Numerical Computing on Evolving Domains

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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.********