  Nonlinear extensions to the active subspaces method have brought remarkable
results for dimension reduction in the parameter space and response surface
design. We further develop a kernel-based nonlinear method. In particular we
introduce it in a broader mathematical framework that contemplates also the
reduction in parameter space of multivariate objective functions. The
implementation is thoroughly discussed and tested on more challenging
benchmarks than the ones already present in the literature, for which dimension
reduction with active subspaces produces already good results. Finally, we show
a whole pipeline for the design of response surfaces with the new methodology
in the context of a parametric CFD application solved with the Discontinuous
Galerkin method.
