New Developments in Geometric and Multiscale Numerical Methods

几何和多尺度数值方法的新进展

• 批准号: 1522337
• 负责人: Thomas Yu
• 金额: $ 23万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522337&HistoricalAwards=false
• 关键词: New; Developments; Geometric; Multiscale; Numerical

This research project will study representation and approximation of complex data structures involving geometrical configurations and shapes. The methods developed in this project will shed light on questions such as: Why do human red blood cells have a bi-concave donut shape instead of, say, a peanut shape? It is known that the lipid bilayer is the most elementary and indispensable structural component of biological membranes that form the boundary of all cells. To understand the amazing variety of different shapes they exhibit, biophysicists strive to find an accurate mathematical model for such bio-membranes. With the mathematical and computational techniques under development in this project, the solution of such models can be obtained accurately and efficiently. Similar mathematical techniques can also be applied to unravel white matter structure from diffusion tensor magnetic resonance images (DT-MRI) of the brain and to predict, in real-time, a life-threatening phenomenon called aerodynamic flutter in air transportation. These will help diagnose psychiatric disorders and save lives of pilots and passengers. The project studies multi-scale representation and approximation of manifold-valued data, reduced order modeling, as well as design of numerical algorithms that respect the geometric or topological characteristics of the underlying problem. In particular, this research addresses the following: I. Approximation of manifold-valued data (scattered manifold-valued data and reduced order modeling, and theory of nonlinear subdivision algorithms); II. New multi-scale geometric modeling tools (numerical simulation of lipid bilayers, subdivision differential forms for genus 0 topology, and 3-D subdivision methods). The research will combine techniques from mathematical analysis, geometry, numerical analysis, optimization theory, and computing to advance the understanding of applied geometry problems.

该研究项目将研究涉及几何配置和形状的复杂数据结构的表示和近似。该项目开发的方法将揭示以下问题：为什么人类红细胞具有双凹甜甜圈形状，而不是花生形状？众所周知，脂质双层是构成所有细胞边界的生物膜最基本和不可缺少的结构成分。为了了解它们所表现出的各种不同形状，生物物理学家努力为此类生物膜找到准确的数学模型。利用该项目正在开发的数学和计算技术，可以准确有效地获得此类模型的解。类似的数学技术也可用于从大脑的扩散张量磁共振图像（DT-MRI）中解开白质结构，并实时预测航空运输中一种称为气动颤振的危及生命的现象。这些将有助于诊断精神疾病并挽救飞行员和乘客的生命。该项目研究流形值数据的多尺度表示和近似、降阶建模以及尊重潜在问题的几何或拓扑特征的数值算法的设计。具体而言，本研究涉及以下内容： 一、流形值数据的逼近（离散流形值数据和降阶建模、非线性细分算法理论）；二.新的多尺度几何建模工具（脂质双层的数值模拟、0 属拓扑的细分微分形式和 3-D 细分方法）。该研究将结合数学分析、几何、数值分析、优化理论和计算技术，以增进对应用几何问题的理解。