Optimal sampling and Christoffel functions on general domains

TL;DR

This paper develops practical sampling strategies for function approximation on general domains using Christoffel functions, ensuring near-optimal sampling and approximation with feasible offline computations.

Contribution

It introduces feasible, offline computable sampling measures based on upper bounds of Christoffel functions, extending optimal sampling methods to arbitrary domains.

Findings

Near best approximation achieved with perturbed sampling measures.

Proposed strategies are effective for multivariate polynomial spaces.

Multilevel approaches preserve optimal sampling efficiency.

Abstract

We consider the problem of reconstructing an unknown function $u\in L^2(D,\mu)$ from its evaluations at given sampling points $x^1,\dots,x^m\in D$, where $D\subset \mathbb R^d$ is a general domain and $\mu$ a probability measure. The approximation is picked from a linear space $V_n$ of interest where $n=\dim(V_n)$. Recent results have revealed that certain weighted least-squares methods achieve near best approximation with a sampling budget $m$ that is proportional to $n$, up to a logarithmic factor $\ln(2n/\varepsilon)$, where $\varepsilon>0$ is a probability of failure. The sampling points should be picked at random according to a well-chosen probability measure $\sigma$ whose density is given by the inverse Christoffel function that depends both on $V_n$ and $\mu$. While this approach is greatly facilitated when $D$ and $\mu$ have tensor product structure, it becomes problematic for domains $D$ with arbitrary geometry since the optimal measure depends on an orthonormal basis of $V_n$ in $L^2(D,\mu)$ which is not explicitly given, even for simple polynomial spaces. Therefore sampling according to this measure is not practically feasible. In this paper, we discuss practical sampling strategies, which amount to using a perturbed measure $\widetilde \sigma$ that can be computed in an offline stage, not involving the measurement of $u$. We show that near best approximation is attained by the resulting weighted least-squares method at near-optimal sampling budget and we discuss multilevel approaches that preserve optimality of the cumulated sampling budget when the spaces $V_n$ are iteratively enriched. These strategies rely on the knowledge of a-priori upper bounds on the inverse Christoffel function. We establish such bounds for spaces $V_n$ of multivariate algebraic polynomials, and for general domains $D$.