Randomized weakly admissible meshes

TL;DR

This paper extends the concept of weakly admissible meshes to hierarchical subspaces beyond polynomials and demonstrates that random sampling can effectively generate such meshes with controlled growth and comparability.

Contribution

It introduces a generalized notion of WAMs for hierarchical subspaces and proves that random sampling can produce WAMs and AMs with quantifiable growth properties.

Findings

WAMs and AMs can be generated by random sampling.

Concrete estimates for growth of meshes and comparability constants.

Generalization to non-polynomial hierarchical subspaces.

Abstract

A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constants must grow in a controlled manner. In this paper, we generalize the notion of a WAM to a hierarchical subspaces of not necessarily polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAM's. Our main results show that WAM's and their stronger variant, admissible meshes (AM's), can be generated by random sampling, and our analysis provides concrete estimates for growth of both the meshes and the discrete-continuum comparability constants.