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Parameter space reduction has been proved to be a crucial tool to speed-upthe execution of many numerical tasks such as optimization, inverse problems,sensitivity analysis, and surrogate models' design, especially when in presenceof high-dimensional parametrized systems. In this work we propose a new methodcalled local active subspaces (LAS), which explores the synergies of activesubspaces with supervised clustering techniques in order to carry out a moreefficient dimension reduction in the parameter space. The clustering isperformed without losing the input-output relations by introducing a distancemetric induced by the global active subspace. We present two possibleclustering algorithms: K-medoids and a hierarchical top-down approach, which isable to impose a variety of subdivision criteria specifically tailored forparameter space reduction tasks. This method is particularly useful for thecommunity working on surrogate modelling. Frequently, the parameter spacepresents subdomains where the objective function of interest varies less onaverage along different directions. So, it could be approximated moreaccurately if restricted to those subdomains and studied separately. We testedthe new method over several numerical experiments of increasing complexity, weshow how to deal with vectorial outputs, and how to classify the differentregions with respect to the local active subspace dimension. Employing thisclassification technique as a preprocessing step in the parameter space, oroutput space in case of vectorial outputs, brings remarkable results for thepurpose of surrogate modelling.

A local approach to parameter space reduction for regression and classification tasks