| Description |
New developments in many applications, such as weather forecasting, airplane
design, tomographic problems, analysis of the stability of structures, design
of chips and other electrical circuits, etc, rely on numerical simulations.
Such simulations require the numerical solution of linear systems or of
eigenvalue problems. The matrices involved are sparse and high dimensional (1
billion is not acceptional). The solution of these linear problems are
normally by far the most time-consuming part of the whole simulation.
Therefore, the development of new solution algorithms is extremely important
and forms a very active area of research.
The course will give an overview of the modern solution algorithms for linear
systems and eigenvalue problems. Modern approaches rely on schemes that
improve approximate solutions iteratively. The course will start with a
review of basic concepts from linear algebra, after which solution methods
for dense systems (LU, QR and Choleski decomposition) will be discussed.
Next, the basic ideas for iterative solution methods of sparse systems will
be explained, which will lead to the main topic of the course: modern Krylov
subspace methods. The main ideas of these methods will be explained and how
they lead to efficient solvers. Solution algorithms for linear systems that
will be discussed include CG, GMRES, CGS, Bi-CGSTAB, Bi-CGSTAB(l) and IDR(s).
Furthermore several preconditioning and deflation techniques will be
explained. For large scale eigenvalue problems the Lanczos methods, Arnoldi's
method and the Jacobi-Davidson method will be treated. |
