A fast Monte–Carlo method with a reduced basis of control variates applied to uncertainty propagation and Bayesian estimation

The reduced-basis control-variate Monte-Carlo method was introduced recently in [S. Boyaval, T. Lelièvre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm, Commun. Math. Sci. 8 (2010) 735–762 (Special issue “Mathematical Issues on Complex Fluids”)] as an improved Monte-Carlo method, for the fast estimation of many parametrized expected values at many parameter values. We provide here a more complete analysis of the method including precise error estimates and convergence results. We also numerically demonstrate that it can be useful to some parametric frameworks in Uncertainty Quantification, in particular (i) the case where the parametrized expectation is a scalar output of the solution to a Partial Differential Equation (PDE) with stochastic coefficients (an Uncertainty Propagation problem), and (ii) the case where the parametrized expectation is the Bayesian estimator 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 is costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C. Le Bris, Y. Maday, N. Nguyen, A. Patera, A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable robin coefficient, Comput. Methods Appl. Mech. Eng. 198 (2009) 3187–3206]. To our knowledge, this is the first combination of various reduced-basis ideas, with a view to reducing as much as possible the computational cost of a simple versatile Monte-Carlo approach to Uncertainty Quantification.