Seminario di Analisi Numerica: Solving PDEs on surfaces with radial basis functions: from global to local methods

Giovedì 10 Aprile 2014, ore 15:30 - Aula 2BC30 - Grady B. Wright

ARGOMENTI: Seminari Dottorato

Seminario di Analisi Numerica

Giovedì 10 Aprile 2014 alle ore 15:30 in aula 2BC30, Grady B. Wright (Department of Mathematics - Boise State University - USA) terrà un seminario dal titolo "Solving PDEs on surfaces with radial basis functions: from global to local methods".

Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using "local" techniques, such as RBF generated finite differences (RBF-FD).
In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations. The method is quite flexible, only requiring a set of "scattered" nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems.
This talk represents joint work with Ed Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).