  Probabilistic variants of Model Order Reduction (MOR) methods have recently
emerged for improving stability and computational performance of classical
approaches. In this paper, we propose a probabilistic Reduced Basis Method
(RBM) for the approximation of a family of parameter-dependent functions. It
relies on a probabilistic greedy algorithm with an error indicator that can be
written as an expectation of some parameter-dependent random variable.
Practical algorithms relying on Monte Carlo estimates of this error indicator
are discussed. In particular, when using Probably Approximately Correct (PAC)
bandit algorithm, the resulting procedure is proven to be a weak greedy
algorithm with high probability. Intended applications concern the
approximation of a parameter-dependent family of functions for which we only
have access to (noisy) pointwise evaluations. As a particular application, we
consider the approximation of solution manifolds of linear parameter-dependent
partial differential equations with a probabilistic interpretation through the
Feynman-Kac formula.
