Localised proper orthogonal decomposition

Abstract

Proper Orthogonal Decomposition (POD) extracts an optimal orthonormal basis for reconstructing a sequence of data. This technique is widely used in fluid mechanics to construct reduced-order models for physical analysis, modelling, estimation, and control. However, when the number of snapshots is small and the spatial domain is large, convergence of the basis with respect to the number of samples is compromised. This often results in a degradation of the quality of the resulting reduced-order models. In this study, we propose a localisation procedure to mitigate convergence artefacts in the context of limited data. Specifically, we adapt Schur product-based localisation – commonly used in operational data assimilation – to the POD framework. We develop an algorithm that operates directly on the snapshots rather than on the large-scale correlation tensor. In addition, we introduce a family of partition-of-unity localisation functions to preserve the reconstruction properties of the localised POD basis. By artificially increasing the rank of the correlation tensor, we demonstrate that the localised POD, computed from a limited number of snapshots, can approximate the POD basis obtained from a larger dataset. This is illustrated using a wave field scattered by a turbulent jet in a rotating shallow water system. We further show that using smooth, overlapping localisation functions enables physically meaningful reconstructions. The reconstruction capabilities of the method are also demonstrated using a realistic internal wave field in the Gulf Stream region, based on a high-resolution numerical simulation.