Probabilistic Methods in Computational Topology

计算拓扑方法越来越成功地用于分析和理解来自实验来源和数值模拟的复杂数据。许多拓扑方法使用同调理论和相关的贝蒂数来关注基本的拓扑信息。通常，在计算目标的同调性之前，对其进行离散化处理。这种离散化过程可能会引入导致拓扑信息不正确的伪影。由于噪声，不确定性也可能在实验环境中发生。在这个项目中，研究者和他的同事们开发了两种机制来定量评估拓扑信息的正确性。第一种方法是基于大多数应用问题涉及随机性的认识，它通过提供显式概率估计来研究随机场同调计算的准确性，该估计给出了适当问题离散化的先验信息。第二种机制开发了一种基于随机细分的严格计算后验正确性的方法。此外，研究人员还讨论了两个各自有趣的具体应用。首先，他们开发了一种寻找动力系统中隔离块的新方法，该方法具有广泛的应用前景，并为研究进化方程的动力学特性提供了一种新的工具。第二个应用侧重于材料科学中的模式分类，并导致多组分合金中可能的相分离现象的分类。计算科学的最新进展导致了生成数据量的巨大增加，而从这些数据中快速可靠地提取基本信息的新技术的发展已成为一个中心问题。该项目研究开发了拓扑数据分析的概率方法，包括定量可靠性评估。该项目可应用于材料科学领域，如研究工业级金属合金的性能和设计新材料。拓扑学方法已经被证明可以为这些材料的研究提供重要的见解，并且在这个项目中开发的技术被应用于材料表征和设计的背景下。这导致了多组分合金中相分离现象的分类，并有可能导致系统反应图，然后可以用于材料的设计，例如设计动脉支架的药物释放控制涂层。该项目还导致使用同源理论对随机场进行更广泛的研究，从长远来看，这在医学成像中有应用。该项目涉及本科生和研究生阶段的学生研究。

Computational topological methods are increasingly and successfully being used to analyze and understand complex data, provided both from experimental sources and numerical simulations. Many of these topological methods use homology theory and the related Betti numbers to concentrate on essential topological information. Usually the objects of interest are discretized before their homology can be computed. This process of discretization can introduce artifacts which cause the resulting topological information to be incorrect. Uncertainty can also occur in experimental settings due to noise.In this project, the investigator and his colleagues develop two mechanisms for a quantitative assessment of the correctness of topological information.The first approach is based on the realization that most applied problems involve randomness, and it studies the accuracy of homology computations for random fields by providing explicit probability estimates which give a-priori information on suitable problem discretizations. The second mechanism develops a method for rigorous computational a-posteriori correctness based on randomized subdivisions. In addition, the investigators address two specific applications which are interesting in their own right. First, they develop a new method for finding isolating blocks in dynamical systems, which has a wide range of possible applications and provides a novel tool for studying dynamical properties of evolution equations. The second application focuses on pattern classification in materials science and leads to a classification of possible phase separation phenomena in multi-component alloys.Recent advances in computational sciences have led to an immense increase in the amount of generated data, and the development of novel techniques to quickly and reliably extract essential information from this data has become a central issue. The project research develops probabilistic approaches for topological data analysis, including quantitative reliability assessments.The project has applications in the field of materials science, such as in the study of the properties of industrial level metal alloys and the design of new materials. Topological methods have already been shown to provide important insight into such materials studies, and the techniques developed in this project are applied in the context of materials characterization and design. This leads to a classification of phase separation phenomena in multi-component alloys and has the potential to lead to system-response-maps, which can then be used in the design of materials, such as for example the design of controlled drug-release coatings for arterial stents. The project also leads to a broader study of random fields using homology theory, which in the long term has applications in medical imaging. The project involves student research at both the undergraduate and graduate levels.