Topics in Geometric and Multiscale Numerical Methods

几何和多尺度数值方法主题

• 批准号: 1115915
• 负责人: Thomas Yu
• 金额: $ 23.08万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115915&HistoricalAwards=false
• 关键词: Topics; Geometric; Multiscale; Numerical; Methods

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.

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约多尺度表示。这些应用领域正以跨越式的速度发展，相关的应用问题也一直在不断出现。扩散张量成像和协作运动建模是新传感器类型和部署的简单例子，它们会产生大量非线性流形上的数据。我们相信未来会出现更多这样的方法。第一个被提出的项目允许一种非线性数据的多尺度表示，它对这些数据做了小波对图像和信号所做的事情。由此产生的多尺度表示是压缩、特征提取、噪声去除、快速搜索以及利用这些数据所产生的许多其他重要问题的关键。第二个拟议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，尊重基本问题的几何或拓扑特征。第三个拟议的项目是出于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在科学计算的最前沿，就像计算流体力学几十年来一直走在科学计算的最前沿一样。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，称为Kohn-Sham泛函最小化问题。提出了一种具体的几何结构进行研究。我们的最终目标是充分利用光滑流形的结构来提出更有效、更鲁棒、具有可证明的、易于理解的收敛性质的线性尺度方法。我们的近期目标是分析和合成许多新类型的数据，特别是那些在非线性流形上取值的数据，以及自由形式流形上的函数、向量场和微分形式，正好符合寻找有效方法来组织和操作大量复杂的高维几何数据这一广泛而基本的目标。这种方法的需求在科学和工程中无处不在，因此该项目的潜在影响更加广泛。我们还相信，我们在这里集中的努力将最终找到他们的方式进入大规模的科学和工程模拟问题，计算机辅助几何设计和计算机辅助工程领域目前正在相互融合。在另一个不同的方向，我们结合联合收割机严格的几何和数值的想法来攻击一个中心问题，在电子结构计算。该项目的更广泛的影响是显而易见的，从材料科学和纳米技术的世界范围内的利益。这些项目还为研究生提供跨学科研究和培训机会，并促进计算数学家，工程师和科学家之间的合作。我们的研究成果的公开软件实施进一步促进了这种培训和合作。