Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method

Abstract

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.

Multi-fidelity data fusion for the approximation of scalar functions with low intrinsic dimensionality through active subspaces

“…Differently from the previous test case, the one-dimensional response function in the left panel of Figure 3 does not explain well the model: in this case kernel-based active subspaces could be employed to reach a better expressiveness of the surrogate model [30]. Even the scatter plot in the right panel of Figure 3, which shows the correlations between the low-fidelity and high-fidelity levels of the NARGP model, exhibits a worse accuracy in the low-fidelity level with respect to the previous test case.…”

“…The SEIR model for the spread of Ebola depends on 8 parameters and the output of interest is the basic reproduction number R 0 . A complete AS analysis was made in [10], while a kernel-based active subspaces comparison can be found in [30]. The formulation is the following:…”

“…Reduction in parameter space through AS has been proven successful in a diverse range of applications such as: shape optimization [22,9], hydrologic models [13], naval and nautical engineering [41,7,24,37,39], coupled with intrusive reduced order methods in cardiovascular studies [36], in CFD problems in a data-driven setting [8,40]. A kernel-based extension of AS for both scalar and vectorial functions can be found in [30].…”

Hull shape design optimization with parameter space and model reductions, and self-learning mesh morphing

“…The active subspaces (AS) [2,55,36] property is an emerging technique for dimension reduction of parameterized problems. Let us initially assume that the input/output relationship of the problem under study is represented by function f (µ) : Ω ⊂ R n → R. The reduction is performed by computing a linear transformation of the original parameters µ M = Aµ, in which A is an M × n matrix, and M < n. In the last years AS has been extended to vector-valued output functions [55], and to nonlinear transformations of the input parameters using the kernel-based active subspaces (KAS) method [33]. AS has been also coupled with reduced order methods such as POD-Galerkin [45] in cardiovascular studies, and POD with interpolation [8] and dynamic mode decomposition [49] for CFD applications.…”

ATHENA: Advanced Techniques for High Dimensional Parameter Spaces to Enhance Numerical Analysis

“…In this package we focus on techniques which use linear and nonlinear transformations to align the input space along the directions of maximum variation of the function of interest. In particular in the ATHENA package are implemented the interfaces to easily employ Active Subspaces ( [4]), Kernel-based Active Subspaces ( [9]), and the Nonlinear Level-set Learning ( [13]) method. Figure 1 depicts a simple application of two of these methods.…”