Advances in Level Set and Related Methods: New Technology and Applications

NSF提案：DMS-0074735“水平集和相关方法的进展：新技术和应用” 主要研究者：由Osher和Sethian于1988年提出的水平集方法，作为一种数值和理论手段，在表示和分析R^2中曲线和R^3中曲面的运动方面，已经被证明是非常成功的。 一个水平集演算已经开发，最近的扩展包括鬼流体方法，卷积生成的运动，动态表面扩展，变分水平集方法，和更高的余维对象的运动。 最近的应用包括多相流体动力学，岛动力学模型的外延生长，水平集为基础的无组织点的插值，和快速方法在图像恢复。 这项工作得到了我们以前的NSF赠款的部分支持。 本研究的目标是通过以下两个项目来扩展该技术和应用范围：（1）卷积产生的细丝运动。(2)稳态几何Hamilton-Jacobi方程的快速算法及 水平集方法正迅速成为在计算机上模拟许多重要的物理、生物、材料科学、图像处理、计算机视觉、电磁和其他真实的世界问题的首选方法。 特别是纳米技术领域也将受到影响。 用于模拟这些现象的数值方法的改进最终将在计算机芯片的设计、爆炸分析、物体识别和现代技术的许多其他领域中至关重要。 该提议涉及水平集和相关方法的进一步改进。

NSF Proposal: DMS-0074735 "Advances in Level Set and Related Methods: New Technologies and Applications" Principal Investigator: Stanley J. OsherABSTRACTThe level set method devised by Osher and Sethian in 1988 has proven to be phenomenally successful as a numerical and theoretical device for representing and analyzing the motion of curves in R^2 and surfaces in R^3. A level set calculus has been developed, and recent extensions include the ghost fluid method, convolution generated motion, dynamic surface extension, the variational level set approach, and the motion of higher codimensional objects. Recent applications include multiphase fluid dynamics, the island dynamics model for epitaxial growth, level set based interpolation of unorganized points, and fast methods in image restoration. This work was partially supported by our previous NSF grants. The goal of this proposed research is to extend the technology and the range of applications through the following two projects:(1) Convolution generated motion for filaments.(2) Fast algorithms for steady state geometric Hamilton-Jacobi equations and the induced motion of fronts.The level set method is rapidly becoming the method of choice to simulate on the computer a host of important physical, biological, materials science, image processing, computer vision, electromagnetic and other real world problems. In particular areas of nanotechnology will also be impacted. Improvements of the numerical methods used to simulate these phenomena will ultimately be crucial in the design of computer chips, analysis of explosions, recognition of objects and many other areas of modern technology. This proposal addresses further improvements of the level set and related methods.