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 2018.

## Philosophy

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 theoretical results, but also to actively integrate those results into the broader 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 more than 1,100 publications showing 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.

## Impact

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 2018, I am the PI on two grants to develop computational algorithms and implement them in open source software ($1.7M and $595k). 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 years as a faculty total approximately $15M.

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 both the SPEC CPU 2006
and SPEC CPU 2017 benchmarks that are widely
used to measure the speed of computers (the award is $5,000
for each of the two benchmarks).

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 10,000 times per year. As of 2018, more than 1,100 publications reference it as the tool by which
numerical results were generated, and this list is currently growing by
around 200 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 few 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.

## Vision

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
*Cyberinfrastructure for Sustained Scientific Innovation (CSSI)* program (a
$1.7M CSSI grant currently funds the development of *deal.II*;
a previous $1.5M grant supported my work from 2011-2017) 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.