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作用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提高。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建立这些工具。 我建议为大型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问题。主要目标是： 1.扩展我的算法来解决最优运输（OT）问题。数学的基本工具之一是在复集合上强加一种度量结构。Wasserstein距离对于概率测度集就是这样。然而，这个距离的计算并没有明确给出：它是作为一个无限维变分问题的解，即最优运输问题。建立解决这个问题的好方法可以计算这个距离。 2.开发一种使用极值点的数据简化算法。给定二维点的散点图，极值是离中心最远的。在数学上，凸船体用于表示极值。已经有算法来寻找凸壳，但当点的数量或点的维数太大时，它们就失效了。该研究将开发算法来寻找极值点的大集合的高维数据。 3.开发一个算法，多维快速排序，异常检测。您可以在检查每月信用卡账单时执行异常检测。例如，寻找不寻常的项目，无论是因为收取的金额，还是因为供应商没有被识别。拟议的研究将开发算法，以非常快速地对大量数据进行异常检测。通过使用密度函数来简化数据，该密度函数表示我们正在寻找的异常类型的模式。对点进行排序的过程被求解一个涉及密度函数的方程所取代。一旦我们有了方程的解，我们就可以用它来快速找到异常。 4.继续非线性偏微分方程（PDE）求解器的基础工作。 相应的预期成果包括： 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，用于执行形状或图像的配准。 (c)一个国家的最先进的图像扭曲工具，将使用的几个顶级外科医生在美国医院。 2. (a)一个数据简化工具，用于查找高维数据的极值。 (b)改进了直升机的离线和潜在的实时性能监测。 3.股票异常的流式检测。 4.新方程的有效求解器，导致未来的结果，如上面的1-3。

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.