New Developments in Geometric and Multiscale Numerical Methods

这个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。该项目中开发的方法将阐明以下问题：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，脂质双分子层是形成所有细胞边界的生物膜的最基本和不可缺少的结构组分。为了理解它们所表现出的惊人的各种不同形状，生物制药学家努力为这种生物膜找到一个精确的数学模型。随着数学和计算技术的发展，在这个项目中，这样的模型的解决方案可以得到准确和有效的。类似的数学技术也可以应用于从大脑的扩散张量磁共振图像（DT-MRI）中揭示白色物质结构，并实时预测一种称为空气动力学颤振的危及生命的现象。这将有助于诊断精神疾病，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，以及尊重底层问题的几何或拓扑特征的数值算法设计。具体而言，本研究解决了以下问题：I。流形值数据的近似（离散流形值数据和降阶建模，以及非线性细分算法理论）; II.新的多尺度几何建模工具（脂质双层的数值模拟、亏格0拓扑的细分微分形式和3-D细分方法）。研究将结合联合收割机技术，从数学分析，几何，数值分析，优化理论和计算，以促进应用几何问题的理解。

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.