CASUS Hands-on Software Seminar, Michael Hecht, MPI-CBG
Location: CASUS Lecture Room, Görlitz
Across disciplines, current challenges in sciences ask for a relief from computational limitations caused by the complexity of multi-dimensional systems. Recently, we found a notion of unisolvent interpolation nodes P _ Rm. These nodes uniquely determine a polynomial Q interpolating a function f : Rm ! R on P, i.e., f(p) = Q(p) for all p 2 P. Due to this notion we generalized the 1D Newton & Lagrange interpolation schemes (developed 300 years ago) to arbitrary dimensions. As it turns out, this discovery allows to lift the curse of dimensionality from classical computational schemes as multivariate interpolation/ regression, numerical integration, level-set methods, global optimization, fast Fourier transforms, ODE & PDE solvers etc.
We will present the essential theoretical aspects and sketch the resulting approaches for the mentioned applications. As a first insight, below, approximation errors for classical interpolation approaches are benchmarked with our new algorithm MIP.
Michael is a mathematician with various interests. In his PhD he studied topological invariants of Hamiltonian dynamics by linking infinite dimensional Morse theory and finite dimensional Floer theory. As a Postdoc he had the opportunity to work in the bioinformatics group of Peter F. Stadler in Leipzig. Thereby, he provided a feasible solution of the classical NP-complete Feed Back Arc Set Problem. Since 2015 Michael is a member of the Mosaic group headed by Ivo F. Sbalzarini. Michael discovered an characterisation of multi-dimensional unisolvent interpolation nodes allowing to generalise the classical 1D-Newton & Lagrange Interpolation schemes to arbitrary dimensions. Current research aims to lift the curse of dimensionality from multivariate interpolation/regression, numerical integration, global optimisation, fast Fourier transforms and level-set methods by incorporating this discovery.