Some general remarks on my research motivation.
Sophisticated mathematical methods contribute to modeling, simulation and optimization approaches and play a major role in various fields of research. While stochastic and numeric techniques are widely used on their own, I see great potential especially in their combination. Their combination enables more realistic modeling, efficient simulation and opens up great opportunities for highly relevant scientific problems. As important as mathematical methods are, only their implementation on computer systems brings researchers in a position where they can actually benefit from them. Adequate programming approaches and efficient use of computational resources are necessary to gain a maximal level of knowledge from mathematical theory, and will become even more important in the future. On the other hand, even in the applied context, a strong connection to pure mathematics has to be maintained. Conclusions drawn from simulations will only stand if thoroughly justified by abstract results. Further, the powerful tool of abstraction guides the exploration of new methods and approaches in applied mathematics.
What my dissertation is on.
My adviser Richard H. Stockbridge has developed a linear programming approach for stochastic control problems. The underlying stochastic differential equation is transformed into an operator-integral equation, and the considered cost criterion can be expressed as integration of the cost functions against certain measures. This poses a linear program. However, the unknowns are from a space of measures, and the constraints are defined by a function space – both are infinite dimensional objects. I use a finite element type approach to discretize these spaces, and to derive a finite linear program that easily can be solved with standard techniques. In particular, I am investigating the approximation properties of this discretization approach and maintaining a software implementation that can solve various control problems.
If you are interested in this topic, here is a short list of interesting publications.
Helmes, K. and Stockbridge, R.H. (2008). Determining the optimal control of singular stochastic processes using linear programming. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 4:136-153
Helmes, K. and Stockbridge, R.H. (2007). Linear programming approach to the optimal stopping of singular stochastic processes. An International Journal of Probability and Stochastic Processes, 79.
Kaczmarek, P., Kent, S., Rus, G.A., Stockbridge, R.H. and Wade, B.A. (2007). Numerical Solution of a long-term average control problem for singular stochastic processes. Mathematical Methods of Operations Research. 66:451-473.