Combinatorial Vector Fields for Dynamical Systems and Shape Similarity

动力系统和形状相似性的组合矢量场

• 批准号: RGPIN-2016-03663
• 负责人: Kaczynski, Tomasz
• 金额: $ 1.31万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708836
• 关键词: Combinatorial; Vector; Fields; Dynamical; Systems; Combinatorial; Vector; Fields; Dynamical; Systems

My general long-term goal has been to develop Computational Topology in the context of analysis and processing of multidimensional data. In the past decade, the concept of combinatorial vector field, also called discrete vector field (abbreviation DVF) introduced by Forman became a highly successful tool for discretization of continuous problems in Computational Mathematics, Imaging and Visualization. My current research proposal is focused on developing combinatorial vector fields along two axes of applications: in dynamical systems and in the study of shape recognition. The applications of DVF to dynamical systems will be based on a modification of Forman's concept of V-paths and the use of a multivalued mapping dynamics introduced in my joint 2014 work with Mrozek and Wanner. This modification permits studying both periodic and asymptotic trajectories, thus giving access to a full specter of dynamical behavior. Our combinatorial model for the Lorenz attractor shows that DVFs also provide a good framework for describing chaotic dynamics. The primary goal is to establish a one-to-one correspondence between continuous and combinatorial vector field dynamics, in particular on the level of Conley index and Morse decomposition. We expect that our first publication is a starting point for a very promising research direction. The applications of DVF to shape recognition are a continuation of my recently initiated research in collaboration with Allili and Landi. Our matching algorithm is the first tentative of adapting the DVF methods to functions with multidimensional values with the goal of speeding up the computation of multi-filtered persistent homology, that is, the subspace homology system filtered by a lattice rather than by a linear order. This structure will in turn be used for computing a refined shape similarity measure, currently bringing computational challenges. This project involves computer programming as well as a theoretical development of an appropriate extension of the Morse theory to multidimensional functions. Finally, I plan to work, jointly with Mischaikow, Mrozek, and Wanner, on a new edition of the Computational Homology book published by Springer in 2004. We were approached by Springer editors about the second edition several times but we postponed the project because the field developed so much since 2004 that this will mean writing practically a new book. As it was the case with the first edition, we expect that the process of writing the book will motivate the progress in current research and prompt for many student research projects.

我的一般长期目标是在分析和处理多维数据的背景下开发计算拓扑。 在过去的十年中，组合矢量场的概念（也称为离散矢量字段（缩写DVF））被eman引入的概念成为离散计算数学，成像和可视化连续问题的非常成功的工具。我目前的研究建议的重点是沿两个应用轴发展组合矢量场：在动力学系统和形状识别研究中。 DVF在动力系统上的应用将基于对Forman的V-Path概念的修改以及在我与Mrozek和Wanner的联合合作中介绍的多估计映射动力学的使用。这种修饰允许研究周期性和渐近轨迹，从而获得了动态行为的完整幽灵。我们针对洛伦兹吸引子的组合模型表明，DVF还为描述混乱动力学提供了一个良好的框架。主要目标是在连续和组合矢量场动力学之间建立一对一的对应关系，尤其是在Conley索引和Morse分解的水平上。我们希望我们的第一个出版物是一个非常有前途的研究方向的起点。 DVF在塑造识别方面的应用是我与Allili和Landi合作的最近开始的研究的延续。我们的匹配算法是将DVF方法调整为具有多维值的函数的第一个暂定性，目的是加快计算多滤波持续的同源性的计算，即，由lattice而不是线性顺序过滤的子空间同源性系统。此结构又将用于计算精致的形状相似性度量，目前带来了计算挑战。该项目涉及计算机编程以及将莫尔斯理论适当扩展到多维函数的理论发展。 最后，我计划与Mischaikow，Mrozek和Wanner共同合作，并在2004年Springer出版的新版本的《计算同源书》中合作。Springer编辑与Springer编辑联系了几次有关第二版，但我们推迟了该项目，因为自2004年以来，该领域的发展很大，以至于这将在2004年发展，这将意味着实际写一本新书。第一版的情况下，我们希望撰写本书的过程将激发当前研究的进步并促使许多学生研究项目。