Improve the Active Subspace Method by Partitioning the Parameter Space

Abstract

The active subspace method is a powerful tool that can be applied in many fields such as uncertainty quantification, inverse problems and optimisation. However, the standard active subspace method constructs an active subspace over the whole parameter space, which makes the method only applicable to functions that have ridge or near-ridge structures in other words, it only works for a function f such that f (x) g(WTx), where x is an mdimensional parameter space and W is m n, n < m. In this thesis, we propose two families of algorithms that use Voronoi diagrams to (randomly and adaptively) partition the input space and hence construct an active subspace for each region. Our proposed methods work on functions that have local ridges from region to region. Based on the four test functions that we employed in this thesis, we find that our proposed algorithms produce more accurate response surfaces than those generated by the standard active subspace method. To evaluate the accuracy, we test the response surfaces with a separate test set of points and calculate their mean squared error for each response surface. Our proposed algorithms achieve a lower MSE than the active subspace method.We also introduce a new algorithm that may work for more general functions. Show more