Mingzhe Li

In this paper, we present a flexible and probabilistic framework for tracking topological features in time-varying scalar fields using merge trees and partial optimal transport. Merge trees are topological descriptors that record the evolution of connected components in the sublevel sets of scalar fields. We present a new technique for modeling and comparing merge trees using tools from partial optimal transport. In particular, we model a merge tree as a measure network, that is, a network equipped with a probability distribution, and define a notion of distance on the space of merge trees inspired by partial optimal transport. Such a distance offers a new and flexible perspective for encoding intrinsic and extrinsic information in the comparative measures of merge trees. More importantly, it gives rise to a partial matching between topological features in time-varying data, thus enabling flexible topology tracking for scientific simulations. Furthermore, such partial matching may be interpreted as probabilistic coupling between features at adjacent time steps, which gives rise to probabilistic tracking graphs. We derive a stability result for our distance and provide numerous experiments indicating the efficacy of our framework in extracting meaningful feature tracks.

Atmospheric blocking events are quasi-stationary high-pressure systems that disrupt the typical paths of polar and subtropical air currents, often producing prolonged extreme weather events such as summer heat waves or winter cold spells. Despite their critical role in shaping mid-latitude weather, accurately modeling and analyzing blocking events in long meteorological records remains a significant challenge. To address this challenge, we present an uncertainty visualization framework for detecting and characterizing atmospheric blocking events. First, we introduce a geometry-based detection and tracking method, evaluated on both pre-industrial climate model simulations (UKESM) and reanalysis data (ERA5), which represent historical Earth observations assimilated from satellite and station measurements onto regular numerical grids using weather models. Second, we propose a suite of uncertainty-aware summaries: contour boxplots that capture representative boundaries and their variability, frequency heatmaps that encode occurrences, and 3D temporal stacks that situate these patterns in time. Third, we demonstrate our framework in a case study of the 2003 European heatwave, mapping the spatiotemporal occurrences of blocking events using these summaries. Collectively, these uncertainty visualizations reveal where blocking events are most likely to occur and how their spatial footprints evolve over time. We envision our framework as a valuable tool for climate scientists and meteorologists: by analyzing how blocking frequency, duration, and intensity vary across regions and climate scenarios, it supports both the study of historical blocking events and the assessment of scenario-dependent climate risks associated with changes in extreme weather linked to blocking.

Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale using highperformance computing. To address this challenge, recent work has introduced distributed hierarchical contour trees for distributed computation and storage of contour trees. However, effective use of these distributed structures in analysis and visualization requires subsequent computation of geometric properties and branch decomposition to support contour extraction and exploration. In this work, we introduce distributed algorithms for augmentation, hypersweeps, and branch decomposition that enable parallel computation of geometric properties, and support the use of distributed contour trees as query structures for scientific exploration. We evaluate the parallel performance of these algorithms and apply them to identify and extract important contours for scientific visualization.

Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a set of merge trees using techniques from matrix sketching. That is, given a large set T of merge trees, we would like to find a much smaller set of basis trees S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be approximated via matrix sketching techniques such as principal component analysis and column subset selection. However, until now, there has not been any work on sketching a set of merge trees. We develop a framework for sketching a set of merge trees that combines matrix sketching with tools from optimal transport. In particular, we vectorize a set of merge trees into high-dimensional vectors while preserving their structures and structural relations. We demonstrate the applications of our framework in sketching merge trees that arise from time-varying scientific simulations. Specifically, our framework obtains a set of basis trees as representatives that capture the “modes” of physical phenomena for downstream analysis and visualization.