Numerical solution of saddle point problems
Period of duration of course
A strong background in numerical linear algebra is expected. Basic knowledge of numerical methods for PDEs and numerical optimization will also be useful.
The course is intended prinarily for students at the PhD level.
Presentation of the course, introduction to saddle point problems, examples, main assumptions, solvability conditions, algebraic and spectral properties of saddle-point matrices. Overview of solution algorithms, direct vs. iterative methods, Schur complement reduction, null space methods, augmented Lagrangian formulation, factorization of saddle-point matrices, remarks on sparse direct solvers (fill-in, elimination graph, reorderings, minimum degree, band-reducing heuristics). Stationary iterations (Arrow-Hurwicz, Uzawa, inexact and preconditioned variants), Hestenes' Method of Multipliers, introduction to Krylov subspace methods. More on Krylov methods: convergence analysis; flexible, inexact, and preconditioned variants, asymptotic convergence rates for sequences of problems of increasing size, field-of-values analysis. Block preconditioners for saddle point problems (block diagonal and block triangular, constraint preconditioning), spectral analysis, exact vs. inexact, augmented Lagrangian-based. Block diagonal/triangular preconditioners for the Stokes and Navier-Stokes equations; Hermitian-skew Hermitian (HSS) and Modified HSS (MHSS) preconditioning. Dimensional Splitting and Relaxed Dimensional Factorization preconditioning for the Stokes and Navier-Stokes equations. Double/multiple saddle point problems, the coupled Darcy-Stokes problem.
The students will be introdiced to the state-of-the art in the numerical solution of large-scale problems of saddle point type.
References will be given during the course.