  Sobol indices are a widespread quantitative measure for variance-based global
sensitivity analysis, but computing and utilizing them remains challenging for
high-dimensional systems. We propose the tensor train decomposition (TT) as a
unified framework for surrogate modeling and global sensitivity analysis via
Sobol indices. We first overview several strategies to build a TT surrogate of
the unknown true model using either an adaptive sampling strategy or a
predefined set of samples. We then introduce and derive the Sobol tensor train,
which compactly represents the Sobol indices for all possible joint variable
interactions which are infeasible to compute and store explicitly. Our
formulation allows efficient aggregation and subselection operations: we are
able to obtain related indices (closed, total, and superset indices) at
negligible cost. Furthermore, we exploit an existing global optimization
procedure within the TT framework for variable selection and model analysis
tasks. We demonstrate our algorithms with two analytical engineering models and
a parallel computing simulation data set.
