Optimal sampling and Christoffel functions on general domains
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We consider the problem of reconstructing an unknown function u∈ L^2(D,μ) from its evaluations at given sampling points x^1,…,x^m∈ D, where D⊂ℝ^d is a general domain and μ a probability measure. The approximation is picked from a linear space V_n of interest where n=(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/ε), where ε>0 is a probability of failure. The sampling points should be picked at random according to a well-chosen probability measure σ whose density is given by the inverse Christoffel function that depends both on V_n and μ. While this approach is greatly facilitated when D and μ 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,μ) 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 σ 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.
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