This statement summarizes my approach to research, the impact it has had, as well as my vision for the future. A curriculum vitae detailing many of the points made below can be found here. This statement is current as of the summer of 2014.
My research centers around developing methods for scientific computing, specifically for the numerical solution of partial differential equations. Within this field, I have specialized in the development of efficient finite element discretizations and solution methods for the high performance computing solution of complex problems that arise in engineering and the sciences. I also work extensively on providing widely used open source software to the general scientific computing community.
The overall focus of my research is interdisciplinary: it includes (i) the mathematical disciplines of designing and analyzing algorithms, typically using the finite element method, adaptive mesh refinement, sophisticated linear and nonlinear solvers, and their implementation in high performance computing environments; (ii) creating software to solve concrete problems; and (iii) applying the results of my work to real-world challenges in the sciences and engineering. In my collaborations, I often see myself as a knowledge transfer facilitator: I develop numerical methods and apply them to problems in a variety of disciplines; I also draw upon questions from the applied sciences as motivation for research into more fundamental questions.
Working at this intersection of engineering, the applied sciences, and mathematics requires a broad knowledge base across disciplines and a willingness to learn other disciplines' languages and approaches. Philosophically, my facilitation approach is inspired by my conviction that research has an obligation to not only create new results, but also to actively integrate those results into the body of knowledge and to disseminate them to others outside one's own field. By demonstrating the impact of newly developed computational methods in engineering and the applied sciences, I believe that my approach not only advances the state of the art in computing but also in the targeted applications areas.
Numerical methods for partial differential equations are in demand in a broad variety of disciplines and I have worked widely with colleagues in many areas. When entering collaborations, the question that drives my curiosity is "what is needed to solve this problem?" As a result of these interdisciplinary projects, I have written publications in journals in computational mathematics, geophysics, biomedical imaging, nuclear engineering, computer science, petroleum engineering, and mathematical software. (Each of the links above references only one paper in each category, but there are typically multiple.) Furthermore, I am the principal author of the deal.II finite element library that is widely used around the world and that is used in simulations in areas as diverse as plant root modeling and preservation of culturally significant artifacts. My work is motivated by the excitement of interdisciplinary interactions: Collaborating with people from a wide variety of fields, listening to their computer simulation needs and applying state-of-the-art mathematical algorithms to solve their problems.
As listed in my CV, I have been the PI or co-PI on a significant number of grants from a variety of funding agencies: NSF, the Department of Energy (DoE), the National Institutes of Health (NIH), the Department of Homeland Security (DHS), the Sloan Foundation, and an NSF center focused on geophysical research. As of 2014, I am the PI on two grants to develop computational algorithms and implement them in open source software ($814k and $1.5M). I was one of the co-PIs on a grant to develop methods to detect the illicit importation of nuclear materials ($7.5M). A few smaller grants complement this portfolio. All told, the grants on which I am or have been a PI or co-PI over my first 9 years as a faculty total approximately $10.7M.
For my research, I have been awarded the J. H. Wilkinson Prize for Numerical Software (for the creation of the deal.II software library; this personal prize of $3,000 was shared with my then co-authors Guido Kanschat and Ralf Hartmann), as well as a Sloan Research Fellowship (these fellowships support largely unrestricted research expenses). In a competitive process, my software has also become part of the SPEC CPU 2006 benchmark that is widely used to measure the speed of computers (the award is $5,000).
This all said, funding and awards mostly measure the appreciation of the community for one's work, but not immediately impact. The most significant impact of my research work is through the software deal.II of which I am the principal developer. deal.II is downloaded approximately 400 times per month. As of 2014, nearly 600 publications reference it as the tool by which numerical results were generated, and this list is currently growing by around 100 per year. (These numbers are certainly underestimates. The list of publications only shows papers we find through Google Scholar, but this service does not find all papers and not everyone references us despite using our software. For example, the list contains no publications from China although we know that it is used there in teaching, and some 20% of downloads are from China.) The list contains publications from virtually every field in the sciences and engineering (from numerical analysis to the simulation of crystal growth and plastic deformation, to the modeling of ice sheets) and is thus a good indicator of the breadth of applicability of my work.
A different measure of impact is the number of people we interact with when disseminating our research. This includes the many students from at least a dozen departments at our university who go through the very practical MATH 676 course I teach and that helps them develop software for their own research. It also includes the hundreds of students I taught in summer courses around the world.
Scientific computing has established itself as a third branch of the sciences in general, augmenting theory and experimentation. Many of the fields that utilize scientific computing use mathematical models based on partial differential equations, and they are often solved by finite element methods — the focus of my work.
Historically, finite element methods were first developed by practitioners, often in engineering departments. However, over the the past 30 years, development of new methods such as adaptive mesh refinement, hp adaptivity, or multigrid has become a more formal, mathematical enterprise and there is a growing gap between method developers and potential users of these methods. My research is positioned at this chasm and I try to bridge it by working both as a method developer as well as collaborating with applied scientists to use these methods in real-world settings. The impact of my work validates that there is a demand for this research.
My long-term vision is to found a Center for Finite Element Software and Applications. There is no doubt that sophisticated computational modeling will be a part of all aspects of engineering and the sciences in the future. Within the university, such a center — reaching out to other departments and parts of the university — would serve as a focal point for research that requires finite element modeling of a wide variety of processes. It would provide a resource and platform for joint work — preferably externally funded — in developing high performance computing models. Part of the core funding for such a center may come from programs like the NSF Software Infrastructure for Sustained Innovation (SI2) program (a $1.5M SI2 grant currently funds the development of deal.II) or DoE's Advanced Scientific Computing Research (ASCR). It would also serve as a point of contact for industrial collaborations and affiliations.
Within the larger scientific computing and finite element modeling communities, the center described above would also serve as a home for the future development of the deal.II software. As outlined above, this software is widely used in many areas of engineering and the sciences and I expect it to continue growing. I lead this project, but while there are today four co-principal developers, four developers, and some 30 contributors around the world who contribute to its development, there is still a clear need for a central location where software infrastructure work can be done independently of concrete research projects. deal.II has been the mechanism that has enabled most of my externally funded research projects, and I anticipate this also being the case in the future. Having a location where foundational work can be done is therefore an investment into future collaborations and projects.