Igor Tominec
Department of Information
Uppsala University
In this seminar I will make an overview of Radial basis function methods for solving partial differential equations. At first I will discuss the parent method to which the literature often refers as RBF-Direct. When RBF-Direct is applied to a PDE it allows spectral convergence under the node refinement.
Unfortunately the resulting linear system is full which leads to high computational costs when it is solved. In the last two decades researchers have made advances in RBF methods which instead produce sparse linear systems (RBF-FD, RBF-PUM,..). One of them is called RBF-Partition of Unity method (RBF-PUM) on which we put the main focus. I will show how the differential operators are discretized, how the boundary conditions are imposed and which error bounds can be used for analysis. At the end I will present some research challenges when applying RBF-PUM to the PDEs that include nonlinear differential operators. The challenges are motivated by an attempt to solve a PDE model of the diaphragm which is the main muscle of the respiratory cycle.
Some reading material:
E. Larsson, B. Fornberg, A numerical study of some radial basis function based solution methods for elliptic PDEs
(On RBF-Direct)
V. Shcherbakov, Localised Radial Basis Function Methods for Partial Differential Equations
(On RBF-PUM)
S. Milovanovic, L. von Sydow - Radial Basis Function generated Finite Differences for option pricing problems
(On RBF-FD)
S.K. Powers, M.P. Wiggs, K.J. Sollanek, A.J. Smuder, Ventilator-induced diaphragm dysfunction: cause and effect
(On the diaphragm)