High Order Schemes for Gradient Flows and Interfacial Motion

在广泛的科学和工程领域中，许多重要的现象都是用移动的曲线或曲面来描述的。例如，在材料科学中，大多数金属和陶瓷的内部结构揭示了数百万个微小的单个微晶粘在一起。当材料在常见的制造过程中被加热时，这些微小微晶的边界并因此将它们彼此分开的表面网络开始移动，例如锻造或退火。由这种表面网络定义的微晶的形状和大小对材料的重要物理特性有影响，如导电性和屈服强度。另一个非常不同的例子来自计算机视觉，在计算机视觉中，自动将数字图像中的前景对象与背景分开的常见技术是从一条曲线开始，例如包含前景对象的大圆，然后规定一条更新规则，该更新规则缩小圆直到它跑到对象的边缘，在该过程中围绕它收缩并捕捉其轮廓。然后，为了识别目的，可以将轮廓与形状库进行比较。在这两个应用中，就像在其他许多应用中一样，描述所涉及的界面运动的方程通常属于一个重要的类别，称为梯度流，或最陡下降：这种演变可以被描述为减少适当的成本函数或能量的最快方式。该项目将开发高精度和可靠的数值方法，在计算机上模拟这些演变。它包括支持研究生的研究培训，以及为本科生提供暑期研究机会。该项目将开发非常通用的、与问题无关的技术，以提高现有演化方程数值格式的时间精度，这些方程是作为一种能量的梯度流(最陡下降)而产生的。在梯度流的数值分析中，一个自然的稳定性条件是能量稳定性：代价函数是否从一个时间步耗散到下一个时间步。用来提高现有格式精度的新技术将保持理想的稳定性。例如，如果现有格式是一阶时间精度和无条件能量稳定的，则其精度阶数将提高到二阶或更高，但其无条件稳定性将保持不变。此外，只需在每个时间步多次调用原始方案的黑盒实现，就可以实现精度的提高。该项目的主要目标将是将该技术扩展到流行的数值方法(水平集方法、阈值动力学)，用于梯度流引起的界面几何运动，例如平均曲率的多相运动。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many important phenomena in a wide variety of scientific and engineering fields are described by moving curves or surfaces. For example, in materials science, the internal structure of most metals and ceramics reveal millions of tiny individual crystallites stuck together. The network of surfaces that delineate the boundaries of these tiny crystallites and thus separate them from one another begins to move when the material is heated during common manufacturing processes such as forging or annealing. The shapes and sizes of the crystallites, defined by this network of surfaces, have implications for important physical characteristics of the material, such as its conductivity and yield strength. Another, very different example comes from computer vision, where a common technique for automatically separating the foreground object from the background in a digital image is to start with a curve, such as a large circle containing the foreground object, and then prescribe an update rule that shrinks the circle until it runs into the edges of the object, shrink-wrapping around it and capturing its outline in the process. The outline can then be compared to a library of shapes, for recognition purposes. In both applications, as in many others, the equations describing the motion of the interfaces involved often fall into an important class known as gradient flow, or steepest descent: The evolution can be characterized as the fastest way to decrease an appropriate cost function or energy. This project will develop highly accurate and reliable numerical methods for simulating these evolutions on the computer. It includes support for research training of a graduate student, as well as summer research opportunities for undergraduate students.The project will develop very general, problem independent techniques for boosting the order of accuracy in time of existing numerical schemes for evolution equations that arise as gradient flow (steepest descent) for an energy. A natural stability condition in the numerical analysis of gradient flows is energy stability: whether the cost function is dissipated from one time step to the next. The new techniques for boosting the order of accuracy of existing schemes will preserve desirable stability properties. For example, if the existing scheme is first order accurate in time and unconditionally energy stable, its order of accuracy will improve to second order or higher, but its unconditional stability will be preserved. Moreover, the improvement in accuracy will be achieved by merely calling multiple times per time step a black-box implementation of the original scheme. A primary goal of the project will be to extend the technique to popular numerical methods (the level set method, threshold dynamics) for geometric motions of interfaces that arise as gradient flow, such as multiphase motion by mean curvature.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.