Probabilistic Methods in Computational Topology

计算拓扑方法越来越多地被成功地用于分析和理解复杂的数据，这些数据既来自实验来源，也来自数值模拟。许多拓扑学方法利用同调理论和相关的Betti数来集中研究本质的拓扑信息。通常，感兴趣的对象在计算其同源性之前被离散化。这种离散化过程可能会引入伪影，从而导致所得到的拓扑信息不正确。在这个项目中，研究人员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。第一种方法基于大多数应用问题都涉及随机性的认识，它通过提供关于适当问题离散化的先验信息的显式概率估计来研究随机场的同调计算的精度。第二种机制发展了一种基于随机细分的严格计算后验正确性的方法。此外，研究人员还介绍了两个本身就很有趣的具体应用。首先，他们发展了一种在动力系统中寻找孤立块的新方法，该方法具有广泛的可能应用，并为研究发展方程的动力学性质提供了一种新的工具。第二个应用集中在材料科学中的模式分类，并导致对多组分合金中可能的相分离现象的分类。最近计算科学的进步导致产生的数据量的巨大增加，开发新的技术来快速可靠地从这些数据中提取必要的信息已经成为一个中心问题。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。该项目在材料科学领域有应用，如在工业级金属合金性能研究和新材料设计中。拓扑学方法已经被证明为此类材料研究提供了重要的洞察力，本项目中开发的技术被应用于材料表征和设计。这导致了对多组分合金中的相分离现象的分类，并有可能产生系统响应图，然后可以用于材料设计，例如动脉支架的受控药物释放涂层的设计。该项目还导致了使用同调理论对随机场进行更广泛的研究，从长远来看，这一理论在医学成像中有应用。该项目涉及本科生和研究生两个层面的学生研究。

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.