Kernel-based Multilevel Methods for High-dimensional Approximation Problems on Sparse Grids - Derivation, Analysis and Applications to Uncertainty Quantification

• Applicants: Professor Dr. Christian Rieger; Professor Dr. Holger Wendland
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 452806809

In this project, kernel-based multilevel methods for high-dimensional approximation problems will be developed, analysed and implemented. Kernel-based methods have, compared to other methods, the advantage that they do not require the given data to have a particular structure. On the contrary, they work for arbitrarily scattered data. We will in particular study tensor products of lower dimensional scattered points and the sparsification of such tensor products, which leads to generalized sparse grids. Moreover, by an appropriate choice of the kernel, it is possible to easily build high-order approximation spaces. Multilevel methods have, in particular for large data sets, the additional advantage that adaptive versions, versions for data compression and efficient implementations are possible.High-dimensional approximation problems appear naturally in the context of parametric partial differential equations, which often arise when modeling complex systems. Often, the parameters are unknown and are hence modeled stochastically. This means that the parameters and, consequently, also the solution of the partial differential equation are functions on a probability space. Using a series expansion with a subsequent, finite dimensional approximation, this eventually leads to a model where the parameters come from a high-dimensional space.Quantities of interest of such a model can often be described as functionals on the space of parametrized solutions. The approximate computation of such a quantity of interest requires therefore a high-dimensional reconstruction process with input being pairs of parameters and numerically computed solutions of the partial differential equation.The methods will be developed for arbitrary domains with special emphasis on tensor products of intervals and low-dimensional spheres. It is the goal to develop an a priori error analysis for such high-dimensional approximation schemes, which contains all relevant discretization parameters. Moreover, it will be investigated how such methods can be used to determine a Karhunen-Loeve expansion numerically and whether they can be applied in the context of the Design of Experiments or Reduced Order Modeling. Finally, these methods will be applied to specific examples from Uncertainty Quantification.

• DFG Programme: Research Grants