Maximum Likelihood Particle Tracking in Turbulent Flows via Sparse Optimization

Abstract

Lagrangian particle tracking 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 estimation (MLE) framework that explicitly accounts for this non-Gaussian intermittency. By formulating a modified Gaussian process to model the random incremental forcing, we introduce a sparse optimization scheme utilizing a convex 1-norm 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 (Re approx. 310). Results demonstrate that the IRLS approach consistently outperforms state-of-the-art discrete MLE, continuous 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-Reynoldsnumber turbulent flows that baseline methods severely attenuate.

Figures (8)

• Figure 1: Normalized RMSE from IRLS-filtered trajectories as a function of sparsity penalty parameter $\gamma$ for position (first - top), velocity (second), and acceleration (third) in comparison with reconstructed acceleration statistics as a function of $\gamma$ (bottom). All graphs are plotted on log-log scale and the tuned $\gamma$ value is indicated by the dash pink line. In all RMSE graphs the minima occur shortly after the exponential decay (linear in log-log scale) transition of the acceleration statistics demonstrating that tuning the IRLS filter is achievable even in the absence of ground truth data.
• Figure 2: Example of a trajectory. Noisy position data (solid grey) is fed into the IRLS filter (solid pink), the discrete MLE filter (dashed blue), continuous MLE filter (dot-dashed green), and B-splines filter (dotted yellow). Comparing to the DNS data (solid black), the IRLS data significantly outperforms the other filters, particularly in regions of extreme accelerations.
• Figure 3: Error distributions for position, normalized by the standard deviation of the measurement noise $\sigma_w$. In all cases, the IRLS method outperforms the discrete MLE, continuous MLE, and B-splines approaches. Coloring and line styles match those used in Figure \ref{['fig:pos']}.
• Figure 4: Acceleration time series from a representative trajectory (top) and the corresponding acceleration differences (bottom). Differences are evaluated at the sampling interval $\Delta t$ and serve as a discrete proxy for jerk. All quantities are normalized by the standard deviation of the corresponding quantity from the DNS. Original DNS data is shown in black, the discrete MLE reconstruction in blue, and newly-developed IRLS reconstruction in pink.
• Figure 5: PDF of normalized accelerations. The DNS data exhibit the characteristic heavy-tailed structure of turbulent acceleration. The baseline MLE methods and the B-spline approach attenuate the tails of the distribution, underpredicting the probability of extreme acceleration events. In contrast, the IRLS reconstruction more closely follows the DNS distribution, demonstrating improved recovery of the heavy-tail behavior. The inset shows a zoomed view of the core of the distribution near zero acceleration, highlighting differences in the central region. Coloring and line styles match those used in Figure \ref{['fig:pos']}.