High Order Schemes for Gradient Flows and Interfacial Motion

点击翻译按钮获取中文摘要

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.