Gradient free active subspace construction using Morris screening elementary effects

Abstract

Among multivariate functions with high-dimensional input spaces, it is common for functions to vary more strongly in a few dominant directions related to a small number of highly influential parameters. In such cases, the input dimension may be greatly reduced by constructing a low-dimensional response space that is aligned with the directions of strongest dominance; this is the basis behind active subspace methods. Until recently, gradient-based methods have been employed to construct the active subspace. We introduce a gradient-free active subspace construction method that avoids the need to sample from the gradient, which may not be available, via construction of a coarse approximation to the gradient matrix by employing the concept of “elementary effects” from Morris screening procedures. In addition, we introduce the use of adaptive step sizes and directions, when constructing these elementary effects, to allow for more accuracy in locally sensitive regions while still covering a substantial amount of the input space. This increases algorithmic efficiency by avoiding function evaluations in directions in which the gradient is relatively flat. To demonstrate the method, we use an elliptic PDE example with two correlation lengths to illustrate the effects of differing rates of singular value decay. The gradient-free active subspace method is compared to a local sensitivity analysis using coordinate reduction. This problem is then modified to contain a clearly defined 10-dimensional active subspace for verification of our method on a more complex example.