Mathematical Foundations of Complex System Science
Numerical Differential Geometry
Research
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.
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.
Papers:
Thekke Veettil, S.K., Zavalani, G., Hernandez Acosta, U., Sbalzarini, I.F., and Hecht, M. Global polynomial level sets for numerical differential geometry of smooth closed surfaces. SIAM J. Sci. Comput., 45(4):A1995-A2018, 2023.
Zavalani, G., Shehu, E., and Hecht. M. A note on the rate of convergence of integration schemes for closed surfaces Comp. Appl. Math. 43, 92 (2024)., 2024, arXiv:2301.02996.
Zavalani, G., Sander, O., and Hecht, M. High-order integration for regular cubical re- parameterizations of triangulated manifolds reaches super-algebraic approximation rates. arXiv:2311.13909