Numerical Methods for Nonlinear Partial Differential Equations, with applications to Optimal Transportation, and Geometric Data Reduction

应用数学和计算数学的角色正在发生根本性的变化。在科学、技术、商业和医学领域，我们收集和存储数据的能力正在急剧增加。这些发展要求我们改变对数据的理解，并要求开发新的工具。计算数学的技术是唯一适合构建这些工具的技术。*我建议为大数据集构建新的算法，并将这些算法应用于解决重要的现实世界问题。其主要目标是：*1.开发我的算法的扩展以解决最优运输(OT)问题。数学的基本工具之一是将度量结构强加给复数集。沃瑟斯坦距离对于概率度量集是这样做的。然而，这个距离的计算并不是明确的：它是作为一个无限维变分问题--最优运输问题的解来获得的。建立好的方法来解决这个问题，这样就可以计算出这个距离。*2.开发一个使用极值点的数据约简算法。给出一个二维点的散点图，极值是离中心最远的。在数学上，用凸包来表示极值。已经有了寻找凸壳的算法，但当点的数量或点的维度太大时，这些算法就会失效。提出的研究将开发算法来寻找大量高维数据的极值点。*3.开发一种用于异常检测的算法--多维快速排序。当您检查您每月的信用卡账单时，您可以执行异常检测。例如，寻找不寻常的项目，要么是因为收取的金额，要么是因为供应商没有被识别。拟议的研究将开发算法，以非常快速地对海量数据执行异常检测。通过使用密度函数来简化数据，密度函数代表了我们正在寻找的异常类型的模式。通过求解一个涉及密度函数的方程来代替对点进行排序的过程。一旦我们有了方程的解，我们就可以用它来快速地发现异常。*4.继续建立非线性偏微分方程组(PDE)解算器的基础工作。*相应的预期结果包括：*1.(A)一种比较文档或图像的有效方法。*(B)一种几何敏感的方法来执行形状或图像的配准。*(C)一种最先进的图像扭曲工具，供美国医院的几位顶级外科医生使用。*2.(A)用于寻找高维数据极值的数据简化工具。*(B)改进了对直升机的离线和潜在的实时性能监测。*3.股票异常的流媒体检测。*4.新方程的有效解算器，导致类似上述1-3的未来结果。**

A fundamental change is taking place in the role of applied and computational mathematics.****Across science, technology, commerce, and medicine our ability to collect and store data is increasing dramatically. These developments call for a change in our understanding of data and demand the development of new tools. The techniques of computational mathematics are uniquely suited for building these tools. ****I propose to build new algorithms for large data sets and apply these algorithms to solve important real-world problems. The primary objectives are to: ****1. Develop an extension of my algorithm for solving the Optimal Transportation (OT) problem. One of the fundamental tools of mathematics is impose a metric structure on complex sets. The Wasserstein distance does so for the set of probability measures. However, evaluating this distance is not given explicitly: it is obtained as the solution of an infinite dimensional variational problem, the Optimal Transportation problem. Building good methods for solving this problem allow this distance to be computed.***2. Develop an algorithm for data reduction using extremal points. Given a scatterplot of two dimensional points, the extremal values are the farthest out from the centre. Mathematically, the convex hull is used to represent the extremal values. There are already algorithms for finding convex hulls, but they break down when the number of points, or the dimension of the points, is too large. The proposed research will develop algorithms to find extremal points of large set of high dimensional data.***3. Develop an algorithm, Multidimensional Quicksort, for anomaly detection. You may perform anomaly detection when you check your monthly credit card bill. For example, looking for items that are unusual, either because of the amount charged, or because the vendor is not recognized. The proposed research will develop algorithms to perform anomaly detection on huge amounts of data very quickly. The data is simplified by using a density function, which represents the pattern of the types of anomalies we are looking for. The process of sorting the points is replaced by solving an equation involving the density function. Once we have the solution of the equation, we can use it to quickly find the anomalies.***4. Continue foundational work building of nonlinear Partial Differential Equation (PDE) solvers. ****The corresponding expected outcomes include: ***1.(a) An effective method to compare documents or images.*** (b) A geometry-sensitive method to perform registration of shapes or images.*** (c) A state-of-the-art image warping tool, to be used by several top surgeons at US hospitals. ***2.(a) A data reduction tool for finding extreme values of high dimensional data.*** (b) Improved off-line and potential real-time performance monitoring for helicopters. ****3. Streaming detection of stock anomalies. ****4. Effective solvers for new equations, leading to future outcomes like 1-3 above.**