Regularizing nested Monte Carlo Sobol' index estimators to balance the trade-off between explorations and repetitions in global sensitivity analysis of stochastic models
Résumé
Sobol' sensitivity index estimators for stochastic models are
functions of nested Monte Carlo estimators, which are estimators
built from two nested Monte Carlo loops. The outer loop explores
the input space and, for each of the explorations, the inner loop
repeats model runs to estimate conditional expectations. Although
the optimal allocation between explorations and repetitions of one's
computational budget is well-known for nested Monte Carlo
estimators, it is less clear how to deal with functions of nested
Monte Carlo estimators, especially when those functions have
unbounded Hessian matrices, as it is the case for Sobol' index
estimators. To address this problem, a regularization method is
introduced to bound the mean squared error of functions of nested
Monte Carlo estimators. Based on a heuristic, an
allocation strategy that seeks to minimize a bias-variance trade-off
is proposed. The method is applied to Sobol' index estimators for
stochastic models. A practical algorithm that adapts to the level
of intrinsic randomness in the models is given and illustrated on
numerical experiments.
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