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.

翻译：主动子空间方法的非线性扩展已在参数空间降维和响应面设计方面取得了显著成果。我们进一步开发了一种基于核的非线性方法，特别是在更广泛的数学框架中引入该方法，该框架同时也考虑了多变量目标函数在参数空间中的降维问题。本文详细讨论了该方法的实现，并在比文献中已有基准更具挑战性的问题上进行了测试——对于这些基准问题，主动子空间降维已能产生良好结果。最后，我们展示了在采用间断伽辽金法求解的参数化CFD应用背景下，运用新方法设计响应面的完整流程。