Abstract:
Sparse grids allow to circumvent the so-called curse of dimension for numerical tasks such as solving partial differential equations or numerical quadrature. They generalize the classical approaches of Smolyak and Babenko based on tensor products of spaces to a corresponding tensor product construction of single basis functions, thus allowing for an explicit discretization on these grids as well as for a simple adaptive mesh refinement. Furthermore, with the help of hierarchical Lagrangian interpolation, the underlying one-dimensional hierarchical bases - originally piecewise linear - can be generalized to hierarchical bases of piecewise arbitrary polynomial degree.
The talk, first, recapitulates the principles and properties of sparse grids in general and higher order approaches in particular and, then, focuses on recent fields of application such as multivariate numerical quadrature in the moderately or highly dimensional case.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.