Topics in Geometric and Multiscale Numerical Methods

在科学和工程的各个学科中，有一种新兴的兴趣是开发“生活”在黎曼流形或李群上的数据的简洁的多尺度表示。这些应用领域正在突飞猛进地发展，相关的应用问题也一直在出现。扩散张量成像和协同运动建模是新传感器类型和部署的简单例子，它们产生了大量的非线性流形数据。我们相信未来会有更多这样的方法出现。第一个提议的项目允许对非线性数据进行一种多尺度表示，就像小波对图像和信号所能做的那样。由此产生的多尺度表示是数据压缩、特征提取、噪声去除、快速搜索以及在利用此类数据时出现的许多其他重要问题的关键。在任意拓扑的2- d和3-D流形上扩展现有的细分方法以处理函数、向量场、1-形等也有日益增长的需求。第二个提议的项目解决了这些需求的一部分。圣杯是设计数值算法，在适当的意义上，尊重底层问题的几何或拓扑特征。第三个提议项目的动机是对纳米技术的巨大兴趣。据推测，计算纳米科学可能会逐渐走到科学计算的前沿，就像计算流体力学在科学计算中占据了几十年的前沿一样。在这个项目中，我们研究了电子结构计算中的一个中心方法，即Kohn-Shamfunctional最小化问题。提出了一种特殊的几何结构进行研究。最终目标是充分利用问题背后的光滑流形结构，提出更有效、更稳健、具有可证明、易于理解的收敛性质的线性缩放方法。我们的直接目标是分析和综合许多新类型的数据，特别是那些在非线性流形中取值的数据，以及函数、向量场和自由流形上的微分形式，这与寻找有效的方法来组织和处理大量复杂的高维几何数据的广泛和基本目标是一致的。在科学和工程领域，对这些方法的需求无处不在，因此该项目的潜在影响更为广泛。我们也相信，随着计算机辅助几何设计和计算机辅助工程领域目前相互融合，我们在这里的重点努力最终将找到解决大规模科学和工程模拟问题的方法。在不同的方向上，我们结合了严格的几何和数值思想来解决电子结构计算中的一个中心问题。从世界范围内对材料科学和纳米技术的兴趣来看，这个项目的广泛影响是显而易见的。这些项目还为研究生提供了跨学科的研究和培训机会，并促进了计算数学家、工程师和科学家之间的合作。我们的研究成果的公开软件实现进一步促进了这种培训和合作。

There is an emerging interest in various disciplinesof science and engineering to develop parsimonious multiscalerepresentations of data that `lives' on a Riemannian manifold orLie group. Such application areas are growing by leaps and boundsand cognate application problems are arising all the time.Diffusion tensor imaging and collaborative motion modelling aresimple examples of new sensor types and deployments that give riseto massive volumes of data taking values in nonlinear manifolds.We believe that many more such methods are going to be seen in thefuture. The first proposed project allows a kind ofmultiscale representation of nonlinear data that does for suchdata what wavelets were able to do for images and signals. Theresulted multiscale representations are the key to datacompression, feature extraction, noise removal, fast search, andmany other important problems that arise in exploiting such data.There is also an increasing need to extend the current subdivisionmethods to handle also functions, vector fields, 1-forms, etc. on2-D and 3-D manifolds of arbitrary topology. The second proposed projectaddresses part of these needs. The holy grail is to designnumerical algorithms that, in an appropriate sense, respect thegeometric or topological characteristics of the underlyingproblem. The third proposed project is motivated by thevast interests in nanotechnologies. It is speculated thatcomputational nanoscience may gradually take the forefront ofscientific computing in the same way that computational fluiddynamics was at the forefront of scientific computing forseveral decades. In this project we study a central method inelectronic structure computation known as the Kohn-Shamfunctional minimization problem. A specific geometric structure is proposed to be studied. The ultimate goal is to take full advantage of the smooth manifold structure underlying the problem to come up with linear scaling methods that are more efficient, more robust and possess provable, well-understood convergence properties.Our immediate goal of analysis and synthesis of many new types of data, especially those taking values in nonlinear manifolds, as well as functions, vector fields, and differential forms on free-form manifolds, fits right into the broad and fundamental goal of finding efficient ways to organize and manipulate enormous and complex volumes of high-dimensional geometric data. The need of such methods is ubiquitous in science and engineering, so the potential impact of the project is even wider. We also believe that our focused effort here will eventually find their way into large scale scientific and engineering simulation problems, as the fields of computer-aided geometric design and computer-aided engineering are currently converging to each other. In a different direction, we combine rigorous geometric and numerical ideas to attack a centralproblem in electronic structure calculations. The broader impact of this project is evident from the the world-wide interests in material sciences and nanotechnologies. These projects also provide interdisciplinary research and training opportunities for graduate students, and stimulates collaboration among computational mathematicians, engineers and scientists. The publicly available software implementation of our research results further facilitates such training and collaborations.