Global Sensitivity Analysis of Spatial Sets: Finite-Element-Based Estimation and the Role of Observation Windows

RESUMO

In this work we consider numerical models with random input parameters, where the model output is a spatially distributed random field. We carry out a statistical sensitivity analysis of spatial sets which arise for instance when the model output exceeds a critical threshold. We consider two approaches: (i) a kernel-based sensitivity analysis working with the Hilbert--Schmidt Independence Criterion, and (ii) a function-valued sensitivity analysis working with generalized Sobol' indices, where the model output is the indicator function of the spatial set of interest. We develop efficient estimators for the sensitivity indices, especially for finite-element-based numerical models, where we approximate volume integrals in the sensitivity measures by finite element quadrature, thereby avoiding Monte Carlo sampling over the spatial domain. This improves the state-of-the-art for the kernel-based indices and makes these methods practically accessible for expensive numerical models. For the generalized Sobol' indices we utilize the same finite element quadrature for efficient computation. We present numerical experiments for a hydrogen combustion process, modeled by a coupled system of convection-diffusion-reaction equations in a two-dimensional spatial domain, and ask whether the temperature remains below a critical value in selected regions of the combustion domain. Our results show that in this example the kernel-based and the generalized Sobol' indices give qualitatively similar input importance rankings. Moreover, the ranking depends on the chosen observation window and can flip between different windows. The algorithms for implementing the proposed methods are provided in the supplementary materials and in an open source code repository.