Numerical Computing on Evolving Domains

演化领域的数值计算

• 批准号: RGPIN-2016-03757
• 负责人: Macdonald, Colin
• 金额: $ 1.97万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=658225
• 关键词: Numerical; Computing; Evolving; Domains

Partial differential equations (PDEs) are essential tools in science, engineering, and other fields. For example, PDEs are used as mathematical models for fluid flow, weather systems, stock markets, and chemical and biological processes. They are also used extensively in medical imaging, diagnosis, data processing, and computer vision. Because exact solutions are rarely available, practical progress is most often made through scientific computing: namely, the use of accurate and efficient numerical methods for computing approximate solutions.******In many realistic applications, the curved geometry of the domain of the problem plays a significant role: for example, one does not encounter many rectangular structures in biology. Often these curved and realistic domains are changing in time (again, think of biological growth). I propose assembling a team of students to build tools for scientific computing which solve these PDE problems on realistic dynamic and evolving domains. I want these methods to be simple from various points of view:*** • mathematical elegance;*** • efficiency and ease of implementation;*** • extensible, for broad “end-user” applicability;*** • understandable, reliable and verifiable, even by non-experts.***This will require fundamental advancements to the mathematics, numerical analysis and computational practice of dealing with geometry: indeed, the sort of advancements my previous students and I have been making for the past few years.***As mentioned above, the fundamental importance of PDEs over a broad spectrum of human endeavor means new algorithmic ideas can have a wide impact. When these tools are also easy-to-use and scale efficiently from “cartoon” test problems up to large-scale industrial applications, then both researchers and practitioners have the freedom to push their models and their science much further. However, to realize that impact outside of the direct community of people working on numerical methods, any new algorithms must first be used outside of that community. To encourage this to actually happen, I will:****** • further expand my work directly into application areas in biology, materials science, engineering, image processing, and other areas.*** • develop a software tool with a library of examples. This shifts the focus for adoption: instead of needing to read, understand and re-implement my research, anyone can choose and run a simple example and subsequently focus on understanding and how to extend it to her particular area.*** • practice open and reproducible research. For example, all numerical experiments used in my publications would be part of the library of examples.********

偏微分方程(PDE)是科学、工程等领域的重要工具。例如，偏微分方程被用作流体流动、天气系统、股票市场以及化学和生物过程的数学模型。它们还广泛应用于医学成像、诊断、数据处理和计算机视觉。由于精确解很少获得，实际的进展通常是通过科学计算取得的：即使用精确和高效的数值方法来计算近似解。在许多现实应用中，问题域的曲面几何起着重要的作用：例如，在生物学中不会遇到很多矩形结构。通常，这些弯曲而现实的领域正在随着时间的推移而变化(同样，想想生物增长)。我建议召集一个学生团队来构建科学计算工具，这些工具可以解决现实、动态和不断发展的领域中的PDE问题。我希望这些方法从不同的角度看都是简单的：*·数学上的优雅；*·实施的效率和易用性；*·可扩展的，适用于广泛的“最终用户”；*·可理解、可靠和可验证，即使是非专家也是如此。*这将需要在处理几何问题的数学、数值分析和计算实践方面取得根本性的进步：事实上，我和我之前的学生在过去几年里一直在取得的进步。*如上所述，偏微分方程在广泛的人类努力中的根本重要性意味着新的算法思想可以产生广泛的影响。当这些工具也易于使用并有效地从“卡通”测试问题扩展到大规模工业应用时，研究人员和实践者都可以自由地推动他们的模型和科学进一步发展。然而，为了在从事数值方法工作的人的直接社区之外实现这种影响，任何新的算法都必须首先在该社区之外使用。为了鼓励这一点真正发生，我将：*·进一步将我的工作直接扩展到生物学、材料科学、工程学、图像处理和其他领域的应用领域。*·开发一个带有实例库的软件工具。这转移了采用的重点：任何人都可以选择并运行一个简单的例子，然后专注于理解并如何将其扩展到自己的特定领域，而不是需要阅读、理解和重新实施我的研究。*·实践开放和可重复的研究。例如，我的出版物中使用的所有数值实验都将是实例库的一部分。