Two to Six Dimensional Numerical Solutions to the Poisson Eigenvalue Partial Differential Equation Using Generalized Multiquadrics

We solve numerically an eigenvalue elliptic partial differential equation (PDE) ranging from two to six dimensions using the generalized multiquadric (GMQ) radial basis functions (RBFs). Two discretization methods are employed. The first method is similar to the classic mesh-based discretization method requiring n centers per dimension or a total n^d points. The second method is based upon n randomly generated points in ${？}^d$ requiring far fewer data centers than the classic mesh method. Instead of having a crisp boundary, we form a “fuzzy” boundary. Using the analytic or numerical inverse interior and boundary operators, we find the local and global minima and maxima to cull discretization points. We also find that the GMQ-RBF “flatness” can be controlled by increasing the GMQ exponential, β. We perform a search to find the smallest root mean squared error (RMSE) by varying the exponent, the maximum, the minimum range of the GMQ shape parameter, and boundary influence, with the exponential having the most influence. Because the GMQ-RBFs are essentially nonlinear, it follows that the starting point of the simple search influences the end result. The optimal algorithm for high dimensional PDEs is still the subject of much research and may wait for the common place availability of massively parallel quantum computers for even higher dimensional PDEs and integral equations (IEs).