The investigators conduct research on several fundamental topics in computational science, which include optimization methods, numerical optimal control, parameter estimation, inverse problems, evolution partial differential equations with constraints, and parallel adaptive finite element methods. A feature common to all of these topics is the parallel implicit solution of optimization problems with ordinary and partial differential equations constraints. Many fundamental physical systems are described by (generally nonlinearly) constrained ordinary or partial differential equations of motion (also generally nonlinear) in the sense that exact solutions to the equations of motion always satisfy a given set of constraints. Examples include Maxwell's equations, the incompressible Navier-Stokes equations, the Yang-Mills equations, and Einstein's equations, among others. Accurate discretizations of differential equation constraints lead to very large sparse constrained optimization problems, where much of the structure reflects the discretization. The problems are highly nonlinear, with reliability of the optimization being the dominant factor in the formulation of successful methods.
The optimization of functions subject to differential equation constraints arises in many contexts in engineering and scientific computation, since physical reality is often expressed through models involving ordinary and partial differential equations. The investigator's research program is motivated and guided by some challenging applications of computational science in engineering and science. These applications include the design of large neurobiological network models, off-shore petroleum exploration, the numerical modeling of gravitational waves, and trajectory planning for spacecraft and unmanned autonomous vehicles. In addition, the investigators are active in the development and dissemination of computer software that embodies the advanced computation techniques developed as part of their research program. Software developed by the investigators provides an effective method of technology transfer and provides US scientists and engineers with instant access to state-of-the-art methods. Within the Center for Computational Mathematics, the Investigators offer a program of instruction and research that emphasizes the role of computational science in the formulation, modeling, and solution of problems from diverse and changing areas. The program of research conducted by the investigators helps to attract advanced graduate students into the area of computational science, which plays a vital role in the study of systems arising in manufacturing, engineering and the natural sciences.
