# Efficient Probabilistic Visualization of Local Divergence of 2D Vector Fields with Independent Gaussian Uncertainty

arXiv: https://arxiv.org/abs/2510.01190 · HTML: https://mlsanity.com/paper/2510.01190/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 37 references that are in the map.
- First posted on arXiv 21 Aug 2025.

## Builds on (most cited first, 6 of 6)

- [Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps](https://mlsanity.com/paper/1912.06341/) (2019)
- [Direct Volume Rendering with Nonparametric Models of Uncertainty](https://mlsanity.com/paper/2008.13576/) (2020)
- [Uncertainty Visualization of the Marching Squares and Marching Cubes Topology Cases](https://mlsanity.com/paper/2108.03066/) (2021)
- [Accelerated Probabilistic Marching Cubes by Deep Learning for Time-Varying Scalar Ensembles](https://mlsanity.com/paper/2207.07260/) (2022)
- [Glyph-Based Uncertainty Visualization and Analysis of Time-Varying Vector Fields](https://mlsanity.com/paper/2409.00042/) (2024)
- [Uncertainty Visualization of Critical Points of 2D Scalar Fields for Parametric and Nonparametric Probabilistic Models](https://mlsanity.com/paper/2407.18015/) (2024)

## Abstract (arXiv)

This work focuses on visualizing uncertainty of local divergence of two-dimensional vector fields. Divergence is one of the fundamental attributes of fluid flows, as it can help domain scientists analyze potential positions of sources (positive divergence) and sinks (negative divergence) in the flow. However, uncertainty inherent in vector field data can lead to erroneous divergence computations, adversely impacting downstream analysis. While Monte Carlo (MC) sampling is a classical approach for estimating divergence uncertainty, it suffers from slow convergence and poor scalability with increasing data size and sample counts. Thus, we present a two-fold contribution that tackles the challenges of slow convergence and limited scalability of the MC approach. (1) We derive a closed-form approach for highly efficient and accurate uncertainty visualization of local divergence, assuming independently Gaussian-distributed vector uncertainties. (2) We further integrate our approach into Viskores, a platform-portable parallel library, to accelerate uncertainty visualization. In our results, we demonstrate significantly enhanced efficiency and accuracy of our serial analytical (speed-up up to 1946X) and parallel Viskores (speed-up up to 19698X) algorithms over the classical serial MC approach. We also demonstrate qualitative improvements of our probabilistic divergence visualizations over traditional mean-field visualization, which disregards uncertainty. We validate the accuracy and efficiency of our methods on wind forecast and ocean simulation datasets.