I am a doctoral student enrolled in the cross-disciplinary Scientific Computing and Computational Mathematics program at Stanford. My advisor, Claire Tomlin of the Aero/Astro department, studies the synthesis and verification of control laws for systems which are best described by combinations of discrete and continuous variables.

My particular research work involves determining reachability for these "hybrid systems" -- given a set of initial states, a mathematical model describing the behavior of the system, and a set of possible input actions, reachability determines the set of all states that the system can reach in either a fixed or infinite time.

For the nonlinear mathematical models that I am considering, the continuous reachability problem can be written as the solution to a Hamilton-Jacobi partial differential equation. I determine the solution computationally by finite difference approximations on a cartesian grid, using methods borrowed from level set techniques.

While my current examples involve only a few discrete modes with two or three continuous variables, many significant control problems involve four, six, or more continuous states. It is theoretically simple to scale finite difference PDE solvers up to four or more dimensions; however, if I wish to tackle these larger problems, I will clearly need to marshal computing power greater than that of a desktop workstation.

Fortunately, level set techniques are eminently parallelizable. In addition, the computations within a discrete mode of a hybrid system can usually be run simultaneously with those of other discrete modes. Consequently, there is great potential to take advantage of high performance computing in my research.

After the Bay Area Scientific Computing day (hosted by LBL) in Feburary of this year, I identified the A++/P++ library as one tool that might allow me to easily move to more powerful computing platforms. In March, I drafted a proposal to Stanford's School of Engineering which eventually lead to the purchase of a 16 node Beowulf PC cluster for the SCCM program.

I have a considerable background in systems software, hardware and networking from a masters degree in computer science that I received prior to entering Stanford. I attended Supercomputing '97 in San Jose as a student volunteer. While I joined the SCCM program in order to pursue interests in high performance scientific computing, I have not yet be able to take advantage of available parallel computing hardware because of a local lack of suitable software development tools and knowledge.

I would like to attend the ACTS workshop in order to identify more tools (other than A++/P++) which may help me to solve and visualize larger and more realistic reachability problems. I hope that I might also be able to make those tools available to other students in my program -- for example, on the new PC cluster, and through the new parallel methods in numerical analysis course for which I may be course assistant next winter.L Dorset (1996) Acta Crystallographica