Kernel‐based active subspaces with application to computational fluid dynamics parametric problems using the discontinuous Galerkin method

Abstract

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.

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls

“…Future works will focus on improving the accuracy of constraints evaluations, for example with a multi‐fidelity approximation of the scalar output and not only for the reconstruction of the entire field. 23 Another possibility is the exploitation of local information with local active subspaces 64 or nonlinear techniques, based on kernels 65 or level‐sets, 66 , 67 to further improve the regression performance of the low‐fidelity model. Other physical constraints can also be considered such as the position of the center of mass.…”

Multi‐fidelity data fusion through parameter space reduction with applications to automotive engineering

“…It can be computed with the following formula: $$$$ {R}_0=\frac{\beta_1+\frac{\beta_2{\rho}_1{\gamma}_1}{\omega }+\frac{\beta_3}{\gamma_2}\psi }{\gamma_1+\psi } $$$$ with parameters range taken from Reference 46, where they conducted a global sensitivity analysis with AS. For a kernel‐based active subspaces comparison the reader can refer to Reference 31.…”

“…Future research lines should investigate the use of different active subspaces‐based methods, such as kernel AS, 31 or local AS, 32 which exploit kernel‐based and localization techniques, respectively. This multi‐fidelity framework has also the potential to be integrated with other reduced order modeling techniques 54‐56 to further increase the accuracy in the resolution of parametric problems, especially for high‐dimensional surrogate‐based optimization 57 …”

“…Successful applications of parameter space reduction with active subspaces can be found in many engineering fields: naval and nautical problems, 18 shape optimization, [19][20][21][22] car aerodynamics studies, 23 inverse problems, 24,25 cardiovascular studies coupled with intrusive model order reduction, 26 for the study of high-dimensional parametric PDEs, 27 and in CFD problems in a data-driven setting, 28,29 among others. New extensions of AS have also been developed in the recent years such as AS for multivariate vector-valued functions, 30 a kernel approach for AS for scalar and vectorial functions, 31 a localization extension for both regression and classification tasks, 32 and sequential learning of active subspaces. 33 The multi-fidelity setting has been used to find an active subspace given different fidelity models.…”