CAREER: A Measure Theoretic Framework for Topology-Based Visualization

由于计算设备的能力不断增强，多物理场模拟（如双黑洞合并和流体动力学）产生的数据呈指数级增长。与此同时，数据密集型科学依赖于数据的获取、管理、分析和可视化，其空间和时间分辨率不断提高。该项目开发了一套新的方法来支持科学数据可视化的核心任务（如特征跟踪、事件检测、集成分析和交互式可视化），以一种使用测量理论更能反映底层物理的方式。结果将通过一组开源软件工具实例化，这些工具将部署给材料科学和高性能计算领域的合作科学家，以及更大的研究社区。本项目利用几何测量理论、信息论和运输理论的工具进行基于拓扑的可视化，利用拓扑概念来描述、简化和组织数据，以实现科学的理解和交流。该项目侧重于两个技术组成部分。第一个组件将拓扑描述符表示为配备概率度量的度量空间，这支持通过物理量、信息量化和比较分析丰富拓扑描述符。第二个组件使用信息和传输理论来实现各种时变数据和集成的可视化任务。该项目将对应标准与最优运输的优化过程结合起来，以了解感兴趣的特征的演变；结合几何测量在事件检测中的不确定性；并利用度量空间的统计来指导交互可视化。研究者与科学家们密切合作，利用来自天体物理学、材料科学和机械工程的数据来评估和调整框架，以更好地反映潜在的物理学。该项目为本科生和研究生提供了多学科活动和培训机会的独特环境。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Data generated from multiphysics simulations, such as binary black hole mergers and fluid dynamics, have experienced exponential growth because of the growing capabilities of computing facilities. At the same time, data-intensive science relies on the acquisition, management, analysis, and visualization of data with increasing spatial and temporal resolutions. This project develops a new set of approaches to support the core tasks in scientific data visualization (such as feature tracking, event detection, ensemble analysis, and interactive visualization) in a way that is more reflective of the underlying physics using measure theory. The results will be instantiated by a collection of open-source software tools to be deployed for the collaborating scientists in materials science and high-performance computing, and the larger research community. This project leverages tools from geometric measure theory, information theory, and transportation theory for topology-based visualization, which utilizes topological concepts to describe, reduce and organize data for scientific understanding and communication. The project focuses on two technical components. The first component represents topological descriptors as metric spaces equipped with probability measures, which supports their enrichments with physical quantities, information quantification, and comparative analysis. The second component uses information and transportation theory to enable a wide variety of visualization tasks for time-varying data and ensembles. The project couples correspondence criteria with optimization processes from optimal transport to understand the evolution of features of interest; incorporates uncertainty in event detection with geometric measures; as well as utilizes statistics of metric measure spaces to guide interactive visualization. The investigator works closely with scientists using data from astrophysics, materials science, and mechanical engineering to evaluate and tune the framework to better reflect the underlying physics. This project provides a unique environment for multidisciplinary activities and training opportunities for undergraduate and graduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.