Linear solvers for industrial problems

The aim of this course is to present the iterative methods used to approximate the solutions of large linear systems.
Most numerical simulations require the resolution of large linear systems. In particular, we will consider linear systems arising from problems in structural or fluid mechanics.
The cost of solving these systems often represents a significant proportion of the overall computational cost. What's more, these systems are often poorly conditioned, which means that even the most basic methods fail.
The aim is to understand the principles of these methods and to be able to choose the most appropriate one for a given problem.
Outline:

- iterative methods: general principle, relaxation methods (including Jacobi and Gauss-Seidel)
- gradient methods (optimal step gradient, conjugate gradient)
- hollow systems and precondtioning
- least squares (normal equations, QR method)
- introduction to Krylov methods (projection view, examples, principle, mention of GMRES, BiCG, etc.)

- Basic concepts of linear algebra.
- Direct methods: LU (with and without permutations), Cholesky.

- iterative methods: general principle, relaxation methods (including Jacobi and Gauss-Seidel)
- gradient methods (optimal step gradient, conjugate gradient)
- hollow systems and precondtioning
- least squares (normal equations, QR method)
- introduction to Krylov methods (projection view, examples, principle, mention of GMRES, BiCG, etc.)