2020/10/21 by Albert Cohen, Cohen, Albert, Matthieu Dolbeault +1 · 2 citations
Mathematics · #41A10 #41A65 #62E17 #65C50 #93E24 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2010.11040
openalex publication_date 2020/10/21 · openalex created_date 2020/10/29 · openalex updated_date 2026/07/28
We consider the problem of reconstructing an unknown function u∈ L2(D,μ) from its evaluations at given sampling points x1,…,xm∈ D, where D⊂ \mathbb Rd is a general domain and μ a probability measure. The approximation is picked from a linear space Vn of interest where n=dim(Vn). 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 Vn 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 Vn in L2(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 \widetilde 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 Vn 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 Vn of multivariate algebraic polynomials, and for general domains D.