In addition to my previous post about Teaching Philosophy, I have also been working on a Research Statement. I am posting it here for me to refer back to in later months or years, so I can see how things might have changed.
My Research Interests
My primary areas of research interest lie in numerical analysis, specifically numerical solutions to ODE and PDE problems. The most recent areas of emphasis have been in the development of non-conforming finite elements utilizing multi-field methods. I have developed a three dimensional implementation of the three-field method for the partitioning of domains. It has been applied to elliptic and parabolic model problems. Additionally, I have been developing the model equations used in these techniques to include stochastic terms, primarily in the load function. However, I am interested in exploring results when other parameters in the model contain stochastic elements. I, along with current researchers in the Mathematics Department at Texas Tech University, have discussed the possibility of extending the current application to multi-scale models.
Additionally, I am interested in continuing my research in areas of scattered data fitting with B-spline functions. Currently, I am exploring the performance of a variable knot spline algorithm for data fitting which has become much more feasible with the advent of high performance computing technology. Working with researchers at Texas Tech and Arizona State University, we are comparing the performance with other spline algorithms such as penalty splines or smoothing splines.
During my time at Texas Tech University, I worked for the High Performance Computing Center, developing numerous parallel applications for the multi-processor SGI Origin 2000, as well as high-end 3D visual representations of data. Most recently, with the implementation of a computational grid, I developed applications to run in such a grid environment. I would definitely like to continue the development of high performance computing applications as they relate to areas of computational mathematics, such as the finite element method and data fitting algorithms mentioned above.
I have also been involved in interdisciplinary research with the Physical Chemistry department at Texas Tech as a postdoctoral researcher. I developed an implementation of isolating rotational and vibrational energies of a molecule by use of a rotating coordinate system. I feel that such interdisciplinary projects are central to the future of mathematical research, particularly in the area of numerical analysis. Collaboration with other departments is beneficial, not only to each individual researcher and their field, but to the preparation of students involved in those research areas as they head into industry.
Although, in my current position at a teaching university, I have not had the opportunity to obtain grant funding for such research projects as mentioned above. I have participated in grant projects as a researcher and am very interested in actively pursuing external funding for such work.