Sobol' indices for problems defined in non-rectangular domains

TL;DR

This paper introduces a new framework for calculating Sobol sensitivity indices in models with inputs constrained to non-rectangular domains, using two numerical methods validated against analytical benchmarks.

Contribution

It develops a general, efficient approach for Sobol index estimation in constrained input spaces, combining quadrature and Monte Carlo methods with acceptance-rejection sampling.

Findings

Quadrature method effective for low to medium dimensions

Acceptance-rejection Monte Carlo estimators validated against analytical solutions

Framework applicable to models with inequality constraints

Abstract

A novel theoretical and numerical framework for the estimation of Sobol sensitivity indices for models in which inputs are confined to a non-rectangular domain (e.g., in presence of inequality constraints) is developed. Two numerical methods, namely the quadrature integration method which may be very efficient for problems of low and medium dimensionality and the MC/QMC estimators based on the acceptance-rejection sampling method are proposed for the numerical estimation of Sobol sensitivity indices. Several model test functions with constraints are considered for which analytical solutions for Sobol sensitivity indices were found. These solutions were used as benchmarks for verifying numerical estimates. The method is shown to be general and efficient.