Geometric Approximation and Variational Problems

拟议的研究计划解决了几何数值方法中的问题。除了大量的工程应用外，精确的几何信息近似和估计计算方法可以帮助理解和拯救生命。生物膜问题的数值处理是生命科学应用的一个例子。多值数据的逼近方法可以应用于扩散张量图像数据来重建人脑的白质结构，这种技术在精神疾病的诊断中显示出了良好的前景。由于在机器学习、信号处理和控制系统等领域的成功应用，大规模数值优化被认为是工程中的一个关键组成部分，耐心地研究具有实际意义的具体几何问题的优化方法将有助于理解解决大规模优化问题。这项研究中概述的项目还为研究生提供跨学科的研究和培训机会，并促进计算数学家、工程师和科学家之间的合作。我们研究成果的公开软件实施进一步促进了这种培训和合作。一些项目的标题为“几何近似和变分问题”。Wmincon软件的扩展及其在广义相对论几何变分问题中的应用这项工作详细介绍了与几何近似和变分问题有关的一系列研究，包括系统地研究生物膜和机械工程中的双层平板模型的数值解，以及几何数据的近似和分析。这些项目将导致几何学、最优化理论、计算数学以及工程模拟、新几何信号处理和几何机器学习等应用领域的交叉滋养。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The proposed research program address problems in geometric numerical methods. Besides numerous engineering applications, accurate computational methods for approximation and estimation of geometric information can help understanding and saving lives. Numerical treatment of biomembrane problems is one example of application to life sciences. Approximation methods of manifold-valued data can be applied to diffusion tensor image data for reconstructing white matter structure of human brain; such techniques have shown promises in diagnosing psychiatric disorders. Due to the success in applications such as machine learning, signal processing, and control system, large scale numerical optimization is now considered as a key component of engineering, a patient study of optimization methods for specific geometric problems with practical relevance will contribute to the understanding of solving large scale optimization problems. The projects outlined in this research also provide interdisciplinary research and training opportunities for graduate students, and stimulate collaboration among computational mathematicians, engineers and scientists. The publicly available software implementation of our research results further facilitates such training and collaborations.A number of projects under the headline of "geometric approximation and variational problems". Extensions of the Wmincon software with applications to geometric variational problems in general relativity This work details a line of research related to geometric approximation and variational problems, including systematic studies of numerical solution of biomembranes and bilayer plate models from mechanical engineering, as well as approximation and analysis of geometric data. The projects will lead to a cross fertilization of geometry, optimization theory, computational mathematics, as well as application areas such as engineering simulation, processing of novel geometric signals, and geometric machine learning.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.