A co-kurtosis based dimensionality reduction method for combustion datasets
Principal Component Analysis (PCA) is a dimensionality reduction technique widely used to reduce the computational cost associated with numerical simulations of combustion phenomena. However, PCA, which transforms the thermo-chemical state space based on eigenvectors of co-variance of the data, could fail to capture information regarding important localized chemical dynamics, such as the formation of ignition kernels, appearing as \rev{extreme-valued} samples in a dataset. In this paper, we propose an alternate dimensionality reduction procedure, co-kurtosis PCA (CoK-PCA), wherein the required principal vectors are computed from a high-order joint statistical moment, namely the co-kurtosis tensor, which may better identify directions in the state space that represent stiff dynamics. We first demonstrate the potential of the proposed CoK-PCA method using a synthetically generated dataset that is representative of typical combustion simulations. Thereafter, we characterize and contrast the accuracy of CoK-PCA against PCA for datasets representing spontaneous ignition of premixed ethylene-air in a simple homogeneous reactor and ethanol-fueled homogeneous charged compression ignition (HCCI) engine. Specifically, we compare the low-dimensional manifolds in terms of reconstruction errors of the original thermo-chemical state, and species production and heat release rates computed from the reconstructed state. \rev{The latter -- a comparison of species production and heat release rates -- is a more rigorous assessment of the accuracy of dimensionality reduction.} We find that, even using a simplistic linear reconstruction, the co-kurtosis based reduced manifold represents the original thermo-chemical state more accurately than PCA, especially in the regions where chemical reactions are important.