Adaptive Low-Rank Tensor Manifolds for Time-Resolved Velocimetry (1): The Algorithm

Reconstruction of the dense Eulerian flow field from Lagrangian Particle Tracking (LPT) data is an important post-processing step to compute quantities such as derivatives, strain, rotation, and pressure. However, the sparse and scattered nature of the LPT data makes the reconstruction challenging. Currently available techniques to reconstruct a dense Eulerian field from LPT data include numerical interpolation, data assimilation based on physical laws, and machine learning based solutions. While these techniques can yield reasonably accurate Eulerian or continuous reconstructions from LPT data, they often require high particle density to achieve acceptable accuracy and may incur significant computational costs. To address these challenges, we have developed a 4D low-rank tensor approximation algorithm that treats the time-resolved LPT data as a regression problem constrained on a low-rank tensor manifold. This algorithm leverages the spatio-temporal correlations in the data to provide a continuous, grid-less representation of the flow-field. The algorithm is tuning-free and fully adaptive, automatically determining model complexity on the fly with minimal user input. Compared to existing methods, it achieves comparable accuracy using significantly less data and reduces the computational cost of field reconstruction by at least an order of magnitude. In this talk, we present the theoretical foundation of the methodology and the details of the algorithm.