Using Gaussian Eigenfunctions to Solve Boundary Value Problems (Advances in Applied Mathematics and Mechanics, 2021)
Kernel-based methods are popular in computer graphics, machine learning, and statistics, amongother fields; because they do not require meshing of the domain under consideration, higherdimensions and complicated domains can be managed with reasonable effort.  Traditionally, thehigh 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.  Recent advancesin representing Gaussians using eigenfunctions have proven successful at avoiding thisdestabilization in scattered data approximation problems.  This paper will extend thesetechniques to the solution of boundary value problems using collocation.  The method of particularsolutions will also be considered for elliptic problems, using Gaussian eigenfunctions tostably produce an approximate particular solution.