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

TL;DR

This work derives a closed-form approach for highly efficient and accurate uncertainty visualization of local divergence, assuming independently Gaussian-distributed vector uncertainties, and integrates this approach into Viskores, a platform-portable parallel library, to accelerate uncertainty visualization.

Abstract

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.

Figures (3)

• Figure 1: MC (blue curve) vs. analytical (red curve) divergence probability distribution. The mean and standard deviation values used are listed below the plots. The MC results are plotted for $1e+3$ (left column) and $1e+5$ (right column) samples.
• Figure 2: Visualization of divergence of the gradient of velocity magnitude field for the Red Sea dataset with 20 ensemble members. The top row (images a and b) depicts gradient fields for the two ensemble members visualized with the arrow glyphs. The zoomed-in views indicate source (positive divergence) positions in the gradient fields, which are representative of vortex core positions. However, the source positions vary considerably across the two ensemble members. Image c visualizes the divergence of the mean of ensemble of gradient fields with potential sources/vortex core positions (yellow/red regions) delineated using the isocontour (cyan) with the divergence isovalue $0.003$. The mean-field isocontour, however, does not account for uncertainty across the ensemble members. Image d visualizes probabilistic isocontour positions derived with our analytical approach. It reveals new potential positive divergence regions indicative of likely vortex cores that are missed or truncated in the mean-field visualization (as illustrated with the magenta boxes). The results in images e and f computed using the classical MC approach are much less accurate and slower than the analytical result in image d.
• Figure 3: Performance and accuracy plots for the Red Sea dataset. Our analytical method is far more efficient and accurate than the MC sampling method.