Syllabus
1. Introductory remarks. Formulation of saddle-point problem.
2. Applications leading to saddle-point problems. Augmented systems in the least squares problems. Saddle point problems from the discretization of partial differential equations with constraints. Kuhn-Karush-Tucker (KKT) systems in interior-point methods.
3. Properties of saddle point matrices. The inverse of a saddle-point matrices. Spectral properties of saddle-point matrices.
4. Solution approaches for saddle-point problems. Schur complement reduction. Null-space projection method.
5. Direct methods for symmetric indefinite systems. Direct solution of saddle-point methods.
6. Iterative solution of saddle-point problems. Stationary iteration methods. /Krylov subspace methods. Preconditioned Krylov subspace methods.
7. Saddle-point preconditioners.
8. Implementation and numerical behavior of saddle-point solvers.
9. Polluted undeground water flow modelling in porous media.
Annotation
The course is devoted to the solution of large linear saddle-point systems that arise in a wide variety of applications in computational science and engineering.
The aim is to discuss particular properties of such linear systems as well as a large selection of algebraic methods for their solution with emphasis on iterative methods and preconditioning.