A new method to perform a nonlinear reduction in parameter spaces is proposed. By using a kernel approach it is possible to find active subspaces in high-dimensional feature spaces. A mathematical foundation of the method is presented, with several applications to benchmark model functions, both scalar and vector-valued. We also apply the kernel-based active subspaces extension to a CFD parametric problem using the Discontinuous Galerkin method. A full comparison with respect to the linear active subspaces technique is provided for all the applications, proving the better performances of the proposed method. Moreover we show how the new kernel method overcomes the drawbacks of the active subspaces application for radial symmetric model functions.

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 computational fluid dynamics application solved with the discontinuous Galerkin method.

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