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The main focus of this article is to provide a mathematical study of greedy algorithms for the construction of reduced bases so as to approximate a collection of parameter-dependent random variables. For each value of the parameter, the associated random variable belongs to some Hilbert space (say the space of square-integrable random variates for instance). But carrying out an exact greedy algorithm in this context would require the computation of exact expectations or variances of parameter-dependent random variates, which cannot be done in practice. Instead, expectations and variances can only be computed approximately via empirical means and empirical variances involving a finite number of Monte-Carlo samples. The aim of this work is precisely to study the effect of finite Monte-Carlo sampling on the theoretical properties of greedy algorithms. In particular, using concentration inequalities for the empirical measure in Wasserstein distance proved by Fournier and Guillin Probab. Theory Related Fields 162 (2015), pp. 707–738, we provide sufficient conditions on the number of samples used for the computation of empirical variances at each iteration of the greedy procedure to guarantee that the resulting method algorithm is a weak greedy algorithm with high probability. Let us mention here that such an algorithm has initially been proposed by Boyaval and Lelièvre Commun. Math. Sci. 8 (2010), pp. 735–762 with the aim to design a variance reduction technique for the computation of parameter-dependent expectations via the use of control variates constructed using a reduced basis paradigm. The theoretical results we prove here are not fully practical and we therefore propose a heuristic procedure to choose the number of Monte-Carlo samples at each iteration, inspired from this theoretical study, which provides satisfactory results on several numerical test cases.
Blel et al. (Fri,) studied this question.