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The Reduced-Basis Control-Variate Monte-Carlo method was introduced recentlyin [S. Boyaval and T. Leli\`evre, CMS, 8 2010] as an improved Monte-Carlomethod, for the fast estimation of many parametrized expected values at manyparameter values. We provide here a more complete analysis of the methodincluding precise error estimates and convergence results. We also numericallydemonstrate that it can be useful to some parametrized frameworks inUncertainty Quantification, in particular (i) the case where the parametrizedexpectation is a scalar output of the solution to a Partial DifferentialEquation (PDE) with stochastic coefficients (an Uncertainty Propagationproblem), and (ii) the case where the parametrized expectation is the Bayesianestimator of a scalar output in a similar PDE context. Moreover, in each case,a PDE has to be solved many times for many values of its coefficients. This iscostly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C.Le Bris, Nguyen C., Y. Maday and T. Patera, CMAME, 198 2009]. This is the firstcombination of various Reduced-Basis ideas to our knowledge, here with a viewto reducing as much as possible the computational cost of a simple approach toUncertainty Quantification.

A fast Monte-Carlo method with a Reduced Basis of Control Variates applied to Uncertainty Propagation and Bayesian Estimation