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RESEARCH in APPROXIMATION THEORY

I work in approximation theory with a particular focus on multivariate numerical integration and multivariate meshless approximation with radial basis functions. In recent years a particular focus is on approximation from noisy data.

Currently my research focusses on meshless approximation with radial basis functions on the sphere. Interpolation and approximation with radial basis functions (or more general kernel functions) on manifolds is a highly active research area. Apart from purely approximation theoretical problems, radial basis functions (or more general kernel functions) have found important applications in the context of inverse problems and partial differential equations (in the latter context as an alternative to finite elements) and in machine learning.

• Smoothing approximation of Functions (on the Sphere) from Noisy Scattered Data with Radial Basis Functions: In “smoothing approximation” (also called “penalised least squares”) data is approximated (and not interpolated), and the balance between data fitting and the smoothness of the approximating function is controlled by the so-called smoothing parameter. For noisy data, interpolation cannot successfully be used, and smoothing approximation is the appropriate approximation technique. I am particularly interested in error estimates and strategies for choosing the smoothing parameter for smoothing approximation with radial basis functions or with the “hybrid approximation scheme” (a polynomial part\ plus a radial basis function approximation).

My previous research has covered the following topics:

• local numerical integration rules for the sphere and their theoretical properties: Rules for numerical integration over local subsets of the sphere are important for applications in which the data is only given on a subset of the earth's surface (which is in first approximation a sphere). My particular interest was focussed on error estimates in special classes of function spaces (e.g. Sobolev spaces).

• local interpolation with radial basis functions on the sphere: I proved local error estimates for the approximation order in Sobolev spaces.
• energy of point sets on the sphere: This is more a theoretical research topic, concerned with estimating the Coulomb energy and its generalisations for point sets on the sphere with “good” properties.

• hyperinterpolation on the sphere: Hyperinterpolation is a kind of discretised orthogonal projection on the space of polynomials of degree less or equal to n.
• numerical integration on the sphere: In particular, I have derived error estimates for classes of numerical integration rules with “good” properties in a Sobolev space setting. Upper and lower bounds for the error of the same order of convergence guarantee that the error estimates are order optimal.

• modelling of the earth's gravitational potential from satellite data: This was the topic of my doctoral thesis. Modelling the earth's gravitational potential with high accuracy is still one of geodesy's big challenges. This modelling problem can be formulated as a pseudodifferential operator equation and is an exponentially ill-posed / inverse problem, since the gravitational potential is attenuated with increasing distance from the earth's surface. The research involved regularisation methods, approximation methods based on spherical geometries, as well as numerical methods, in particular domain decomposition techniques.

• error estimates for the growth behaviour of solutions to the $\overline{\partial}$-equation on Pseudo-Siegel-Domains: This research in analysis of several complex variables was the topic of my Diplomarbeit (“Diplom” Thesis).