Efficient algorithms for evolving continuum processes on curved surfaces

表面连续处理是各种现代应用的重要组成部分。这些包括基于物理的计算机动画对象建模，形状的理解和表征，物体纹理的应用和增强，以及阿尔茨海默病皮层变化的映射。偏微分方程（PDEs）是为平面和曲面上的连续过程制定数学算法的基本工具。然而，当过程发生在曲面上而不是在标准笛卡尔坐标空间上时，处理这样的方程要复杂得多。因此，解决底层方程所需的算法和软件往往难以理解，效率低下，或者根本不可用。我的长期愿景是为一般几何（静态或移动，开放或封闭，分段平滑或点云，以及某些一般嵌入空间内的任意协维）的一般连续统模型（包括标准以及退化微分算子，约束，积分等）开发有效的算法和软件。与此相一致，我们引入并发展了最近点法。这种方法的优点是将复杂问题显著地简化为插值和连续体演化两个标准问题。迄今为止，关于最近点方法的大多数工作都集中在某些光滑、移动表面上偏微分方程的数值近似和最终用户对该方法的实际应用。在提出的研究中，我们(i)对原始的显式最近点方法进行了第一次详细分析，（ii）分析并开发了CPM的并行算法和软件，（iii）推导出有效的时间进化策略，（iv）将方法扩展到实际兴趣的新流的近似，以及(v)构建曲面之间的映射，从而实现新的，有效的曲面处理方法。该研究项目开发的算法和软件既准确高效，又简单，因为它们在利用现有的3D标准算法和软件的同时，尽可能统一地计算不同连续体模型的解决方案。它提高了当前使用方法的效率，为提高对现有方法和新方法的理解进行了分析，并使目前无法计算的表面过程的数值逼近成为可能。它还开发了第一个领域分解软件，用于并行计算一些最常见的运动表面问题的解决方案。因此，在这项资助下开发的方法和软件将使研究人员和最终用户能够以高精度的数值研究复杂移动表面上一般连续过程的新的和现实的模型。

Continuum processes on surfaces are essential components to a remarkable variety of modern applications.  These include the physics-based modelling of computer-animated objects, the understanding and characterization of shape, the application and enhancement of texture on objects, and the mapping of cortical change in Alzheimer's disease.  Partial differential equations (PDEs) are the fundamental tools for formulating mathematical algorithms for continuum processes on flat spaces and curved surfaces.  However, working with such equations is much more complicated when the processes occur on curved surfaces rather than on standard Cartesian coordinate spaces.  As a consequence, the algorithms and software needed to solve the underlying equations are often poorly understood, inefficient, or simply unavailable. My long term vision is the development of efficient algorithms and software for general continuum models (involving standard as well as degenerate differential operators,  constraints, integrals, etc.) on general geometries (static or moving, open or closed, piecewise smooth or point cloud, and of arbitrary co-dimension within some general embedding space).  Consistent with this, we have introduced and developed closest point methods.  Such methods have the advantage of dramatically simplifying complex problems into the two standard problems of interpolation and continuum evolution.  To date, most work on closest point methods has focused on the numerical approximation of PDEs on certain smooth, moving surfaces and on the practical application of the method by end-users.  In the proposed research, we (i) conduct the first detailed analysis of the original explicit closest point method, (ii) analyze and develop parallel algorithms and software for the CPM, (iii) derive efficient time-evolution strategies, (iv) extend methods to the approximation of new flows of practical interest, and (v) construct maps between surfaces thereby enabling new, efficient methods for the processing of surfaces. The program of research develops algorithms and software that are accurate and efficient, yet are simple in the sense that they compute solutions to different continuum models as uniformly as possible while leveraging the use of existing standard algorithms and software in 3D. It improves the efficiency of methods in current use, conducts analysis for the improved understanding of existing and new methods, and enables the numerical approximation of surface processes that cannot presently be computed. It also develops the first domain decomposition software for the parallel computing of solutions to some of the most frequently occurring problems on moving surfaces.  As a consequence, the methods and software developed under this grant will enable researchers and end-users to numerically investigate new and realistic models of general continuum processes on complex moving surfaces with high accuracy.