Tensor Methods and Visualization for 
Global Sensitivity Analysis

Sobol indices and other, more recent quantities of interest are of great aid in sensitivity analysis, uncertainty quantification, and model interpretation. Unfortunately, computing as well as visualizaing such indices is still challenging for high-dimensional systems. We propose the tensor train decomposition (TT) as a unified framework for surrogate modeling and sensitivity analysis of independently distributed variables, and introduce the Sobol tensor train (Sobol TT) data structure, a compressed data format that can quickly and approximately answer sophisticated queries over exponential-sized sets of Sobol indices. Furthermore, we propose a novel visualization tool that leverages this new Sobol TT representation. Our approach efficiently captures the complete global sensitivity information of high-dimensional scalar models, allows interactive aggregation and subselection operations, and we are able to obtain related Sobol indices and other related quantities at low computational cost. In our three-stage visualization, variable sets to be analyzed can be added or removed interactively. Additionally, a novel hourglass-like diagram presents the relative importance for any single variable or combination of input variables with respect to any composition of the rest of the input variables. We showcase our visualization with several example models, whereby we demonstrate the high expressive power and analytical capability made possible with the proposed Sobol TT method.