Collaborative Research-ITR-High Order Partial Differential Equations: Theory, Computational Tools, and Applications in Image Processing, Computer Graphics, Biology, and Fluids

该项目旨在开发一个全面的研究和教育计划，在高阶偏微分方程（PDEs）描述的物理系统的计算方法和模拟领域。该计划将统一算法、可视化、理论和实验工作，并解决科学和技术领域的应用，包括计算机图形学、图像处理、生物学和流体。该项目推进了高阶偏微分方程领域的知识，特别强调曲面数据，并产生了解决生物学、图像处理、计算机图形学和一般流体中的基本问题的技术。新的科学是在计算技术，实验研究和不同的应用解决由一个多学科的团队。该项目汇集了计算机科学、应用数学、机械工程、物理以及电气和计算机工程五个领域。拟议活动的更广泛影响随着对高阶偏微分方程的兴趣日益增加，该项目产生的计算工具和经验的影响超出了本建议中的特定应用。学生们将接受不同寻常的广泛跨学科培训，计划中的研讨会将进一步汇集不同领域的专家。新的公共领域软件结合了开发的算法，使来自不同领域的研究人员能够使用高阶偏微分方程执行最先进的数值模拟和图形渲染他们感兴趣的应用程序。该研究项目的教育内容包括：(1)通过不同领域的pi共同指导，培养新的跨学科研究生和博士后；(2)开设计算机图形学、数值分析、高阶偏微分方程描述的物理现象建模/模拟等跨学科新课程；(3)首次将使用高阶偏微分方程的不同科学研究人员聚集在一起的研讨会。

This project seeks to develop a comprehensive research and education program in the area of computationalmethods and simulations of physical systems described by high order Partial Differential Equations (PDEs). The program will unify algorithmic, visualization, theoretical, and experimental efforts as well as address applications in areas of science and technology, including computer graphics, image processing, biology, and fluids.Intellectual merit of the proposed activityThis project advances knowledge in the area of high order PDEs, with particular emphasis on curved surface data, and produces enabling technology to address fundamental problems in biology, image processing, computer graphics, and fluids in general. The novel science is in the computationaltechniques, experimental research, and diverse applications addressed by a multi-disciplinary team. Thisproject brings together the five fields of computer science, applied mathematics, mechanical engineering, physics, and electrical and computer engineering.Broader impacts of the proposed activityWith the increasing interest in high order PDEs, the computational tools and experience resulting from this project impact beyond the particular applications in this proposal. Students will receive unusually broadinterdisciplinary training and the workshop planned further brings experts from different fields together. New public domain software incorporating the developed algorithms enables researchers from different fields using higher order PDEs to perform state-of-the-art numerical simulations and graphics rendering of their application of interest.Educational initiatives of this research program include: (1) new interdisciplinary training of graduate students and postdocs through co-mentoring by PIs in different fields; (2) new interdisciplinary courses in computer graphics, numerical analysis, and modeling/simulation of physical phenomena described by higher order PDEs; (3) a workshop bringing together for the first time diverse scientific researchers using high order PDEs.