Sampling for approximation in non-orthogonal bases

Astrid Herremans (KU Leuven, Belgio)

Abstract: Non-orthogonal bases arise naturally in many areas of computational mathematics, including Trefftz methods, approximation on irregular domains and data-driven or learned representations. While these building blocks efficiently capture the behaviour of the target function, they also introduce numerical challenges. In particular, the influence of numerical rounding errors on the approximation error can be non-negligible. In this talk, we investigate the stability and accuracy of least squares approximation in non-orthogonal bases from a sampling-theoretic perspective. We show how the interplay between floating-point arithmetic and sampling strategies affects the quality of approximation. Building on connections with row subsampling problems in numerical linear algebra, we develop an efficient and robust algorithm for discretizing such problems. The results provide both theoretical insights and practical tools for working with non-orthogonal systems. This is joint work with Daan Huybrechs and Ben Adcock.