Mathematik-Gebäude, Seminarraum M614/616

Multilevel methods have been one class of powerful iterative methods for solving a linear system of equations. Some examples are multigrid and domain decomposition methods. They are often referred to as subspace correction methods, because of error reduction processes that are carried out in a subset of the solution subspace. Multigrid, for instance, employs coarse-grid correction performed at a coarse grid to reduce the components of error that cannot be effectively reduced by a fixed-point iterative method on the fine (actual) computational grid. These components typically correspond to the slow varying components of the error function in the Fourier space. The fast varying, highly oscillatory components can however be easily handled by a fixed-point iteration. An efficient multigrid method is obtained via an effective interplay between the reduction of error on the fine grid - called smoothing - and the coarse-grid correction.
In this seminar, another method to perform a multilevel iteration is presented. Different from multigrid, the method utilizes a Krylov method as the basis in the error reduction process. Krylov methods however do not have any smoothing property like the usual multigrid smoothers (Jacobi or Gauss-Seidel), and hence the fast and low varying components of error are not distinguishable. As a consequence, a simple replacement of a smoother by a Krylov method will not lead to a useful multilevel method.
We shall show how to build such an iteration properly, which leads to a method that is as effective as an efficient multigrid, and requires almost the same setting as multigrid. In fact, the method can achieve a fast convergence typical for multigrid for problems that standard multigrid methods do not even work. Numerical results for some class of problems including the Helmholtz equation, the biharmonic equation, and involving singular matrices will be presented.
Some parts of this talk are a join work with Reinhard Nabben (TU Berlin) and/or Kees Vuik (TU Delft)

Yogi Ahmad Erlangga

Alfaisal University, Riyadh