High-dimensional interpolation on the Grassmann manifold using Gaussian processes

Abstract

This paper proposes a novel method for performing interpolation of high-dimensional systems. The proposed method projects the high-dimensional full-field solution into a lower-dimensional space where interpolation is computationally more tractable. The method combines the spectral clustering technique, which refers to a class of machine learning techniques that utilizes the eigen-structure of a similarity matrix to partition data into disjoint clusters based on the similarity of the points, in order to effectively identify areas of the parameter space where sharp changes of the solution field are resolved. In order to do this, we derive a similarity matrix based on the pairwise distances between the high-dimensional solutions of the stochastic system projected onto the Grassmann manifold. The distances are calculated using appropriately defined metrics and the similarity matrix is used in order to cluster the data based on their similarity. Points that belong to the same cluster are projected onto the tangent space (which is an inner-product flat space) defined at the Karcher mean of these points and a Gaussian process is used for interpolation on the tangent of the Grassmann manifold in order to predict the solution without requiring full model evaluations.