Ridge functions have recently emerged as a powerful set of ideas for subspace-based dimension reduction. In this paper we begin by drawing parallels between ridge subspaces, sufficient dimension reduction and active subspaces, contrasting between techniques rooted in statistical regression and those rooted in approximation theory. This sets the stage for our new algorithm that approximates what we call a Gaussian ridge function-the posterior mean of a Gaussian process on a dimensionreducing subspace-suitable for both regression and approximation problems. To compute this subspace we develop an iterative algorithm that optimizes over the Stiefel manifold to compute the subspace, followed by an optimization of the hyperparameters of the Gaussian process. We demonstrate the utility of the algorithm on two analytical functions, where we obtain near exact ridge recovery, and a turbomachinery case study, where we compare the efficacy of our approach with three well-known sufficient dimension reduction methods: SIR, SAVE, CR. The comparisons motivate the use of the posterior variance as a heuristic for identifying the suitability of a dimensionreducing subspace.

Motivated by the idea of turbomachinery active subspace performance maps, this paper studies dimension reduction in turbomachinery 3D CFD simulations. First, we show that these subspaces exist across different blades—under the same parametrisation—largely independent of their Mach number or Reynolds number. This is demonstrated via a numerical study on three different blades. Then, in an attempt to reduce the computational cost of identifying a suitable dimension reducing subspace, we examine statistical sufficient dimension reduction methods, including sliced inverse regression, sliced average variance estimation, principal Hessian directions and contour regression. Unsatisfied by these results, we evaluate a new idea based on polynomial variable projection—a non-linear least-squares problem. Our results using polynomial variable projection clearly demonstrate that one can accurately identify dimension reducing subspaces for turbomachinery functionals at a fraction of the cost associated with prior methods. We apply these subspaces to the problem of comparing design configurations across different flight points on a working line of a fan blade. We demonstrate how designs that offer a healthy compromise between performance at cruise and sea-level conditions can be easily found by visually inspecting their subspaces.

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Motivated by the idea of turbomachinery active subspace performance maps, this paper studies dimension reduction in turbomachinery 3D CFD simulations. First, we show that these subspaces exist across different blades-under the same parametrization-largely independent of their Mach number or Reynolds number. This is demonstrated via a numerical study on three different blades. Then, in an attempt to reduce the computational cost of identifying a suitable dimension reducing subspace, we examine statistical sufficient dimension reduction methods, including sliced inverse regression, sliced average variance estimation, principal Hessian directions and contour regression. Unsatisfied by these results, we evaluate a new idea based on polynomial variable projection-a non-linear least squares problem. Our results using polynomial variable projection clearly demonstrate that one can accurately identify dimension reducing subspaces for turbomachinery functionals at a fraction of the cost associated with prior methods. We apply these subspaces to the problem of comparing design configurations across different flight points on a working line of a fan blade. We demonstrate how designs that offer a healthy compromise between performance at cruise and sea-level conditions can be easily found by visually inspecting their subspaces.