifted QR, Arnoldi's method, implicit restarting, ARPACK
and EIGS Shift-Invert spectral transformation, the generalized
eigenvalue problem
Jacobi-Davidson
type methods, pre-conditioning for eigenvalue computation. Large scale
matrix equations: Lyapunov and Sylvester equations
