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We consider the problem of approximating an unknown function $u\inL^2(D,\rho)$ from its evaluations at given sampling points $x^1,\dots,x^n\inD$, where $D\subset \mathbb{R}^d$ is a general domain and $\rho$ is aprobability measure. The approximation is picked in a linear space $V_m$ where$m=\dim(V_m)$ and computed by a weighted least squares method. Recent resultsshow the advantages of picking the sampling points at random according to awell-chosen probability measure $\mu$ that depends both on $V_m$ and $\rho$.With such a random design, the weighted least squares approximation is provedto be stable with high probability, and having precision comparable to that ofthe exact $L^2(D,\rho)$-orthonormal projection onto $V_m$, in a near-linearsampling regime $n\sim{m\log m}$. The present paper is motivated by theadaptive approximation context, in which one typically generates a nestedsequence of spaces $(V_m)_{m\geq1}$ with increasing dimension. Although themeasure $\mu=\mu_m$ changes with $V_m$, it is possible to recycle thepreviously generated samples by interpreting $\mu_m$ as a mixture between$\mu_{m-1}$ and an update measure $\sigma_m$. Based on this observation, wediscuss sequential sampling algorithms that maintain the stability andapproximation properties uniformly over all spaces $V_m$. Our main result isthat the total number of computed sample at step $m$ remains of the order$m\log{m}$ with high probability. Numerical experiments confirm this analysis.

Sequential sampling for optimal weighted least squares approximations in hierarchical spaces