Kernel-based active subspaces with application to computational fluid dynamics parametric problems using the...

Area of Science:

• Computational mathematics
• Applied mathematics
• Scientific computing

Background:

• Active subspaces method offers effective dimension reduction for parameter spaces.
• Nonlinear extensions have shown promise in improving response surface design.
• Existing methods face challenges with complex, high-dimensional problems.

Purpose of the Study:

• To develop and present a novel kernel-based nonlinear method for dimension reduction.
• To extend the mathematical framework for parameter space reduction of multivariate functions.
• To demonstrate the method's efficacy on challenging benchmarks and in a practical application.

Main Methods:

• Kernel-based nonlinear dimension reduction.
• Extension of active subspaces methodology.
• Application to parametric computational fluid dynamics (CFD) using the discontinuous Galerkin method.

Main Results:

• The proposed method achieves effective dimension reduction in parameter spaces.
• Successful implementation and testing on challenging benchmarks, outperforming existing methods.
• Demonstration of a complete pipeline for response surface design in CFD.

Conclusions:

• The kernel-based nonlinear method provides a powerful tool for dimension reduction and response surface design.
• The methodology offers significant advantages for complex parametric modeling, particularly in CFD.
• This work advances the field of dimension reduction for scientific and engineering applications.