Optimal Sampling and Christoffel Functions on General Domains

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 $$\mu $$ probability measure. The approximation picked linear space $$V_n$$ interest $$n=\dim (V_n)$$ . Recent results (Cohen Migliorati in SMAI J Comput Math 3:181–203, 2017, Doostan Hampton Methods Appl Mech Eng 290:73–97, 2015, Jakeman et al. 86:1913–1947, 2017) have revealed that certain weighted least-squares methods achieve near best (or instance optimal) with budget m proportional to n, up logarithmic factor $$\ln (2n/\varepsilon $$\varepsilon >0$$ failure. should be random according well-chosen measure $$\sigma whose density by inverse Christoffel depends both on While this approach greatly facilitated when D tensor product structure, it becomes problematic for domains arbitrary geometry since optimal orthonormal basis $$L^2(D,\mu which not explicitly given, even simple polynomial spaces. Therefore, practically feasible. One computational solution recently proposed Adcock Huybrechs (Approximating smooth, multivariate functions irregular domains, forum mathematics, sigma, Cambridge University Press, Cambridge, 2020) relies using restrictions defined simpler bounding original turn giving $$m\sim n$$ In paper, we discuss practical strategies, amounts perturbed $$\widetilde{\sigma }$$ can computed offline stage, involving measurement u, as Cardenas (SIAM Data Sci 2:607–630, (IMA Numer Anal, 2020. https://doi.org/10.1093/imanum/draa023 ). show attained resulting method near-optimal multilevel approaches preserve optimality cumulated spaces are iteratively enriched. These strategies rely knowledge a-priori upper bounds B(n) D. establish form $$\mathcal O(n^r)$$ algebraic polynomials total degree, exact growth rate r regularity domain, particular $$r=2$$ Lipschitz boundaries $$r=\frac{d+1}{d}$$ smooth domains.