Numerical Computing on Evolving Domains

偏微分方程（PDEs）是科学、工程和其他领域的重要工具。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。它们也广泛应用于医学成像、诊断、数据处理和计算机视觉。由于精确的解很少可用，实际的进展往往是通过科学计算：即使用精确和有效的数值方法来计算近似解。******在许多实际应用中，问题的曲面几何域起着重要的作用：例如，人们在生物学中不会遇到许多矩形结构。通常，这些曲线和现实的领域是随时间变化的（再一次，想想生物生长）。我建议组建一个学生团队来构建科学计算工具，以解决现实动态和发展领域中的这些PDE问题。我希望这些方法从各个角度来看都是简单的：*** *数学上的优雅；*** *** *** *** *** *** *** *** **** *可扩展，适用于广泛的“最终用户”；* * *•可以理解的,可靠的和可核查的,甚至非专家。***这需要在数学、数值分析和处理几何的计算实践方面取得根本性的进步：事实上，这是我和我以前的学生在过去几年里所取得的进步。如上所述，偏微分方程在广泛的人类活动中具有根本的重要性，这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进一步推动他们的模型和他们的科学。然而，要意识到在直接从事数值方法工作的社区之外的影响，任何新算法都必须首先在该社区之外使用。为了鼓励这种情况发生，我将：******•进一步将我的工作直接扩展到生物学、材料科学、工程、图像处理和其他领域的应用领域。开发一个带有示例库的软件工具。这转移了采用的焦点：而不是需要阅读，理解和重新实施我的研究，任何人都可以选择和运行一个简单的例子，然后专注于理解和如何将其扩展到她的特定领域。*** *实践开放和可重复的研究。例如，在我的出版物中使用的所有数值实验将是示例库的一部分。********

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.********