TY - JOUR
T1 - Using Gaussian Eigenfunctions to Solve Boundary Value Problems
AU - McCourt , Michael
JO - Advances in Applied Mathematics and Mechanics
VL - 4
SP - 569
EP - 594
PY - 2013
DA - 2013/08
SN - 5
DO - http://doi.org/10.4208/aamm.13-13S08
UR - https://global-sci.org/intro/article_detail/aamm/86.html
KW - Meshless method, method of particular solutions, boundary value problem.
AB - <p style="text-align: justify;">Kernel-based methods are popular in computer graphics, machine learning, and statistics, among
other fields; because they do not require meshing of the domain under consideration, higher
dimensions and complicated domains can be managed with reasonable effort. &nbsp;Traditionally, the
high order of accuracy associated with these methods has been tempered by ill-conditioning,
which arises when highly smooth kernels are used to conduct the approximation. &nbsp;Recent advances
in representing Gaussians using eigenfunctions have proven successful at avoiding this
destabilization in scattered data approximation problems. &nbsp;This paper will extend these
techniques to the solution of boundary value problems using collocation. &nbsp;The method of particular
solutions will also be considered for elliptic problems, using Gaussian eigenfunctions to
stably produce an approximate particular solution.</p>