"Sparse grid methods for problems in two and three dimensions"23. Mai 2016
23.05.2016 17:00 Uhr
(bis 18 Uhr)
FernUniversität, TGZ-Gebäude, Universitätsstr. 33, 58097 Hagen, 4. OG, Raum E08
Lehrgebiet Numerische Mathematik
Dr. Niall Madden (NUI Galway, Irland)
Prof. Dr. Torsten Linß
(Tel.: 02331 - 9872683)
Finite element methods are a class of algorithms used to solve partial differential equations. Like most other standard methods, they suffer from the infamous "curse of dimensionality": for fixed accuracy, the computational cost grows exponentially with the number of dimensions. Sparse grid methods are a family of schemes for which the cost grows only slowly with the number of dimensions. These methods have been discovered, and rediscovered, over the past 30 years or so, with many applications including function approximation, quadrature, and numerical solution of PDEs.
For PDEs, popular variants include the celebrated hierarchical basis approach of Zenger (1991), which is highly efficient, and the so-called two-scale method developed by Zhou and co-authors, which is less efficient, but (arguably) much simpler in both theory and practice.
In this talk, I will explain some of the background to the hierarchical method, and its relationship with multigrid linear solvers.
I will then outline the two-scale method, and describe the results of recent work on our efforts to combine its simplicity with the optimality of the hierarchical approach.
Our motivation will be the numerical solution of singularly perturbed problems (SPPs). However, the focus will be on the general applicability of the schemes, rather than technicalities associated with SPPs.
This is joint work with Stephen Russell. For more information, see our pre-print at http://arxiv.org/abs/1511.07193 which also has links to MATLAB/Octave programs for exploring these methods.