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

TL;DR

This paper enhances a Monte-Carlo method using reduced basis control variates for efficient uncertainty propagation and Bayesian estimation in PDE models, providing error analysis and demonstrating significant computational savings.

Contribution

It offers a comprehensive analysis and convergence results for the reduced-basis control variate Monte-Carlo method, combining multiple reduced-basis techniques for the first time in this context.

Findings

Provides precise error estimates and convergence results.

Demonstrates efficiency in uncertainty propagation and Bayesian estimation.

Achieves significant computational cost reduction.

Abstract

The Reduced-Basis Control-Variate Monte-Carlo method was introduced recently in [S. Boyaval and T. Leli\`evre, CMS, 8 2010] 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 parametrized 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, Nguyen C., Y. Maday and T. Patera, CMAME, 198 2009]. This is the first combination of various Reduced-Basis ideas to our knowledge, here with a view to reducing as much as possible the computational cost of a simple approach to Uncertainty Quantification.