Sensitivity analysis (SA) plays a central role in mathematical modeling, providing valuable insights into how variations in input parameters affect a model’s output. This tool allows practitioners to evaluate the robustness and reliability of their models by quantifying the response to changes in key variables. Historically, SA methods have been tailored for scalar-valued models. However, some physical problems involve more complex outputs, such as set-valued models, where each model evaluation represents a subspace within a larger space X. Traditional SA methods are not directly applicable or adaptable to set-valued models, posing a challenge in quantifying uncertainties in the set-valued response.
To fill this gap, we propose three different approaches for defining sensitivity indices when dealing with set-valued models. The first two are Sobol-like indices based on random set tools [3] and universal indices [2] adapted to set outputs. The third approach uses kernel-based sensitivity indices, specifically HSIC and HSIC-ANOVA [1], with sets. This requires the introduction of a kernel defined on sets, which we thoroughly investigate. In particular, we demonstrate its characteristic property, based on a result from [4], an essential property within the HSIC framework. These indices are studied and compared on toy sets and pollutant concentration maps, which allow for detecting the determinant factors in pollutant dispersion. Another application of these indices lies in chance-constrained problems and robust optimization, where they are used to quantify the impact of uncertain inputs on excursion sets and help to simplify steps within Bayesian robust optimization.
