Mathematical Foundations of Complex System Science

We have developed a computational scheme

Numerical Differential Geometry

We have developed a computational scheme that derives a global polynomial level set parametrisation (GPLS) for smooth closed surfaces from a regular surface-point set.

The GPLS enables to approximate a broad class of smooth surfaces by affine algebraic varieties. Posterior, differential-geometric quantities like mean and Gauss curvature can be efficiently and accurately computed. Even 4th-order terms such as the Laplacian of mean curvature are approximated with high precision. The accuracy performance results in a gain of computational efficiency, significantly reducing the number of surface points required compared to classic alternatives that rely on surface meshes or embedding grids. Fig.1 shows a GPLS fit of the prominent Stanford bunny.

Visiting Professor Dr Michael Hecht

We further develop the technique and apply it to a vast class of computational tasks, arising in numerical differential geometry. This includes a High Order Surface Quadrature (HOSQ) technique, capable for integrating highly varying surface integrals, Fig2.

Publications

Global polynomial level sets for numerical differential geometry of smooth closed surfaces

A note on the rate of convergence of integration schemes for closed surfaces

High-order integration for regular cubical re- parameterizations of triangulated manifolds reaches super-algebraic approximation rates