Weighted approximate Fekete points: Sampling for least-squares polynomial approximation
This work provides a practical and theoretically grounded sampling strategy for improving the stability and accuracy of polynomial approximations, particularly beneficial for computational scientists using least-squares methods.
The paper proposes a weighted greedy method for selecting sample points for least-squares polynomial approximation, which achieves optimal conditioning in 1D and outperforms both randomized and deterministic designs in low and high dimensions.
We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on $L^2$ spaces weighted by a probability density function. Our procedure is a particular weighted version of the approximate Fekete points method, with the weight function chosen as the (inverse) Christoffel function. Our procedure has theoretical advantages: when linear systems with optimal condition number exist, the procedure finds them. In the one-dimensional setting with any density function, our greedy procedure almost always generates optimally-conditioned linear systems. Our method also has practical advantages: our procedure is impartial to compactness of the domain of approximation, and uses only pivoted linear algebraic routines. We show through numerous examples that our sampling design outperforms competing randomized and deterministic designs when the domain is both low and high dimensional.