Greedy kernel approximation

Prof. Dr. Bernard Haasdonk

Dr. Gabriele Santin

01/11/2015

30/09/2019

This project deals with greedy algorithms for kernel-based approximation. The goal is to use data samples to construct sparse surrogate models of possibly high dimensional functions.

Kernel methods are flexible and powerful techniques which allow for the reconstruction of functions on high-dimensional spaces from scattered samples. As opposite to other methods, such techniques scale well with the input space dimension and are capable of approximating vector-valued functions.

Among other techniques, we are interested in greedy algorithms. They are designed to incrementally select relevant samples from a given dataset, and to compute sparse approximants based on these data. This incremental construction produces approximants which can be shown to be, in several cases, as good as the full model, while being very fast to evaluate.

These features make greedy kernel algorithms well suited for the construction of surrogate models which can be used to replace expensive components in complex simulations, such as coupling of submodels or nonlinear relations.

In this project we address theoretical aspects and in particular convergence properties of the algorithms, as well as efficient implementation and application to real-world simulations scenarios.

Within the setting of the present project, we employ these algorithms as surrogate models in different simulation scenarios:

• We are collaborating with Tobias Köppl (Department of Hydromechanics and Modelling of Hydrosystems) on the efficient simulation of the bloodflow in a vascular network.
• We are participating in a project with Markus Köppel, Ilja Kröker (IANS), Sergey Oladyshkin (Institute for Modelling Hydraulic and Environmental Systems), Fabian Franzelin (Institute for Parallel and Distributed Systems) to compare different model reduction techniques for Uncertainty Quantification.