  Parameter space reduction has been proved to be a crucial tool to speed-up
the execution of many numerical tasks such as optimization, inverse problems,
sensitivity analysis, and surrogate models' design, especially when in presence
of high-dimensional parametrized systems. In this work we propose a new method
called local active subspaces (LAS), which explores the synergies of active
subspaces with supervised clustering techniques in order to carry out a more
efficient dimension reduction in the parameter space. The clustering is
performed without losing the input-output relations by introducing a distance
metric induced by the global active subspace. We present two possible
clustering algorithms: K-medoids and a hierarchical top-down approach, which is
able to impose a variety of subdivision criteria specifically tailored for
parameter space reduction tasks. This method is particularly useful for the
community working on surrogate modelling. Frequently, the parameter space
presents subdomains where the objective function of interest varies less on
average along different directions. So, it could be approximated more
accurately if restricted to those subdomains and studied separately. We tested
the new method over several numerical experiments of increasing complexity, we
show how to deal with vectorial outputs, and how to classify the different
regions with respect to the local active subspace dimension. Employing this
classification technique as a preprocessing step in the parameter space, or
output space in case of vectorial outputs, brings remarkable results for the
purpose of surrogate modelling.
