A meshless data-tailored approach to compute statistics from scattered data with adaptive radial basis functions

TL;DR

The paper addresses reconstructing flow statistics from scattered measurements using constrained RBF regression by introducing gradient-informed, anisotropic adaptation and adaptive sampling. The approach combines local gradient estimation, density-aware sub-sampling, and an anisotropic metric to tailor basis functions to flow directionality, augmented with gradient penalties for stability. Across DNS-channel and 3D-PTV jet datasets, the method achieves higher accuracy, reduces spurious oscillations near gradients, and lowers basis counts and computation time compared to isotropic formulations. This meshless, linear framework enables robust, high-fidelity reconstruction of turbulent statistics in complex geometries and supports efficient statistical analysis from sparse data.

Abstract

Constrained radial basis function (RBF) regression has recently emerged as a powerful meshless tool for reconstructing continuous velocity fields from scattered flow measurements, particularly in image-based velocimetry. However, existing formulations based on isotropic kernels often suffer from spurious oscillations in regions with sharp gradients or strong flow anisotropy. This work introduces an anisotropic, gradient-informed, and adaptively sampled extension of the constrained RBF framework for regression of scattered data. Gradient information is estimated via local polynomial regression at collocation points, smoothed, and used to (1) re-sample data, maximizing sampling density near steep gradients while downsampling in smooth regions, and (2) construct a local anisotropic metric that shapes each basis function according to the flow directionality. In addition, a gradient-informed regularization is introduced by embedding observed gradients into the least-squares system as weighted soft constraints. The resulting formulation is fully meshless, linear, and computationally efficient, while significantly improving reconstruction quality in challenging regions. The method is evaluated on both synthetic and experimental datasets, including direct numerical simulation (DNS) data of a turbulent channel and time-resolved particle tracking velocimetry of a turbulent jet. Results show that the proposed approach outperforms isotropic and gradient-free RBF formulations in accuracy, smoothness, and physical consistency -- particularly near shear layers and boundaries -- while reducing the number of bases by an order of magnitude. To support the application, we have created a repository (https://github.com/mendezVKI/SPICY_VKI) that provides access to the investigated datasets.

Figures (11)

• Figure 1: Test case 1: (a) Scatter plot of the velocity ensemble for the regression of average velocity, coloured by streamwise velocity component, with the vertical slice region at $x/h=0.45$ highlighted in gray. (b) Data points extracted from the highlighted slice with the time-average DNS velocity profile (solid, red line)
• Figure 2: Test case 2: (a) Scatter plot of the velocity ensemble for the regression of average velocity, coloured by streamwise velocity component, with the horizontal slice region at $x/h=1$ highlighted in gray. (b) Data points extracted from the highlighted slice
• Figure 3: Comparison between the ground truth gradient from the DNS, the estimated gradient and the gradient smoothed with $\sigma_S=0.05$
• Figure 4: Test case 1: Number of data samples along each position $y/h$ from the adaptive sampling approach
• Figure 5: Test case 1: Spatial locations of the collocation points, coloured by the number of supporting points using the adaptive data sampling for the isotropic uniform (a), isotropic adaptive (b) and anisotropic adaptive basis (c). Aspect ratio (AR) of the anisotropic basis elements (d)