Radial Basis Functions for Solving PDEs: Advances Over The Last 25 Years

So I'm going to call the Euclidean distance. Here, R is going to be centered somewhere and I'm going to evaluate it someplace else. So here, I have a chart with a spatial dimension and let's put the center at zero just for simplicity, okay? If I put R, if I put the center at zero, then in 1D, it's just the absolute derivative of X, which is a radial basis function. I notice that if I go to 2D, I just take this absolute value of X and rotate it around, and the formula becomes X squared plus Y squared square root, the Euclidean norm. Notice that now when I go to 3D, all I have to add (and of course, I couldn't graph it unless I did an iso surface) is the Z component. If I went to 4D and let's say called it the 4DT, it would be plus T squared. So, one thing to note is that it's a very simple method and all RBFs are built on this concept. All RBFs are built on the Euclidean distance. The argument of the RBF, notice, is independent of the coordinate system. Yes, I chose Cartesian here, but I could have chosen a spherical coordinate system or any other coordinate system to measure the Euclidean distance between two points. Also, notice that the algebraic complexity does not really increase with the dimension. That's why I can take a 2D code and easily transition to a 3D code without much trouble. So, like I said, all RBFs are built on this. Now, RBFs come in two different flavors. One is the infinitely differentiable RBFs, which give you special convergence.