Regularizing nested Monte Carlo Sobol' index estimators to balance the trade-off between explorations and repetitions in global sensitivity analysis of stochastic models

TL;DR

This paper introduces a regularization method for nested Monte Carlo Sobol' index estimators to optimize the exploration-repetition trade-off, improving sensitivity analysis of stochastic models.

Contribution

It proposes a novel regularization technique and an adaptive allocation strategy for nested Monte Carlo estimators, specifically addressing Sobol' indices with unbounded Hessians.

Findings

The method effectively bounds mean squared error in Sobol' index estimation.

The adaptive algorithm improves estimation accuracy in stochastic models.

Numerical experiments demonstrate practical benefits of the approach.

Abstract

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.