Mathematics Colloquium: Some Remarks on the Construction of Designer Kernels and Their Applications

Positive definite reproducing kernels (or covariance kernels) play a central role in many applications in numerical analysis, spatial statistics, as well as statistical learning.  They appear in methods known, e.g., as radial basis functions, kriging, Gaussian processes, or simply kernel-based methods.  Some kernels, such as Gaussian kernel, multiquadric kernel or the family of Matern kernels, are very popular and are often used in a "one-size-fits-all" general purpose strategy.  In this talk I will emphasize a different approach; that of custom-built designer kernels that have certain desirable built-in properties such as, e.g., periodicity, satisfaction of boundary conditions, or non-stationarity.  After introducing a few different types of designer kernels I will illustrate their use with some examples from data fitting, the numerical solution of PDEs, and electrical power demand forecasting. 

Free