A technique for characterising feature size and quality of manifolds

Effective dimension reduction is a key factor in facilitating large-scale simulation of high-dimensional dynamical systems. The behaviour of low-dimensional surrogate models often relies on accurate reconstruction of quantities that can be nonlinear functions of the original parameters. For instance, in low-dimensional combustion models, source terms representative of complex chemical kinetics must be modelled accurately in the reduced dimensional space in order to yield accurate predictions. Features such as sharp gradients or non-uniqueness in a quantity of interest (QoI) may be introduced through parameterisation and pose difficulties for reconstruction techniques. Many existing manifold quality assessments do not consider these features and limit examination to the original parameters and low-dimensional embedding. We have developed a technique for quantitatively assessing manifold quality through characterising the feature size of QoIs by monitoring the change in variance over an increasing filter width. Through the identification of variance at small scales, this technique detects undesirable sharp gradients and non-uniqueness of QoIs. Our technique is not limited to a specific reduction method and can be used to compare or assess manifold parameterisations in arbitrary dimensions. We demonstrate our technique on combustion data from both simulation and experiment.