Influence of Monte-Carlo sampling on the convergence rates of greedy algorithms for reduced-basis methods
Virginie Ehrlacher  1@  , Mohammed-Raed Blel  2  , Tony Lelièvre  3  
Ecole des Ponts ParisTech, L'Institut National de Recherche en Informatique et e n Automatique (INRIA)
Ecole Nationale des Ponts et Chaussées
Ecole Nationale des Ponts et Chaussées INRIA

In this talk will be presented recent results about the mathematical study of an algorithm where a variance reduction technique for the computation of parameter-dependent
expectations is appliedusing a reduced basis paradigm. We study the effect of Monte-Carlo sampling on the theoretical properties of greedy algorithms. In particular, using concentration inequalities for the empirical measure in Wasserstein distance, 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. These theoretical
results 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.

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