Locating good points for multivariate polynomial approximation

Locating good points for multivariate polynomial approximation, in particular interpolation, is an open challenging problem, even in standard domains. One set of points that is always good, in theory, is the so-called Fekete points. They are defined to be those points that maximize the (absolute value of the) Vandermonde determinant on the given compact set. However, these are known analytically in only a few instances (the interval and the complex circle for univariate interpolation, the cube for tensor product interpolation), and are very difficult to compute, requiring an expensive and numerically challenging nonlinear multivariate optimization. Recently, a new insight has been given by the theory of “Admissible Meshes” of Calvi and Levenberg [9], which are nearly optimal for least-squares approximation and contain interpolation sets (Discrete Extremal Sets) nearly as good as Fekete points of the domain. Such sets, termed Approximate Fekete Points and Discrete Leja Points, are computed using only basic tools of numerical linear algebra, namely QR and LU factorizations of Vandermonde matrices. Admissible Meshes and Discrete Extremal Sets allow us to replace a continuous compact set by a discrete version, that is “just as good” for all practical purposes.