### CAS Seminar

## "On preconditioners for generalized sparse grid generating systems"

** Date: ** |
Wed, February 22, 2012 |

** Time: ** |
9:45 |

** Place: ** |
Research I Seminar Room |

**Abstract:** Tensor products of one-dimensional multilevel systems can be used to
represent multivariate functions. Then, by a proper truncation of the
resulting series expansion, we can construct problem-dependent sparse
grids, which allow us to efficiently approximate higher-dimensional
problems for various norms and smoothness classes.
We discuss additive Schwarz preconditioners for the corresponding
systems of linear equations. The problem of finding the optimal diagonal
scaling for the generating system subspaces can be solved by means of
Linear Programming. For e.g. H^{1}-elliptic problems, an optimally
scaled regular sparse grid generating system exhibits a condition number
of the order O(k^{d-2}) for level k in d dimensions.
This is suboptimal compared to the O(1) condition numbers realized by
prewavelet discretizations that directly rely on multiresolution norm
equivalences. However, we will discuss an approach that likewise
realizes O(1) condition numbers in the generating system without
specifically discretising the detail spaces via more complicated basis
functions.
This is joint work with M. Griebel (University of Bonn) and P. Oswald
(Jacobs University Bremen).