@Article{AAMM-5-4,
author = {Michael, McCourt},
title = {Using Gaussian Eigenfunctions to Solve Boundary Value Problems},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2013},
volume = {5},
number = {4},
pages = {569--594},
abstract = {<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>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.13-13S08},
url = {https://global-sci.com/article/73507/using-gaussian-eigenfunctions-to-solve-boundary-value-problems}
}