Maximum likelihood reconstruction of particle trajectories in turbulent flows via sparse optimization

Abstract

Extracting Lagrangian particle dynamics is essential for characterizing turbulent flows, but inferring particle acceleration from inherently noisy position data remains a significant challenge. Fluid particles in turbulence experience extreme, intermittent accelerations, resulting in heavy-tailed probability density functions (PDFs) that deviate strongly from Gaussian predictions. Existing filtering techniques, such as Gaussian kernels and penalized B-splines, implicitly assume Gaussian-distributed jerk, thereby penalizing sparse, high-magnitude acceleration changes and artificially suppressing the intermittent tails. In this work, we develop a novel maximum likelihood framework for filtering noisy particle tracks in turbulence that explicitly accounts for this non-Gaussian intermittency. By modeling the jerk with a sparsity-promoting Bernoulli–Gaussian prior in place of the usual Gaussian one, we obtain a sparse optimization problem, which we render tractable through a convex \(\ell _1\) -relaxation. To overcome the numerical stiffness associated with high-order difference operators, the problem is efficiently solved using an iteratively reweighted least squares (IRLS) algorithm. The proposed filter is evaluated against direct numerical simulation (DNS) data of homogeneous, isotropic turbulence ( \(\textrm{Re}_{\lambda } \simeq 433\) ) for the Johns Hopkins Turbulence Database. Results demonstrate that the IRLS approach consistently outperforms state-of-the-art continuous MLE, discrete MLE, and B-spline methods, yielding systematic reductions in root-mean-squared error (RMSE) across position, velocity, and acceleration. Most importantly, the proposed framework succeeds in better recovering the heavy-tailed statistical structure of both acceleration and acceleration differences (jerk) across temporal scales, preserving the physical intermittency characteristic of high-Reynolds-number turbulent flows that baseline methods severely attenuate.