High Order Schemes for Gradient Flows and Interfacial Motion

梯度流和界面运动的高阶方案

基本信息

项目摘要

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.
许多科学和工程领域中的重要现象都可以用运动曲线或曲面来描述。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微小的单个微晶粘在一起。当材料在锻造或退火等常见制造过程中被加热时,描绘这些微小微晶边界并因此将它们彼此分离的表面网络开始移动。由这种表面网络定义的微晶的形状和尺寸对材料的重要物理特性(例如其导电性和屈服强度)具有影响。另一个非常不同的例子来自计算机视觉,在数字图像中自动将前景对象从背景中分离出来的常用技术是从一条曲线开始,例如包含前景对象的大圆,然后规定一个更新规则,缩小圆直到它进入对象的边缘,在此过程中收缩包裹并捕获其轮廓。然后可以将轮廓与形状库进行比较,以用于识别目的。在这两种应用中,与许多其他应用一样,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:演化可以被描述为降低适当成本函数或能量的最快方式。该项目将开发高度精确和可靠的数值方法,在计算机上模拟这些演变。该项目将开发非常通用的、独立于问题的技术,用于提高现有演化方程数值格式的时间精度,这些格式是作为能量的梯度流(最陡下降)出现的。梯度流数值分析中的一个自然稳定性条件是能量稳定性:从一个时间步到下一个时间步,代价函数是否耗散。提高现有方案精度的新技术将保持理想的稳定性。例如,如果现有的格式在时间上是一阶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被保持。此外,准确性的提高将通过仅在每个时间步长多次调用原始方案的黑盒实现来实现。该项目的一个主要目标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),用于梯度流界面的几何运动,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价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
High order, semi-implicit, energy stable schemes for gradient flows
  • DOI:
    10.1016/j.jcp.2021.110688
  • 发表时间:
    2020-07
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Alexander Zaitzeff;S. Esedoglu;K. Garikipati
  • 通讯作者:
    Alexander Zaitzeff;S. Esedoglu;K. Garikipati
High order schemes for gradient flow with respect to a metric
相对于度量的梯度流的高阶方案
  • DOI:
    10.1016/j.jcp.2023.112516
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    4.1
  • 作者:
    Han, Saem;Esedoḡlu, Selim;Garikipati, Krishna
  • 通讯作者:
    Garikipati, Krishna
A Monotone, Second Order Accurate Scheme for Curvature Motion
曲率运动的单调二阶精确方案
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Selim Esedoglu其他文献

Selim Esedoglu的其他文献

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{{ truncateString('Selim Esedoglu', 18)}}的其他基金

Computational Tools for Polycrystalline Materials
多晶材料的计算工具
  • 批准号:
    1719727
  • 财政年份:
    2017
  • 资助金额:
    $ 27万
  • 项目类别:
    Standard Grant
Algorithms for Multiple Phases
多阶段算法
  • 批准号:
    1317730
  • 财政年份:
    2013
  • 资助金额:
    $ 27万
  • 项目类别:
    Continuing Grant
Collaborative Research: ATD (Algorithms for Threat Detection): Inverse Problems Methods in Chemical Threat Detection
合作研究:ATD(威胁检测算法):化学威胁检测中的反问题方法
  • 批准号:
    0914567
  • 财政年份:
    2009
  • 资助金额:
    $ 27万
  • 项目类别:
    Continuing Grant
CAREER: Analysis and Modeling for Image Processing Problems
职业:图像处理问题的分析和建模
  • 批准号:
    0748333
  • 财政年份:
    2008
  • 资助金额:
    $ 27万
  • 项目类别:
    Standard Grant
New Models and Algorithms in Image Processing with Partial Differential Equations
偏微分方程图像处理的新模型和算法
  • 批准号:
    0713767
  • 财政年份:
    2007
  • 资助金额:
    $ 27万
  • 项目类别:
    Standard Grant
Geometric and Multiscale Aspects of Image Denoising Models
图像去噪模型的几何和多尺度方面
  • 批准号:
    0605714
  • 财政年份:
    2005
  • 资助金额:
    $ 27万
  • 项目类别:
    Standard Grant
Geometric and Multiscale Aspects of Image Denoising Models
图像去噪模型的几何和多尺度方面
  • 批准号:
    0410085
  • 财政年份:
    2004
  • 资助金额:
    $ 27万
  • 项目类别:
    Standard Grant

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特征p中的类群方案岩泽理论
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Mathematical analyses on one-bit secret sharing schemes and their extensions
一位秘密共享方案及其扩展的数学分析
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