Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

Mohamed-Raed Blel (1) , Virginie Ehrlacher (1, 2) , Tony Lelièvre (1, 2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS

The main focus of this article is to provide a mathematical study of the algorithm proposed in [6] where the authors proposed a variance reduction technique for the computation of parameter-dependent expectations using 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 proved in [14], 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.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Mohamed-Raed Blel, Virginie Ehrlacher, Tony Lelièvre. Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables. 2021. ⟨hal-03238244v2⟩